1 Introduction

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), \(n \in {{\mathbb {N}}}\), be a sequence of linear operators from a Hilbert space \({{\mathfrak {H}}}\) to a Hilbert space \({{\mathfrak {K}}}_n\), which satisfy

$$\begin{aligned} \mathrm{dom\,}T_{n+1} \subset \mathrm{dom\,}T_n \quad \text{ and } \quad \Vert T_n f\Vert \le \Vert T_{n+1}f\Vert , \,\, f \in \mathrm{dom\,}T_{n+1}. \end{aligned}$$
(1.1)

Here and elsewhere the notation \({{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) indicates the class of all linear relations between the Hilbert spaces \({{\mathfrak {H}}}\) and \({{\mathfrak {K}}}\). It will be shown that there exists a limit of this sequence, namely a linear operator \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), whose domain is given by

$$\begin{aligned} \mathrm{dom\,}T= \left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert < \infty \right\} , \end{aligned}$$

while, furthermore,

$$\begin{aligned} \Vert T_n f\Vert \nearrow \Vert Tf\Vert \quad \text{ for } \text{ all } \quad f \in \mathrm{dom\,}T. \end{aligned}$$

The limit is uniquely determined up to partial isometries. Moreover, it will be shown that closability and closedness of the operators \(T_n\) are preserved in the limit. The main idea about the existence of the limit is the notion of a representing map that was described by Szymański [14]. In the present paper the emphasis is on how to construct the limit of the sequence of operators and to discuss analogous sequences of linear relations. There is a close connection with similar convergence results in the context of nonnegative forms by Simon [13] (see also [12]), but the details will be left for a treatment in [9] in terms of Lebesgue decompositions and Lebesgue type decompositions of semibounded forms.

The monotonicity in (1.1) can also be discussed for the case of linear relations \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) by requiring that \(T_{n+1}\) contractively dominates \(T_n\), i.e., there are contractions \(C_n \in {{\textbf{L}}}({{\mathfrak {H}}}_{n+1}, {{\mathfrak {H}}}_n)\) which satisfy

$$\begin{aligned} C_n T_{n+1} \subset T_n. \end{aligned}$$
(1.2)

Likewise, this kind of monotonicity is preserved under closures \(T_n^{**}\) and under taking regular parts \(T_{n, \textrm{reg}}\) of the relations \(T_n\) (see below). In general there is no convergence result as for operators. However, the regular parts \(T_{n, \textrm{reg}}\) form a nondecreasing sequence of closable operators (as in (1.1)) and one may apply the above mentioned results for operators. Thanks to the condition (1.2) the sequence of nonnegative selfadjoint relations \(T_n^*T_n^{**}\) is nondecreasing in the usual sense and the monotonicity principle may be applied. This connects the various forms of convergence.

As mentioned above, in the present paper regular parts of operators or relations play an important role. The regular part \(T_{\textrm{reg}}\) of a linear relation \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) shows up in its Lebesgue decomposition, as follows

$$\begin{aligned} T=T_{\textrm{reg}}+T_{\textrm{sing}} \quad \text{ with } \quad T_\textrm{reg}=(I-P)T, \,\,T_{\textrm{sing}}=PT, \end{aligned}$$

where P stands for the orthogonal projection from \({{\mathfrak {K}}}\) onto \(\mathrm{mul\,}T^{**}\); see [4, 5]. Hence \(T_{\textrm{reg}}\) is a closable operator, while \(T_{\textrm{sing}}\) is singular in the sense that its closure in the graph sense is the product of closed linear subspaces; note in particular that \(\mathrm{ran\,}T_{\textrm{reg}} \perp \mathrm{mul\,}T^{**}\). The regular part \(T_{\textrm{reg}}\) is the largest closable operator that is dominated by T in the sense of contractive domination. There is an interplay with the closure \(T^{**}\) of T, given by the formula

$$\begin{aligned} (T^{**})_{\textrm{reg}}=(T_{\textrm{reg}})^{**}, \end{aligned}$$
(1.3)

see Sect. 1. If the relation T is closed, then \(\mathrm{mul\,}T^{**}=\mathrm{mul\,}T\) and \(T_{\textrm{reg}}\) is the usual closed orthogonal operator part of T, often denoted by \(T_{\textrm{op}}\). In this case, clearly, \(T_{\textrm{reg}} \subset T\) and T has the decomposition

$$\begin{aligned} T=T_{\textrm{reg}} {{\,\mathrm{\, \widehat{+} \,}\,}}(\{0\} \times \mathrm{mul\,}T), \end{aligned}$$

where the sum is componentwise. Note that the left-hand side of the identity (1.3) stands for the orthogonal operator part of \(T^{**}\). In the general case the following identity

$$\begin{aligned} T^*T^{**}= (T_{\textrm{reg}})^*(T_{\textrm{reg}})^{**} \end{aligned}$$

expresses the nonnegative selfadjoint relation on the left-hand side in terms of a similar product of closable operators.

The case of a sequence of nonincreasing linear operators will also be discussed with the same methods. Now closability is not preserved so that the main result is about a nonincreasing sequence of closed linear operators.

The paper is organized as follows. In Sect. 2 there is brief review of the notion of contractive domination for relations and operators. For the convenience of the reader the relevant facts for the monotonicity principle are reviewed in Sect. 3. The representing map is discussed in Sect. 4 in an appropriate context. The convergence results are treated next. The general case of sequences of linear operators can be found in Sect. 5, the special case of sequences of closable operators is treated in Sect. 6, and the general case of sequences of linear relations is given in Sect. 7. In this last section one can also find the connection with the monotonicity principle. In Sect. 8 a simple example shows the different behaviours of the various sequences that have been considered. The approximation of closed linear operators is considered in Sect. 9. A brief discussion about nonincreasing sequences of linear operators or relations can be found in Sect. 10. Finally, in Sect. 11 there is a collection of facts concerning the regular part of the relations \(T^*T\) and \(T^*T^{**}\) which are used throughout this paper.

In the present paper the interest is in monotone sequences of linear operators or relations in a Hilbert space. The above mentioned results have a close connection to work on sequences of operators in the literature; see [11, 12, Supplement to VIII.7], and [13]. The present work also connects sequences which are monotone in the sense of contractive domination with the monotonicity principle in its version for semibounded selfadjoint relations [2]. Related results in the context of sequences of semibounded quadratic forms will be discussed in [9] (including the connections to [13] and [1]).

2 Contractive Domination for Linear Relations

The notion of domination for linear relations was introduced in [6]. The definition and some basic properties are given here. The notation \({{\textbf{B}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) will be used to indicate the class of all bounded everywhere defined linear operators between the Hilbert spaces \({{\mathfrak {H}}}\) and \({{\mathfrak {K}}}\).

Definition 2.1

Let \({{\mathfrak {H}}}_A\), \({{\mathfrak {H}}}_B\), and \({{\mathfrak {H}}}\) be Hilbert spaces, let \(A \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_A)\) and let \(B \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_B)\). Then B is said to contractively dominate A, denoted by \(A \prec _c B\), if there exists a contraction \(C \in {\textbf{B}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\) such that

$$\begin{aligned} CB \subset A. \end{aligned}$$
(2.1)

It follows from \(C \in {\textbf{B}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\) that \(CB=\{ \{f,Cf'\}:\, \{f,f'\} \in B\}\). Therefore, (2.1) implies

$$\begin{aligned} {\left\{ \begin{array}{ll}{} \,\,\mathrm{dom\,}B \subset \mathrm{dom\,}A, &{} \mathrm{ker\,}B \subset \mathrm{ker\,}A, \\ \,\, C(\mathrm{ran\,}B) \subset \mathrm{ran\,}A, &{} C(\mathrm{mul\,}B) \subset \mathrm{mul\,}A. \end{array}\right. } \end{aligned}$$
(2.2)

Observe that Definition 2.1 implies that the contraction \(C \in {\textbf{B}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\) is only fixed as a mapping form \(\mathrm{ran\,}B\) to \(\mathrm{ran\,}A\). In fact, the boundedness of C implies that C takes \(\mathrm{{\overline{ran}}\,}B\) into \(\mathrm{{\overline{ran}}\,}A\). Hence, it may and will be assumed that

$$\begin{aligned} C ((\mathrm{ran\,}B)^\perp ) =\{0\}. \end{aligned}$$

Note that if A and B are linear relations which satisfy \(B \subset A\), then B contractively dominates A with \(C=I_{\mathrm{ran\,}B}\). In particular, A contractively dominates \(A^{**}\). Finally, the notion of contractive domination is transitive:

$$\begin{aligned} \begin{array}{l} A \prec _c B\quad \text {and}\quad B \prec _c C\quad \Rightarrow \quad A \prec _c C. \end{array} \end{aligned}$$

If \(A \prec _c B\) with a contraction \(C \in {\textbf{B}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\), then it follows from (2.1) and [2, Proposition 1.3.9] that

$$\begin{aligned} A^{*} \subset B^{*} C^{*} \quad \text{ and } \quad CB^{**} \subset A^{**}. \end{aligned}$$
(2.3)

In other words, the second inclusion in (2.3) shows that the contractive domination in (2.1) is preserved with the same operator C. In particular, if \(A \prec _c B\), then the following inclusions are valid: \(\mathrm{ran\,}A^*\subset \mathrm{ran\,}B^*\) and \(\mathrm{dom\,}B^{**}\subset \mathrm{dom\,}A^{**}\). Recall that in the particular case when A and B in Definition 2.1 are linear operators it is possible to give an equivalent characterization of contractive domination: \(A \prec _c B\) if and only if

$$\begin{aligned} \mathrm{dom\,}B \subset \mathrm{dom\,}A \quad \text {and} \quad \Vert A f\Vert \le \Vert B f\Vert , \quad f\in \mathrm{dom\,}B. \end{aligned}$$

The following result shows that contractive domination is preserved by the regular parts. This observation goes back to [13] for the case of nonnegative forms and to [4]. Furthermore, it is shown that there is a converse statement in the case of closed linear relations.

Lemma 2.2

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_A)\) and \(B \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_B)\) be linear relations. Then

$$\begin{aligned} A \prec _c B \quad \Rightarrow \quad A_{\textrm{reg}} \prec _c B_\textrm{reg}. \end{aligned}$$

Moreover, if the linear relations A and B are closed, then

$$\begin{aligned} A \prec _c B \quad \Leftrightarrow \quad A_{\textrm{reg}} \prec _c B_\textrm{reg}. \end{aligned}$$

Proof

Assume that \(CB \subset A\) with a contraction \(C \in {\textbf{B}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\). By (2.2) the operator C maps \(\mathrm{mul\,}B^{**}\) into \(\mathrm{mul\,}A^{**}\). Let \(P_B\) be the orthogonal projection onto \(\mathrm{mul\,}B^{**}\) and let \(P_A\) be the orthogonal projection onto \(\mathrm{mul\,}A^{**}\). Let \(\{f,f'\} \in B\) and write \(\{f,f'\}=\{f,(I-P_B)f'+P_Bf' \}\) (i.e., the Lebesgue decomposition of B). Here \(P_B f' \in \mathrm{mul\,}B^{**}\) and one concludes that

$$\begin{aligned} \{f, Cf'\} = \{f, C(I - P_B )f' + CP_B f'\} \in A, \end{aligned}$$

where \(CP_B f' \in \mathrm{mul\,}A^{**}\). Now observe that

$$\begin{aligned} \{f, (I - P_B )f'\} \in B_{\textrm{reg}} \quad \text{ and } \quad \{f, (I - P_A)C(I - P_B )f'\} \in A_{\textrm{reg}}. \end{aligned}$$

Equivalently, this leads to \( [(I - P_A)C]B_{\textrm{reg}} \subset A_{\textrm{reg}}\), and since (\(I - P_A)C\) is a contraction this implies \(A_{\textrm{reg}} \prec _c B_{\textrm{reg}}\).

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_A)\) and \(B \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_B)\) be closed linear relations. Then \(A_{\textrm{reg}}\) and \(B_{\textrm{reg}}\), belonging to \({{\textbf{B}}}({{\mathfrak {H}}}_A, {{\mathfrak {H}}}_B)\), are the closed linear operator parts. Assume the inequality \(A_{\textrm{reg}}\prec B_{\textrm{reg}}\). Then there exists a contraction \(C \in {{\textbf{B}}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}}_A)\) such that \(C B_\textrm{reg} \subset A_{\textrm{reg}}\). Without loss of generality one may take \(C {\upharpoonright \,}(\mathrm{ran\,}B_{\textrm{reg}})^\perp =0\). Then, in particular, \(C {\upharpoonright \,}\mathrm{ran\,}P_B=\{0\}\) and it follows from the Lebesgue decomposition \(B=B_{\textrm{reg}}+B_{\textrm{sing}}\) that

$$\begin{aligned} CB=CB_{\textrm{reg}} \subset A_{\textrm{reg}}. \end{aligned}$$

Since A is closed, one sees that \(A_{\textrm{reg}}\subset A\). Therefore, \(CB \subset A\) and \(A \prec _c B\). \(\square \)

The equivalence in the above theorem is restricted to closed linear relations. By modifying the notion of domination the condition that the relations are closed can be relaxed by introducing a weaker form of the Lebesgue decomposition; cf. [4, 10].

Contractive domination of closed linear relations can be characterized in terms of the corresponding nonnegative selfadjoint relations; see [6, Theorem 4.4]. Recall from [2, Definition 5.2.8] that two nonnegative relations \(H_1\) and \(H_2\) in \({{\textbf{L}}}({{\mathfrak {H}}})\) satisfy \(H_1 \le H_2\) when

$$\begin{aligned} \mathrm{dom\,}H_2^{\frac{1}{2}}\subset \mathrm{dom\,}H_1^{\frac{1}{2}}\quad \text{ and } \quad \Vert (H_{1, \textrm{reg}})^{{\frac{1}{2}}} f\Vert \le \Vert (H_{2, \textrm{reg}})^{\frac{1}{2}}{ f}\Vert , \,\, f \in \mathrm{dom\,}H_2^{\frac{1}{2}}. \end{aligned}$$
(2.4)

With this definition the following theorem is clear.

Theorem 2.3

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_A)\) and \(B \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}_B)\) be closed linear relations. Then the following statements are equivalent

  1. (i)

    \(A^*A \le B^*B\);

  2. (ii)

    \(A \prec _c B\) or, equivalently, \(A_{\textrm{reg}} \prec _c B_{\textrm{reg}}\).

Proof

Let \(H_1=A^*A\) and \(H_2=B^*B\). By Lemma 11.2 it follows that there exist partial isometries \(U_1 \in {{\textbf{L}}}({{\mathfrak {H}}}_A, {{\mathfrak {H}}})\) and \(U_2 \in {{\textbf{L}}}({{\mathfrak {H}}}_B, {{\mathfrak {H}}})\), such that.

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{dom\,}H_1^{\frac{1}{2}}=\mathrm{dom\,}A, \quad (H_{1, \textrm{reg}})^{\frac{1}{2}}=U_1 A_{\textrm{reg}}, \\ \mathrm{dom\,}H_2^{\frac{1}{2}}=\mathrm{dom\,}B, \quad (H_{2, \textrm{reg}})^{\frac{1}{2}}=U_2 B_\textrm{reg}. \end{array} \right. \end{aligned}$$

Therefore by means of (2.4) this shows that \(A^*A \le B^*B\), i.e., \(H_1 \le H_2\), is equivalent to the assertions

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{dom\,}B \subset \mathrm{dom\,}A, \\ \Vert A_{\textrm{reg}} h\Vert \le \Vert B_{\textrm{reg}} h\Vert , \quad h \in \mathrm{dom\,}B. \end{array} \right. \end{aligned}$$

In other words, the inequality \(A^*A \le B^*B\) in (i) is equivalent to the inequality \(A_{\textrm{reg}} \prec _c B_{\textrm{reg}}\) in (ii). \(\square \)

This characterization makes it possible to apply the monotonicity principle in the next section.

3 The Monotonicity Principle

A linear relation \(H \in {{\textbf{L}}}({{\mathfrak {H}}})\) is called the strong graph limit of a sequence of linear relations \(H_n \in {{\textbf{L}}}({{\mathfrak {H}}})\), \(n \in {{\mathbb {N}}}\), if for each \(\{h,h'\} \in H\) there exists a sequence \(\{h_n, h_n'\} \in H_n\) such that \(\{h_n,h_n'\} \rightarrow \{h,h'\}\); see [2, Definition 1.9.1]. The strong graph limit is automatically closed, see [2, p. 80]. Clearly, if all \(H_n\) are symmetric, then H is symmetric. In particular, if all \(H_n\) are nonnegative, then H is nonnegative.

Lemma 3.1

Let \(H_n \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a sequence of nonnegative selfadjoint relations and let its strong graph limit \(H_\infty \) be nonnegative and selfadjoint. Then for every \(f \in \mathrm{dom\,}H_{\infty }\) there exists a sequence \(f_n \in \mathrm{dom\,}H_{n}\) such that

$$\begin{aligned} f_n \rightarrow f \quad \text{ and } \quad \Vert (H_{n, \textrm{reg}})^{\frac{1}{2}}f_n\Vert \rightarrow \Vert (H_{\infty , \textrm{reg}})^{\frac{1}{2}}f\Vert . \end{aligned}$$

Proof

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}})\) be any nonnegative selfadjoint relation with square root \(A^{\frac{1}{2}}\). Recall that \(\mathrm{mul\,}A^{\frac{1}{2}}=\mathrm{mul\,}A\), so that \((A^{\frac{1}{2}})_{\textrm{reg}}= (A_{\textrm{reg}})^{\frac{1}{2}}\). If \(\{f,f'\} \in A\), then there exists an element \(h \in {{\mathfrak {H}}}\) such that \(\{f,h\} \in A^{\frac{1}{2}}\) and \(\{h, f'\} \in A^{\frac{1}{2}}\), which gives

$$\begin{aligned} (f',f)=\Vert h\Vert ^2. \end{aligned}$$
(3.1)

Since \(h \in \mathrm{dom\,}A^{\frac{1}{2}}\subset (\mathrm{mul\,}A)^\perp \), one sees that \(h= (A_{\textrm{reg}})^{\frac{1}{2}}f\). Therefore, it is clear that (3.1) may be written as

$$\begin{aligned} (f',f)=(A_{\textrm{reg}} f,f)=\Vert (A_{\textrm{reg}})^{\frac{1}{2}}f\Vert ^2. \end{aligned}$$
(3.2)

Now let \(f \in \mathrm{dom\,}H_\infty \), then \(\{f,f'\} \in H_\infty \) for some \(f' \in {{\mathfrak {H}}}\). By the strong graph convergence there exists a sequence \(\{f_n, f_n'\} \in H_n\) such that \(f_n \rightarrow f\) and \(f_n' \rightarrow f'\). Therefore, by definition, there exist elements \(h_n \in {{\mathfrak {H}}}\) such that

$$\begin{aligned} \{f_n, h_n\} \in (H_n)^{\frac{1}{2}}\quad \text{ and } \quad \{h_n, f_n'\} \in (H_n)^{\frac{1}{2}}, \end{aligned}$$

and, likewise, there exists an element \(h \in {{\mathfrak {H}}}\) such that

$$\begin{aligned} \{f,h\} \in (H_\infty )^{\frac{1}{2}}\quad \text{ and } \quad \{h,f'\} \in (H_\infty )^{\frac{1}{2}}. \end{aligned}$$

Then clearly

$$\begin{aligned} \Vert h_n\Vert ^2=(f_n',f_n) \rightarrow (f',f)=\Vert h\Vert ^2, \end{aligned}$$

or, equivalently, using (3.2),

$$\begin{aligned} \Vert (H_{n, \textrm{reg}})^{\frac{1}{2}}f_n\Vert \rightarrow \Vert (H_{\infty , \textrm{reg}})^{\frac{1}{2}}f\Vert . \end{aligned}$$

\(\square \)

In the case of a nondecreasing sequence of nonnegative selfadjoint relations \(H_n\) there is a much stronger result. First observe that

$$\begin{aligned} H_m \le H_n \quad \Leftrightarrow \quad (H_m)^{\frac{1}{2}}\prec _c (H_n)^{\frac{1}{2}}, \end{aligned}$$

due to Theorem 2.3, so that if \(H_n\) is nondecreasing, one also has

$$\begin{aligned} (H_{m, \textrm{reg}})^{\frac{1}{2}}\prec _c (H_{n, \textrm{reg}})^{\frac{1}{2}}. \end{aligned}$$

The following monotonicity principle will be recalled from [3, Theorem 3.5], [2, Theorem 5.2.11].

Theorem 3.2

Let \(H_n \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a sequence of nonnegative selfadjoint relations and assume they satisfy

$$\begin{aligned} H_m \le H_n, \quad m \le n. \end{aligned}$$

Then there exists a nonnegative selfadjoint relation \(H_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) with

$$\begin{aligned} H_n \le H_\infty , \quad n \in {{\mathbb {N}}}. \end{aligned}$$

In fact, \(H_n \rightarrow H_\infty \) in the strong resolvent sense or, equivalently, in the strong graph sense. Moreover, the square root of \(H_\infty \) satisfies

$$\begin{aligned} \mathrm{dom\,}(H_\infty )^{\frac{1}{2}}=\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}(H_n)^{\frac{1}{2}}:\, \sup _{n \in {{\mathbb {N}}}} \Vert (H_{n, \textrm{reg}})^{\frac{1}{2}}\varphi \Vert < \infty \right\} \end{aligned}$$
(3.3)

and, furthermore,

$$\begin{aligned} \Vert (H_{n,\textrm{reg}})^{\frac{1}{2}}\varphi \Vert \nearrow \Vert (H_{\infty , \textrm{reg}})^{\frac{1}{2}}\varphi \Vert , \quad \varphi \in \mathrm{dom\,}(H_\infty )^{\frac{1}{2}}. \end{aligned}$$
(3.4)

Note that the multivalued parts of the relations \(H_n\) in Theorem 3.2 form a nondecreasing sequence. Of course, if all relations \(H_n\) in Theorem 10.1 are operators, then the limit \(H_\infty \) may still be a linear relation with a nontrivial multivalued part; see the example below.

Example 3.3

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a nonnegative selfadjoint operator or relation. Then it is clear that the sequence \(H_n=nA\) of nonnegative selfadjoint operators or relations is nondecreasing. Hence there exists a nonnegative selfadjoint relation \(H_\infty \) such that \(H_n \rightarrow H_\infty \) is the strong graph sense. To determine \(H_\infty \), let \(\{f,g\} \in H_\infty \), then there exists a sequence \(\{f_n, g_n\} \in H_n\) such that \(f_n \rightarrow f\) and \(g_n \rightarrow g\). Here \(g_n=n h_n\) with \(\{f_n, h_n\} \in A\) and, clearly, \(h_n \rightarrow 0\). Since A is closed, this implies \(\{f,0\} \in A\). Furthermore, note that \(h_n \in \mathrm{ran\,}A \subset (\mathrm{ker\,}A)^\perp \). Hence \(g_n \in (\mathrm{ker\,}A)^\perp \) which implies \(g \in (\mathrm{ker\,}A)^\perp \). Therefore, it follows that

$$\begin{aligned} H_\infty =\mathrm{ker\,}A \times (\mathrm{ker\,}A)^\perp , \end{aligned}$$

since both relations are selfadjoint. Furthermore one has \(\mathrm{dom\,}(H_\infty )^{\frac{1}{2}}=\mathrm{ker\,}A\) and \((H_\infty )_{\textrm{reg}}=\mathrm{ker\,}A \times \{0\}\) (as in (3.3) and (3.4)).

For sequences of closed relations which are nondecreasing in the sense of domination there are close connections with Theorem 3.2 via Theorem 2.3.

4 Semi-inner Products and Representing Maps

Let \({{\mathfrak {H}}}\) be a Hilbert space with inner product \((\cdot , \cdot )\) and let \({{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) be a linear subspace which is provided with a semi-inner product \((\cdot , \cdot )_+\). In the following lemma it will be shown that such a subspace is generated by a so-called representing map. The assertion is inspired by [14].

Lemma 4.1

Let \({{\mathfrak {H}}}\) be a Hilbert space with inner product \((\cdot , \cdot )\). Let \({{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) be a linear subspace which is provided with a semi-inner product \((\cdot , \cdot )_+\). Then there exists a representing map \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), where \({{\mathfrak {K}}}\) is a Hilbert space, such that

$$\begin{aligned} (\varphi , \psi )_+=(T \varphi , T \psi )_{{\mathfrak {K}}}, \quad \varphi , \psi \in {{\mathfrak {D}}}=\mathrm{dom\,}T. \end{aligned}$$

If \(T' \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}')\), where \({{\mathfrak {K}}}'\) is a Hilbert space, is another representing map with \(\mathrm{dom\,}T'={{\mathfrak {D}}}\), then there exists a partial isometry \(V \in {{\textbf{B}}}({{\mathfrak {K}}}, {{\mathfrak {K}}}')\) with initial space \(\mathrm{{\overline{ran}}\,}T\) and final space \(\mathrm{{\overline{ran}}\,}T'\), such that \(T'=VT\).

Proof

Let \({{\mathfrak {N}}}\) be the set of neutral elements in \({{\mathfrak {D}}}\):

$$\begin{aligned} {{\mathfrak {N}}}=\big \{ \varphi \in {{\mathfrak {D}}}:\, (\varphi , \varphi )_+=0\,\big \}. \end{aligned}$$

Due to the Cauchy-Schwarz inequality the space \({{\mathfrak {N}}}\) is linear. Hence, one may introduce an inner product on the quotient space \({{\mathfrak {D}}}/{{\mathfrak {N}}}\) by

$$\begin{aligned}{}[\varphi +{{\mathfrak {N}}}, \psi +{{\mathfrak {N}}}]=(\varphi , \psi )_+, \quad \varphi , \psi \in {{\mathfrak {D}}}. \end{aligned}$$

The completion of this quotient space is indicated by \({{\mathfrak {K}}}\), so that \({{\mathfrak {K}}}\) is a Hilbert space. Denote the inner product on \({{\mathfrak {K}}}\) by \((\cdot , \cdot )_{{\mathfrak {K}}}\), so that \((\varphi +{{\mathfrak {N}}}, \psi +{{\mathfrak {N}}})_{{\mathfrak {K}}}= [\varphi +{{\mathfrak {N}}}, \psi +{{\mathfrak {N}}}]\) for \(\varphi , \psi \in {{\mathfrak {D}}}\). Next define the operator T from \({{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) to \({{\mathfrak {K}}}\) by

$$\begin{aligned} T \varphi = \varphi +{{\mathfrak {N}}}, \quad \varphi \in {{\mathfrak {D}}}. \end{aligned}$$

Then it follows that

$$\begin{aligned} (T\varphi , T \psi )_{{\mathfrak {K}}}=[\varphi +{{\mathfrak {N}}}, \psi +{{\mathfrak {N}}}] =(\varphi , \psi )_+, \quad \varphi , \psi \in {{\mathfrak {D}}}, \end{aligned}$$

which is the first assertion of the lemma.

If \(T' \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}')\), where \({{\mathfrak {K}}}'\) is a Hilbert space, is another representing map with \(\mathrm{dom\,}T'={{\mathfrak {D}}}\), then

$$\begin{aligned} (T' \varphi , T'\psi )=(\varphi , \psi )_+, \quad \varphi , \psi \in {{\mathfrak {D}}}=\mathrm{dom\,}T'. \end{aligned}$$

Then the linear relation V from \({{\mathfrak {K}}}\) to \({{\mathfrak {K}}}'\), defined by

$$\begin{aligned} \big \{ \{T \varphi , T' \varphi \}:\, \varphi \in {{\mathfrak {D}}}\big \}, \end{aligned}$$

is an isometric operator from \(\mathrm{ran\,}T\) onto \(\mathrm{ran\,}T'\), which can be extended as an isometric operator from \(\mathrm{{\overline{ran}}\,}T\) onto \(\mathrm{{\overline{ran}}\,}T'\), such that \(T'f=VTf\) holds for all \(f\in {{\mathfrak {D}}}\). To get the desired partial isometry V it remains to continue the isometric map to \((\mathrm{ran\,}T)^\perp \) as a zero mapping. This gives the desired result. \(\square \)

Let \({{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) be a linear subspace as in Lemma 4.1. A sequence \(\varphi _n \in {{\mathfrak {D}}}\) is said to converge to \(\varphi \in {{\mathfrak {H}}}\) in the sense of \({{\mathfrak {D}}}\), in notation \(\varphi _n \rightarrow _{{\mathfrak {D}}}\varphi \), if

$$\begin{aligned} \varphi _n \rightarrow \varphi \quad \text{ in } \quad {{\mathfrak {H}}}\quad \text{ and } \quad \Vert \varphi _n-\varphi _m\Vert _+ \rightarrow 0. \end{aligned}$$

Then \({{\mathfrak {D}}}\) is called closable if for any sequence \(\varphi _n \in {{\mathfrak {D}}}\) one has

$$\begin{aligned} \varphi _n \rightarrow _{{\mathfrak {D}}}0 \quad \Rightarrow \quad \Vert \varphi _n\Vert _+ \rightarrow 0, \end{aligned}$$

and, likewise, \({{\mathfrak {D}}}\) is called closed if for any sequence \(\varphi _n \in {{\mathfrak {D}}}\) one has

$$\begin{aligned} \varphi _n \rightarrow _{{\mathfrak {D}}}\varphi \quad \Rightarrow \quad \varphi \in {{\mathfrak {D}}}\quad \text{ and } \quad \Vert \varphi _n -\varphi \Vert _+ \rightarrow 0. \end{aligned}$$

These definitions take a more familiar form in terms of the representing map T in Lemma 4.1 One sees immediately for a sequence \(\varphi _n \in {{\mathfrak {D}}}\) that

$$\begin{aligned} \varphi _n \rightarrow _{{\mathfrak {D}}}\varphi \quad \Leftrightarrow \quad \varphi _n \rightarrow \varphi \quad \text{ in } \quad {{\mathfrak {H}}}\quad \text{ and } \quad \Vert T(\varphi _n-\varphi _m)\Vert \rightarrow 0. \end{aligned}$$

Therefore, \({{\mathfrak {D}}}\) is closable if and only if T is closable, and, likewise, \({{\mathfrak {D}}}\) is closed if and only if T is closed.

An example of a representing map appears in the following construction that will be used in [8]. Let \(A \in {{\textbf{B}}}({{\mathfrak {K}}})\) be a nonnegative contraction in a Hilbert space \({{\mathfrak {K}}}\). The range space \({{\mathfrak {A}}}=\mathrm{ran\,}A^{\frac{1}{2}}\), as a subspace of \({{\mathfrak {K}}}\), is provided with the semi-inner product

$$\begin{aligned} (A^{\frac{1}{2}}h, A^{\frac{1}{2}}k)_{{\mathfrak {A}}}=(\pi h, \pi k)_{{\mathfrak {K}}}, \quad h,k \in {{\mathfrak {K}}}, \end{aligned}$$
(4.1)

where \(\pi \) is the orthogonal projection in \({{\mathfrak {K}}}\) onto \(\mathrm{{\overline{ran}}\,}A^{\frac{1}{2}}=(\mathrm{ker\,}A^{\frac{1}{2}})^\perp \). Then it is clear that the operator \(T \in {{\textbf{L}}}({{\mathfrak {K}}}, {{\mathfrak {H}}})\) defined by

$$\begin{aligned} A^{\frac{1}{2}}h \mapsto \pi h, \quad h \in {{\mathfrak {H}}}, \end{aligned}$$

with \(\mathrm{dom\,}T={{\mathfrak {A}}}\), is actually a representing map as follows from (4.1).

5 Nondecreasing Sequences of Linear Operators

It will be shown that a sequence of linear operators, that is nondecreasing in the sense of contractive domination, as in Definition 2.1, has a linear operator as limit. The limit will be constructed by means of representing maps. Moreover, it will be shown that closability and closedness of the operators are preserved in the limit.

Theorem 5.1

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of linear operators which satisfy

$$\begin{aligned} T_m \prec _c T_n, \quad m \le n. \end{aligned}$$
(5.1)

Then there exists a linear operator \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), where \({{\mathfrak {K}}}\) is a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}T= \left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert < \infty \right\} \end{aligned}$$
(5.2)

and which satisfies

$$\begin{aligned} T_n \prec _c T \quad \text{ and } \quad \Vert T_n \varphi \Vert \nearrow \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$
(5.3)

Moreover, the following statements hold:

  1. (a)

    if \(T_n\) is closable for all \(n \in {{\mathbb {N}}}\), then T is closable;

  2. (b)

    if \(T_n\) is closed for all \(n \in {{\mathbb {N}}}\), then T is closed.

Proof

Let \(T_n\) be a sequence of operators that satisfies (5.1). Then it is seen by Cauchy’s inequality that the right-hand side \({{\mathfrak {D}}}\) in (5.2) is a linear space. Next the existence of the operator T will be shown. For each \(\varphi \in {{\mathfrak {D}}}\) define

$$\begin{aligned} \Vert \varphi \Vert _+= \sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert . \end{aligned}$$

Then \(\Vert \cdot \Vert _+\) is clearly a well defined seminorm on \({{\mathfrak {D}}}\) and let \((\cdot , \cdot )_+\) be the corresponding semi-inner product. By Lemma 4.1 there exists a linear operator T defined on \(\mathrm{dom\,}T={{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) to a Hilbert space \({{\mathfrak {K}}}\) such that

$$\begin{aligned} (\varphi , \psi )_+=(T\varphi , T \psi ), \quad \varphi \in {{\mathfrak {D}}}. \end{aligned}$$

This shows the assertion in (5.3).

  1. (a)

    Assume that \(T_n\), \(n \in {{\mathbb {N}}}\), is closable. To show that T is closable, it suffices to show that \(T=T_{\textrm{reg}}\). By (5.3) one has

    $$\begin{aligned} T_n \prec _c T. \end{aligned}$$

    Hence there exist contractions \(C_n\in {{\textbf{B}}}({{\mathfrak {K}}},{{\mathfrak {K}}}_{n})\), such that \(C_n T \subset T_n\) for all \(n\in {{\mathbb {N}}}\). This implies that

    $$\begin{aligned} C_n T^{**} \subset T_n^{**}; \end{aligned}$$

    see (2.3). In particular, if \(\{0,\varphi \} \in T^{**}\), then \(\{0,C_{n} \varphi \} \in T_{n}^{**}\), so that \(C_{n}\varphi =0\). Thus one concludes that \(\mathrm{mul\,}T^{**}\subset \mathrm{ker\,}C_n\). Let P be the orthogonal projection from \({{\mathfrak {K}}}\) onto \(\mathrm{mul\,}T^{**}\), then \(C_nP=0\). By means of the Lebesgue decomposition \(T=(I-P)T+PT\), this leads to

    $$\begin{aligned} C_n T_{\textrm{reg}}=C_n(I-P)T=C_n[(I-P)T+PT]=C_n T \subset T_n. \end{aligned}$$

    Hence, \(C_n T_{\textrm{reg}} \subset T_n\) for all \(n\in {{\mathbb {N}}}\) and thus

    $$\begin{aligned} \Vert T_n \varphi \Vert =\Vert C_n T_{\textrm{reg}} \varphi \Vert \le \Vert T_{\mathrm{{reg}}} \varphi \Vert \le \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$
    (5.4)

    Taking the supremum over \(n \in {{\mathbb {N}}}\) in (5.4) and combining with (5.3) gives

    $$\begin{aligned} \Vert T \varphi \Vert =\Vert T_{\mathrm{{reg}}} \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$

    This implies that \(T_{\mathrm{{sing}}}=0\) and hence T is closable.

  2. (b)

    Assume that \(T_n\), \(n \in {{\mathbb {N}}}\), is closed. To show that T is closed, let \(\varphi _n\) be a sequence in \(\mathrm{dom\,}T\) such that

    $$\begin{aligned} \varphi _n \rightarrow \varphi \, \text{ in } \,{{\mathfrak {H}}}\quad \text{ and } \quad T(\varphi _n-\varphi _m) \rightarrow 0 \, \text{ in } \, {{\mathfrak {K}}}. \end{aligned}$$
    (5.5)

    Due to (5.3) one sees that \(T_k(\varphi _n-\varphi _m) \rightarrow 0\). Since for each \(k \in {{\mathbb {N}}}\) the operator \(T_k\) is closed one obtains that \(\varphi \in \mathrm{dom\,}T_k\) and \(T_k(\varphi _n-\varphi ) \rightarrow 0\) as \(n \rightarrow \infty \). In particular, \(\varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n\). In order to verify that \(\varphi \in \mathrm{dom\,}T\), first observe that the inequality

    $$\begin{aligned} \left| \Vert T\varphi _n\Vert -\Vert T \varphi _m\Vert \right| \le \Vert T(\varphi _n-\varphi _m)\Vert , \end{aligned}$$

    implies, via (5.5), that \(\sup _{m \in {{\mathbb {N}}}} \Vert T \varphi _m\Vert <\infty \). Now it follows from \(T_n \varphi _m \rightarrow T_n \varphi \) and (5.3) that

    $$\begin{aligned} \Vert T_n \varphi \Vert = \lim _{m \rightarrow \infty } \Vert T_n \varphi _m\Vert \le \lim _{m \rightarrow \infty } \Vert T \varphi _m\Vert \le \sup _{m \in {{\mathbb {N}}}} \Vert T \varphi _m\Vert < \infty . \end{aligned}$$

    Since this holds for all \(n\in {{\mathbb {N}}}\), one concludes that \(\sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert < \infty \). Therefore, \(\varphi \in \mathrm{dom\,}T\). Since by (a) the operator T is closable, it now follows from (5.5) that T is closed.

\(\square \)

The existence of the limit in Theorem 6.1 has been established; however it is clear that there is no uniqueness. In fact, this question has been already addressed in Lemma 4.1. The corollary below is easily verified directly.

Corollary 5.2

Assume the conditions from Theorem  5.1 and let \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be the limit. If \(T' \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}')\), where \({{\mathfrak {K}}}'\) is a Hilbert space, is another limit with \(\mathrm{dom\,}T'=\mathrm{dom\,}T\), then there exists a partial isometry \(V \in {{\textbf{B}}}({{\mathfrak {K}}}, {{\mathfrak {K}}}')\) with initial space \(\mathrm{{\overline{ran}}\,}T\) and final space \(\mathrm{{\overline{ran}}\,}T'\), such that \(T'=VT\).

The following simple result is that an operator that dominates the sequence also dominates the limit. This fact will have important consequences.

Corollary 5.3

Assume the conditions from Theorem  5.1 and let \(T' \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}')\), where \({{\mathfrak {K}}}'\) is a Hilbert space, be a linear operator. Then

$$\begin{aligned} T_n \prec _c T', \quad n \in {{\mathbb {N}}}\quad \Rightarrow \quad T \prec _c T'. \end{aligned}$$

Proof

The inequality \(T_n \prec _c T'\) implies that \(\mathrm{dom\,}T' \subset \mathrm{dom\,}T_n\) and \(\Vert T_n \varphi \Vert \le \Vert T' \varphi \Vert \) for \(\varphi \in \mathrm{dom\,}T'\). Since this holds for all \(n \in {{\mathbb {N}}}\), one sees that

$$\begin{aligned} \mathrm{dom\,}T' \subset \mathrm{dom\,}T \quad \text{ and } \quad \Vert T \varphi \Vert =\sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert \le \Vert T' \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T', \end{aligned}$$

in other words \(T \prec T'\). \(\square \)

6 Nondecreasing Sequences of Closable Operators

It is a consequence of Theorem 5.1 that a sequence of closable linear operators which satisfy (5.1) has a closable limit. The description of the limit of the closures is of interest.

Proposition 6.1

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of linear operators for which (5.1) holds and assume that \(T_n\), \(n \in {{\mathbb {N}}}\), is closable. Let T be the closable limit of \(T_n\) in (5.2) and (5.3). Then the closures \(T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) of \(T_n\) satisfy

$$\begin{aligned} T_m^{**} \prec _{c} T_n^{**}, \quad m \le n, \quad \text{ and } \quad T_n^{**} \prec _c T^{**}. \end{aligned}$$
(6.1)

Consequently, there exists a closed linear operator \(S \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{\textrm{c}})\), where \({{\mathfrak {K}}}_{\textrm{c}}\) is a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}S=\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n^{**}:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_n^{**} \varphi \Vert < \infty \right\} \end{aligned}$$
(6.2)

and which satisfies

$$\begin{aligned} T_n^{**} \prec _c S \prec _c T^{**} \quad \text{ and } \quad \Vert T_n^{**} \varphi \Vert \nearrow \Vert S \varphi \Vert , \quad \varphi \in \mathrm{dom\,}S. \end{aligned}$$
(6.3)

In fact, \(\mathrm{dom\,}T^{**} \subset \mathrm{dom\,}S\), while \(\Vert S \varphi \Vert =\Vert T^{**}\varphi \Vert \) for all \(\varphi \in \mathrm{dom\,}T^{**}\).

Proof

The sequence \(T_n\) is assumed to satisfy (5.1), thus it follows that \(T_m^{**} \prec _c T_n^{**}\) for \(m \le n\), by (2.3). Moreover, by Theorem 5.1 one has \(T_n \prec _c T\), so that also \(T_n^{**} \prec _c T^{**}\) by (2.3). Hence (6.1) holds and, in particular, Theorem 5.1 may be applied to the sequence of closed operators \(T_n^{**}\).

Recall from Theorem 5.1 that the right-hand side in (6.2) is a linear space. Moreover, by the same theorem there exists a closed linear operator S defined on \(\mathrm{dom\,}S\) in (6.2) for which (6.3) holds; observe that \(S \prec _c T^{**}\) by Corollary 5.3.

Now it follows from (5.3) and (6.3) that \(\Vert T\varphi \Vert =\Vert S \varphi \Vert \) for all \(\varphi \in \mathrm{dom\,}T\). Here the operator S is closed and T is closable, and \(S \prec _c T^{**}\) means that \(CT^{**}\subset S\) for some contraction \(C \in {\textbf{B}}({{\mathfrak {K}}},{{\mathfrak {K}}}_{\textrm{c}})\). One concludes that \(\Vert S \varphi \Vert =\Vert CT^{**}\varphi \Vert =\Vert T^{**}\varphi \Vert \) holds in fact for all \(\varphi \in \mathrm{dom\,}T^{**}\). \(\square \)

A special case of Theorem 5.1, where all \(T_n\) are bounded everywhere defined operators, is worth mentioning separately.

Corollary 6.2

Let \(T_n\in {{\textbf{B}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{n})\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, such that

$$\begin{aligned} \Vert T_m \varphi \Vert \le \Vert T_n \varphi \Vert , \quad \varphi \in {{\mathfrak {H}}}, \quad m \le n. \end{aligned}$$

Then there exists a closed linear operator \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_\textrm{c})\), where \({{\mathfrak {K}}}_{\textrm{c}}\) a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}T= \left\{ \varphi \in {{\mathfrak {H}}}:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert < \infty \right\} \end{aligned}$$
(6.4)

and which satisfies

$$\begin{aligned} T_n \prec _c T \quad \text{ and } \quad \Vert T_{n} \varphi \Vert \nearrow \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$
(6.5)

Proof

This is just an application of Theorem 5.1, as \(\bigcap _{n=1}^\infty \mathrm{dom\,}T_n={{\mathfrak {H}}}\). Hence there exists a linear operator \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) for which (6.4) and (6.5) hold. Since \(T_n\in {{\textbf{B}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{n})\) one observes that \(T_n\), \(n \in {{\mathbb {N}}}\), is closed, which implies that T is closed. \(\square \)

Remark 6.3

If in Corollary 6.2 one has \(\sup _{n \in {{\mathbb {N}}}} \Vert T_n\Vert < \infty \), then \(\mathrm{dom\,}T={{\mathfrak {H}}}\) and \(T \in {{\textbf{B}}}({{\mathfrak {H}}},{{\mathfrak {K}}})\) by the closed graph theorem. However, if \(\sup _{n \in {{\mathbb {N}}}} \Vert T_n\Vert =\infty \), then by the uniform boundedness principle there is an element \(\varphi \in {{\mathfrak {H}}}\) for which \(\sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert =\infty \) and \(\mathrm{dom\,}T\) is a proper subset of \({{\mathfrak {H}}}\). Note that \(\mathrm{dom\,}T\) is closed if and only if T is a bounded operator.

7 Nondecreasing Sequences of Linear Relations

In this section the emphasis will be on nondecreasing sequences of linear relations in the general case, i.e., the relations are not necessarily operators or not necessarily closed. However, also the regular parts and the closures form nondecreasing sequences. In particular, one may apply Theorem 2.3, which leads to a connection with the monotonicity principle in Theorem 3.2.

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of linear relations which satisfy

$$\begin{aligned} T_m \prec _c T_n \quad m \le n. \end{aligned}$$
(7.1)

Observe that the regular parts \(T_{n, \textrm{reg}} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) of the relations \(T_n\) are closable operators which satisfy

$$\begin{aligned} T_{m, \textrm{reg}} \prec _c T_{n, \textrm{reg}}, \quad m \le n, \end{aligned}$$
(7.2)

see Lemma 2.2. Hence, by Theorem 5.1, there exists a closable linear operator \(T_{\textrm{r}} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{\textrm{r}})\), where \({{\mathfrak {K}}}_{\textrm{r}}\) is a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}T_{\textrm{r}} =\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_{n, \textrm{reg}} \varphi \Vert < \infty \right\} \end{aligned}$$
(7.3)

and which satisfies

$$\begin{aligned} T_{n, \textrm{reg}} \prec _c T_{\textrm{r}} \quad \text{ and } \quad \Vert T_{n, \textrm{reg}} \varphi \Vert \nearrow \Vert T_{\textrm{r}} \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T_{\textrm{r}}. \end{aligned}$$
(7.4)

Moreover, the closures \((T_{n, \textrm{reg}})^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) are closed linear operators which satisfy

$$\begin{aligned} (T_{m, \textrm{reg}})^{**} \prec _c (T_{n, \textrm{reg}})^{**}, \quad m \le n, \quad \text{ and } \quad (T_{n, \textrm{reg}})^{**} \prec _c (T_{\textrm{r}})^{**}, \end{aligned}$$
(7.5)

see Proposition 6.1. By the same proposition, there exists a closed linear operator \(S_{\textrm{r}} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{\textrm{c}})\), where \({{\mathfrak {K}}}_{\textrm{c}}\) is a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}S_{\textrm{r}} =\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n^{**}:\, \sup _{n \in {{\mathbb {N}}}} \Vert (T_{n, \textrm{reg}})^{**} \varphi \Vert < \infty \right\} \end{aligned}$$
(7.6)

and which satisfies

$$\begin{aligned} (T_{n, \textrm{reg}})^{**} \prec _c S_{\textrm{r}} \prec _c(T_{\textrm{r}})^{**} \quad \text{ and } \quad \Vert (T_{n, \textrm{reg}})^{**} \varphi \Vert \nearrow \Vert S_{\textrm{r}} \varphi \Vert , \quad \varphi \in \mathrm{dom\,}S_{\textrm{r}}. \end{aligned}$$
(7.7)

In fact, \(\mathrm{dom\,}(T_{\textrm{r}})^{**} \subset \mathrm{dom\,}S_{\textrm{r}}\), while \(\Vert S_{\textrm{r}} \varphi \Vert =\Vert (T_{\textrm{r}})^{**}\varphi \Vert \) for \(\varphi \in \mathrm{dom\,}(T_{\textrm{r}})^{**}\).

It follows from (7.1) that the closures \(T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) of \(T_n\) are closed relations which satisfy

$$\begin{aligned} T_m^{**} \prec _c T_n^{**} \quad m \le n, \end{aligned}$$
(7.8)

see (2.3). Of course, by Lemma 2.2 also the regular parts of \(T_n\) satisfy such an inequality; but this gives again (7.5), due to the identity

$$\begin{aligned} ((T_n)^{**})_{\textrm{reg}}=(T_{n, \textrm{reg}})^{**}, \end{aligned}$$

see (1.3). Since the relation \(T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) is closed, it follows that the product

$$\begin{aligned} H_n=T_n^*T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}) \end{aligned}$$

is a nonnegative selfadjoint relation and by Theorem 2.3 one sees that (7.8) implies

$$\begin{aligned} H_m \le H_n, \quad m \le n. \end{aligned}$$

Thus according to Theorem 3.2 there exists a nonnegative selfadjoint relation \(H_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) which is the limit of the relations \(H_n\) in the strong resolvent sense or, equivalently, in the strong graph sense.

Theorem 7.1

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of linear relations which satisfy (7.1). Let \(H_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) be the nonnegative selfadjoint relation, which is the limit of the nondecreasing sequence of nonnegative selfadjoint relations \(T_n^* T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}})\). Then \(H_\infty \) satisfies

$$\begin{aligned} \mathrm{dom\,}(H_\infty )^{\frac{1}{2}}=\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n^{**}:\, \sup _{n \in {{\mathbb {N}}}} \Vert (T_{n, \textrm{reg}})^{**} \varphi \Vert < \infty \right\} \end{aligned}$$
(7.9)

and, furthermore,

$$\begin{aligned} (T_{n, \textrm{reg}})^{**} \varphi \Vert \nearrow \Vert (H_{\infty , \textrm{reg}})^{\frac{1}{2}}\varphi \Vert , \quad \varphi \in \mathrm{dom\,}(H_\infty )^{\frac{1}{2}}. \end{aligned}$$
(7.10)

Moreover, the limit \(S_{\textrm{r}} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_{\textrm{c}})\) of the sequence \((T_{n, \textrm{reg}})^{**} \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) in (7.6) and (7.7) satisfies

$$\begin{aligned} \Vert (T_{n, \textrm{reg}})^{**} \varphi \Vert \nearrow \Vert S_{\textrm{r}} \varphi \Vert , \quad \varphi \in \mathrm{dom\,}S_{\textrm{r}}=\mathrm{dom\,}(H_\infty )^{\frac{1}{2}}. \end{aligned}$$
(7.11)

Consequently, there exists a partial isometry \(U \in {{\textbf{L}}}({{\mathfrak {K}}}_\textrm{c}, {{\mathfrak {H}}})\) such that

$$\begin{aligned} (H_{\infty , \textrm{reg}})^{\frac{1}{2}}=U S_{\textrm{r}} \quad \text{ and } \quad H_{\infty , \textrm{reg}} = (S_{\textrm{r}})^* S_{\textrm{r}}. \end{aligned}$$
(7.12)

Proof

It is clear that the product \(H_n=T_n^{*} T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}})\) is a nonnegative selfadjoint relation. Furthermore, the closures \(T_n^{**}\) of \(T_n\) satisfy the inequalities (7.8). Therefore, the nonnegative selfadjoint relations \(H_n=T_n^{*} T_n^{**} \in {{\textbf{L}}}({{\mathfrak {H}}})\) form a nondecreasing sequence thanks to Theorem 2.3. Thus by Theorem 3.2 there exists a nonnegative selfadjoint relation \(H_\infty \) such that (3.3) and (3.4) hold. Remember that

$$\begin{aligned} H_n=T_n^{*} T_n^{**}= (T_{n, \textrm{reg}})^*(T_{n, \textrm{reg}})^{**}, \end{aligned}$$

so that there exists a partial isometry \(U_n \in {{\textbf{L}}}({{\mathfrak {K}}}_n, {{\mathfrak {H}}})\), such that

$$\begin{aligned} (H_{n, \textrm{reg}})^{\frac{1}{2}}= U_n \,(T_{n, \textrm{reg}})^{**}. \end{aligned}$$

In other words, (3.3) and (3.4) lead to (7.9) and (7.10). Similarly, a comparison of (7.3) and (7.4) with (7.9) and (7.10) shows that (7.11) holds. Therefore, there exists a partial isometry \(U \in {{\textbf{L}}}({{\mathfrak {L}}}, {{\mathfrak {H}}})\) such that \((H_{\infty , \textrm{reg}})^{\frac{1}{2}}= U S_{\textrm{r}}\), which is the first assertion in (7.12). This identity shows that also the second assertion in (7.12) holds. \(\square \)

Assume that the sequence \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) in Theorem 7.1 has an upper bound, i.e., there exists a linear relation \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), where \({{\mathfrak {K}}}\) is a Hilbert space, such that

$$\begin{aligned} T_n \prec _c T, \quad m \le n. \end{aligned}$$
(7.13)

For instance, if the sequence \(T_n\) consists of operators then T may be taken as the limit of \(T_n\) by Theorem 5.1. It follows from (7.13) that

$$\begin{aligned} T_{n, \textrm{reg}} \prec _c T_{\textrm{reg}} \quad \text{ and } \quad (T_{n, \textrm{reg}})^{**} \prec _c (T_{\textrm{reg}})^{**}. \end{aligned}$$

With these upper bounds it follows for the closable limit \(T_\textrm{r}\) of \(T_{n, \textrm{reg}}\) that

$$\begin{aligned} T_{\textrm{r}} \prec _c T_{\textrm{reg}} \quad \text{ and } \text{ hence } \quad (T_\textrm{r})^{**} \prec _c (T_{\textrm{reg}})^{**}. \end{aligned}$$

Consequently, for the closed limit \(S_{\textrm{r}}\) of \((T_{n, \textrm{reg}})^{**}\) one has via (7.5)

$$\begin{aligned} S_{\textrm{r}} \prec _c (T_{\textrm{r}})^{**} \prec _c (T_{\textrm{reg}})^{**}. \end{aligned}$$

8 An Example of a Nondecreasing Sequence

In order to illustrate the various possibilities of convergence a simple example of a nondecreasing sequence will be presented. Let \(R \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be a linear operator and define the sequence of linear operators \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), \(n \in {{\mathbb {N}}}\), by

$$\begin{aligned} T_n=\sqrt{n}\, R. \end{aligned}$$
(8.1)

Then it is clear from (8.1) that

$$\begin{aligned} \bigcap _{n=1}^\infty \mathrm{dom\,}T_n=\mathrm{dom\,}R \quad \text{ and } \quad T_n \prec _c T_{n+1}, \quad n \in {{\mathbb {N}}}, \end{aligned}$$

so that (5.1) is satisfied. Hence one can apply Theorem 5.1 to determine the limit T of the sequence \(T_n\). It follows from (5.2) and (5.3) that

$$\begin{aligned} \mathrm{dom\,}T =\mathrm{ker\,}R \quad \text{ and } \quad T=O_{\,\mathrm{ker\,}R}. \end{aligned}$$
(8.2)

In fact, it is clear that T is closable and singular, simultaneously, and that

$$\begin{aligned} T^{**}=O_{\,{\overline{\mathrm{ker\,}}}\, R}. \end{aligned}$$
(8.3)

Moreover, observe that it follows from (8.2) and (8.3) that

$$\begin{aligned} T^*T= \mathrm{ker\,}R \times (\mathrm{ker\,}R)^\perp \quad \text{ and } \quad T^*T^{**}= \mathrm{{\overline{ker}}\,}R \times (\mathrm{ker\,}R)^\perp . \end{aligned}$$

Note that in the special case where \(R \in {{\textbf{B}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) this illustrates [2, Corollary 5.2.13]. If, in addition, the operator \(R \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) is closable, then all \(T_n\) in (8.1) are closable. The closures \(T_{n}^{**}\) of \(T_n\) are given by

$$\begin{aligned} T_{n}^{**} = \sqrt{n} \,R^{**}, \end{aligned}$$

and it is clear that (6.1) is satisfied. Hence one can apply Proposition 6.1 to obtain the closed limit S of the sequence \(T_{n}^{**}\). It follows from (6.2) and (6.3) that

$$\begin{aligned} \mathrm{dom\,}S =\mathrm{ker\,}R^{**} \quad \text{ and } \quad S=O_{\,\mathrm{ker\,}R^{**}}. \end{aligned}$$
(8.4)

One sees directly from (8.3) that \(T^{**} \subset S\), which illustrates the situation in Proposition 6.1. The inclusion \(T^{**} \subset S\) is strict precisely when \({\overline{\mathrm{ker\,}}} R \subset \mathrm{ker\,}R^{**}\) is strict. As an example where the inclusion is strict, let R be an operator such that \(R^{-1}\) is an operator that is not closable, in which case \(\mathrm{ker\,}R=\{0\}\) and \(\mathrm{ker\,}R^{**} \ne \{0\}\). Note that the nonnegative selfadjoint relation \(S^*S\) is given by

$$\begin{aligned} S^*S= \mathrm{ker\,}R^{**} \times (\mathrm{ker\,}R^{**})^\perp . \end{aligned}$$

as follows from (8.4).

Next consider the Lebesgue decomposition of R which is given by

$$\begin{aligned} R=R_{\textrm{reg}}+R_{\textrm{sing}}, \quad R_{\textrm{reg}}=(I-P)R, \quad R_\textrm{sing}=PR, \end{aligned}$$

where P be the orthogonal projection form \({{\mathfrak {K}}}\) onto \(\mathrm{mul\,}R^{**}\). Then the regular parts \(T_{n, \textrm{reg}}\) of \(T_n\) in (8.1) are given by

$$\begin{aligned} T_{n, \textrm{reg}}=\sqrt{n}\, R_{\textrm{reg}}, \end{aligned}$$

and it is clear that (7.2) is satisfied. For the closable limit \(T_{\textrm{r}}\) of the sequence \(T_{n, \textrm{reg}}\) it follows from (7.3) and (7.4) that

$$\begin{aligned} \mathrm{dom\,}T_{\textrm{r}} =\mathrm{ker\,}R_{\textrm{reg}} \quad \text{ and } \quad T_\textrm{r}=O_{\,\mathrm{ker\,}R_{\textrm{reg}}}. \end{aligned}$$

Since \(T_{\textrm{reg}}=O_{\mathrm{\mathrm{ker\,}R}}\) one sees directly that \(T_{\textrm{r}} \prec _c T_{\textrm{reg}}\), which is the general situation. The inequality is strict precisely when \(\mathrm{ker\,}R \subset \mathrm{ker\,}R_{\textrm{reg}}\) is strict. Observe that

$$\begin{aligned} (T_{\textrm{r}})^* T_{\textrm{r}} = \mathrm{ker\,}R_{\textrm{reg}} \times (\mathrm{ker\,}R_\textrm{reg})^\perp \quad \text{ and }\quad (T_{\textrm{r}})^* (T_{\textrm{r}})^{**} = \mathrm{{\overline{ker}}\,}R_{\textrm{reg}} \times (\mathrm{ker\,}R_{\textrm{reg}})^\perp . \end{aligned}$$

The closures of \(T_{n, \textrm{reg}}\) are given by

$$\begin{aligned} (T_{n, \mathrm reg})^{**}=\sqrt{n} (R_{\textrm{reg}})^{**}, \end{aligned}$$

and it is clear that (7.5) is satisfied. For the closed limit \(S_{\textrm{r}}\) of the sequence \((T_{n, \textrm{reg}})^{**}\) it follows from (7.6) and (7.7) that

$$\begin{aligned} \mathrm{dom\,}S_{\textrm{r}}=\mathrm{ker\,}(R_{\textrm{reg}})^{**} \quad \text{ and } \quad S_\textrm{r}=O_{\,\mathrm{ker\,}(R_{\textrm{reg}})^{**}}. \end{aligned}$$

Therefore, one sees that

$$\begin{aligned} (S_{\textrm{r}})^* S_{\textrm{r}} = \mathrm{ker\,}(R_{\textrm{reg}})^{**} \times (\mathrm{ker\,}(R_{\textrm{reg}})^{**})^\perp . \end{aligned}$$
(8.5)

Finally consider \(T_n\) as in (8.1) with a general operator \(R \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\). Then the product relation \(H_n=T_n^* T_n^{**}\) is given by

$$\begin{aligned} H_n=n R^* R^{**}=n (R_{\textrm{reg}})^{*} (R_{\textrm{reg}})^{**}. \end{aligned}$$

Since \(\mathrm{ker\,}(R_{\textrm{reg}})^*(R_{\textrm{reg}})^{**}=\mathrm{ker\,}(R_\textrm{reg})^{**}\), it follows from Example 3.3 that the limit \(H_\infty \) of \(H_n\) is given by \(H_\infty = \mathrm{ker\,}(R_{\textrm{reg}})^{**} \times (\mathrm{ker\,}(R_{\textrm{reg}})^{**})^\perp \), which agrees with (8.5).

9 A Description of Closed Linear Operators

Let \(T_n \in {{\textbf{B}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) be a sequence of operators that satisfy (5.1). According to Corollary 6.2 there is a closed limit \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) which satisfies (6.4) and (6.5). This section contains some variations on this theme.

First it is shown that any nonnegative selfadjoint operator is rougly speaking the limit of a certain class of nonnegative bounded linear operators.

Lemma 9.1

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a nonnegative selfadjoint operator. Then there exists a sequence of nonnegative selfadjoint operators \(A_n \in {{\textbf{B}}}({{\mathfrak {H}}})\) such that

$$\begin{aligned} (A_m \varphi , \varphi ) \le (A_n \varphi , \varphi ), \quad \varphi \in {{\mathfrak {H}}}, \quad m \le n, \end{aligned}$$
(9.1)

and

$$\begin{aligned} (A_n \varphi , \varphi ) \nearrow \Vert A^{\frac{1}{2}}\varphi \Vert ^2, \quad \varphi \in \mathrm{dom\,}A^{\frac{1}{2}}. \end{aligned}$$
(9.2)

Proof

Consider the spectral representation of the nonnegative selfadjoint operator A the Hilbert space \({{\mathfrak {H}}}\):

$$\begin{aligned} A=\int _0^\infty \lambda \,dE_\lambda . \end{aligned}$$

By means of this representation let the nonnegative selfadjoint operators \(A_n\in {{\textbf{B}}}({{\mathfrak {H}}})\) be defined by

$$\begin{aligned} A_n=\int _0^n \lambda \,dE_\lambda , \quad n\in {{\mathbb {N}}}. \end{aligned}$$

Then is is clear that \((A_m \varphi , \varphi ) \le (A_n \varphi , \varphi )\), \(m \le n\), for all \(\varphi \in {{\mathfrak {H}}}\). This gives (9.1). By the construction of the sequence \(A_n\) one obtains

$$\begin{aligned} (A_n \varphi , \varphi ) \nearrow \Vert A^{\frac{1}{2}}\varphi \Vert ^2, \quad \varphi \in \mathrm{dom\,}A^{\frac{1}{2}}, \end{aligned}$$

which gives (9.2). \(\square \)

As a consequence of Lemma 9.1 there is some kind of converse of Corollary 6.2.

Proposition 9.2

Let \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be a closed linear operator. Then there exists a sequence of linear operators \(T_n \in {{\textbf{B}}}(\mathrm{{\overline{dom}}\,}T, {{\mathfrak {H}}})\), such that

$$\begin{aligned} \Vert T_m \varphi \Vert \le \Vert T_n \varphi \Vert , \quad \varphi \in \mathrm{{\overline{dom}}\,}T, \quad m \le n, \end{aligned}$$
(9.3)

and

$$\begin{aligned} \Vert T_{n} \varphi \Vert \nearrow \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$
(9.4)

Proof

The product relation \(H=T^*T\) is nonnegative and selfadjoint in \({{\mathfrak {H}}}\) with \(\mathrm{mul\,}H=\mathrm{mul\,}T^*=(\mathrm{dom\,}T)^\perp \). Then \(H= A {{\,\mathrm{\, {{\widehat{\oplus }}} \,}\,}}(\{0\} \times (\mathrm{dom\,}T)^\perp \), where \(A=H_{\textrm{reg}}\) is a nonnegative selfadjoint operator in \(\mathrm{{\overline{dom}}\,}T\). Then there exists a sequence of nonnegative selfadjoint operators \(A_n \in {{\textbf{B}}}(\mathrm{{\overline{dom}}\,}T)\) such that

$$\begin{aligned} (A_m \varphi , \varphi ) \le (A_n \varphi , \varphi ), \quad \varphi \in \mathrm{{\overline{dom}}\,}T, \quad m \le n, \end{aligned}$$

and

$$\begin{aligned} (A_n \varphi , \varphi ) \rightarrow \Vert A^{\frac{1}{2}}\varphi \Vert ^2, \quad \varphi \in \mathrm{dom\,}A^{\frac{1}{2}}=\mathrm{dom\,}T \subset \mathrm{{\overline{dom}}\,}T. \end{aligned}$$

Due to \(H=T^*T\) and \(A=H_{\textrm{reg}}\) one sees that

$$\begin{aligned} \Vert A^{\frac{1}{2}}\varphi \Vert =\Vert T\varphi \Vert , \quad \varphi =\mathrm{dom\,}A^{\frac{1}{2}}=\mathrm{dom\,}T, \end{aligned}$$

see Lemma 11.2. Finally define \(T_n=A_n^{\frac{1}{2}}\), so that (9.3) and (9.4) are satisfied. \(\square \)

The last result in this section is a direct consequence of Proposition 9.2; it describes the closability of an operator in terms of a sequence of bounded linear operators; see for the original statement [4, Theorem 8.8, Theorem 8.9].

Corollary 9.3

Let \(S \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be a linear operator. Then the following statements are equivalent:

  1. (i)

    S is closable;

  2. (ii)

    there exists a sequence of linear operators \(T_n \in {{\textbf{B}}}(\mathrm{{\overline{dom}}\,}S, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, such that

    $$\begin{aligned} \Vert T_m \varphi \Vert \le \Vert T_n \varphi \Vert , \quad \varphi \in \mathrm{{\overline{dom}}\,}S, \quad m \le n, \end{aligned}$$
    (9.5)

    and

    $$\begin{aligned} \Vert T_{n} \varphi \Vert \nearrow \Vert S \varphi \Vert , \quad \varphi \in \mathrm{dom\,}S. \end{aligned}$$
    (9.6)

Proof

(i) \(\Rightarrow \) (ii) Let \(S \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be a closable operator and denote its closure by T. Then \(T \in {{\textbf{L}}}({{\mathfrak {H}}},{{\mathfrak {K}}})\) is a closed operator which extends S, such that \(\mathrm{{\overline{dom}}\,}T=\mathrm{{\overline{dom}}\,}S\). Now apply Proposition 9.2.

(ii) \(\Rightarrow \) (i) Let \(T_n \in {{\textbf{B}}}(\mathrm{{\overline{dom}}\,}S, {{\mathfrak {K}}}_n)\) be a sequence for which (9.5) holds. Then by Corollary 6.2 there exists a closed linear operator \(T \in {{\textbf{L}}}(\mathrm{{\overline{dom}}\,}S, {{\mathfrak {K}}})\), such that

$$\begin{aligned} \mathrm{dom\,}T= \left\{ \varphi \in \mathrm{{\overline{dom}}\,}S:\, \sup _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert < \infty \right\} , \end{aligned}$$

and which satisfies

$$\begin{aligned} \Vert T_{n} \varphi \Vert \nearrow \Vert T \varphi \Vert , \quad \varphi \in \mathrm{{\overline{dom}}\,}S. \end{aligned}$$

Thanks to (9.6) one has \(\Vert S \varphi \Vert =\Vert T \varphi \Vert \) for all \(\varphi \in \mathrm{dom\,}S\). Since T is closed if follows that S is closable. \(\square \)

An application of these results can be found in [7, Theorem 6.4], where pairs of bounded linear operators are classified in terms of almost domination.

10 Nonincreasing Sequences of Linear Operators

In this section there is a brief review for the situation of nonincreasing sequences of linear operators in the sense of contractive domination. First recall the analog of the monotonicity principle in Theorem 3.2 for nonincreasing sequences; see [3, Theorem 3.7].

Theorem 10.1

Let \(K_n \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a sequence of nonnegative selfadjoint relations and assume they satisfy

$$\begin{aligned} K_n \le K_m, \quad m \le n. \end{aligned}$$

Then there exists a nonnegative selfadjoint relation \(K_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) with

$$\begin{aligned} K_\infty \le K_n, \quad n \in {{\mathbb {N}}}. \end{aligned}$$
(10.1)

In fact, \(K_n \rightarrow K_\infty \) in the strong resolvent sense or, equivalently, in the strong graph sense. Moreover, the square root of \(K_\infty \) satisfies

$$\begin{aligned} \mathrm{ran\,}(K_\infty )^{\frac{1}{2}}=\left\{ \varphi \in \bigcap _{n \in {{\mathbb {N}}}} \mathrm{ran\,}(K_n)^{\frac{1}{2}}:\, \lim _{n \rightarrow \infty } \Vert ((K_{n})^{-{\frac{1}{2}}})_{\textrm{reg}} \varphi \Vert < \infty \right\} \end{aligned}$$
(10.2)

and, furthermore,

$$\begin{aligned} \Vert ((K_{n})^{-{\frac{1}{2}}})_{\textrm{reg}} \varphi \Vert \nearrow \Vert ((K_{\infty })^{-{\frac{1}{2}}})_{\textrm{reg}} \varphi \Vert , \quad \varphi \in \mathrm{ran\,}(K_\infty )^{\frac{1}{2}}. \end{aligned}$$
(10.3)

Proof

A short proof is included for completeness. By antitonicity, the sequence \((K_n)^{-1} \in {{\textbf{L}}}({{\mathfrak {H}}})\) is nondecreasing; cf. [2, Corollary 5.2.8]. Hence, by Theorem 3.2, there exists a nonnegative selfadjoint relation, say, \((K_\infty )^{-1} \in {{\textbf{L}}}({{\mathfrak {H}}})\), such that \((K_\infty )^{-1}\) is the limit of the sequence \((K_n)^{-1} \in {{\textbf{L}}}({{\mathfrak {H}}})\) in the strong resolvent sense or, equivalently, in the strong graph sense, and \((K_n)^{-1} \le (K_\infty )^{-1}\). Then, again by antitonicity, \(K_\infty \le K_n\) and, moreover, \(K _\infty \) is the limit of the sequence \(K_n\) in the strong graph sense. The rest of the statements is a direct translation of similar statements in Theorem 3.2. \(\square \)

Note that the multivalued parts of the relations \(K_n\) in Theorem 3.2 form a nonincreasing sequence. If one of the relations \(K_n\) in Theorem 10.1 is an operator, then all of its successors are operators and, ultimately, the limit \(K_\infty \) is an operator.

Example 10.2

Let \(A \in {{\textbf{L}}}({{\mathfrak {H}}})\) be a nonnegative selfadjoint operator or relation. Then it is clear that the sequence \(K_n=\frac{1}{n}A\) of nonnegative selfadjoint operators or relations is nonincreasing. Hence there exists a nonnegative selfadjoint relation \(K_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) such that \(K_n \rightarrow K_\infty \) is the strong graph sense. By means of Example 3.3 one sees immediately that

$$\begin{aligned} K_\infty =\mathrm{{\overline{dom}}\,}A \times \mathrm{mul\,}A. \end{aligned}$$

The following result is the analog of Theorem 5.1 for nonincreasing sequences of linear operators. Due to the sequence being nonincreasing there are no further convergence restrictions for the limit as in Theorem 5.1.

Theorem 10.3

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of linear operators which satisfy

$$\begin{aligned} T_n \prec _c T_m, \quad m \le n. \end{aligned}$$
(10.4)

Then there exists a linear operator \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), where \({{\mathfrak {K}}}\) is a Hilbert space, such that

$$\begin{aligned} \mathrm{dom\,}T=\bigcup _{n \in {{\mathbb {N}}}} \mathrm{dom\,}T_n, \end{aligned}$$
(10.5)

and which satisfies

$$\begin{aligned} T \prec _c T_n \quad \text{ and } \quad \Vert T_n \varphi \Vert \searrow \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$
(10.6)

Proof

Denote the right-hand side of (10.5) by \({{\mathfrak {D}}}\). Now let \(\varphi \in {{\mathfrak {D}}}\), then clearly \(\varphi \in \mathrm{dom\,}T_N\) for some \(N \in {{\mathbb {N}}}\). For all \(n \ge N\) one has \(T_n \prec _c T_N\), which implies that \(\varphi \in \mathrm{dom\,}T_n\) for all \(n \ge N\) and \(\lim _{n \rightarrow \infty } \Vert T_n \varphi \Vert \) exists by (10.4). Hence for each \(\varphi \in {{\mathfrak {D}}}\) one may define

$$\begin{aligned} \Vert \varphi \Vert _+= \lim _{n \rightarrow \infty } \Vert T_n \varphi \Vert . \end{aligned}$$

Then \(\Vert \cdot \Vert _+\) generates a well-defined seminorm on the linear subspace \({{\mathfrak {D}}}\). Let \((\cdot , \cdot )_+\) be the corresponding semi-inner product. By Lemma 4.1 there exists a linear operator T defined on \(\mathrm{dom\,}T={{\mathfrak {D}}}\subset {{\mathfrak {H}}}\) to a Hilbert space \({{\mathfrak {K}}}\) such that

$$\begin{aligned} (\varphi , \psi )_+=(T\varphi , T \psi ), \quad \varphi \in {{\mathfrak {D}}}. \end{aligned}$$

This shows the assertion in (10.6). \(\square \)

Now Theorem 10.3 will be applied under the assumption that the linear operators \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\) are closed. Then the corresponding relations \(K_n=T_n^* T_n \in {{\textbf{L}}}({{\mathfrak {H}}})\) are nonnegative and selfadjoint.

Theorem 10.4

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}}_n)\), where \({{\mathfrak {K}}}_n\) are Hilbert spaces, be a sequence of closed linear operators which satisfy (10.4) and let \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\), where \({{\mathfrak {K}}}\) is a Hilbert space, be the limit operator satisfying (10.5) and (10.6). Let \(K_\infty \in {{\textbf{L}}}({{\mathfrak {H}}})\) be the nonnegative selfadjoint relation, which is the limit of the nonincreasing sequence of nonnegative selfadjoint relations \(K_n=T_n^* T_n \in {{\textbf{L}}}({{\mathfrak {H}}})\), so that \(K_\infty \) satisfies (10.2) and (10.3). Then \(K_\infty \) and T are connected via

$$\begin{aligned} K_\infty =T^* T^{**}. \end{aligned}$$
(10.7)

Consequently, there exists a partial isometry \(U \in {{\textbf{B}}}({{\mathfrak {K}}},{{\mathfrak {H}}})\) such that

$$\begin{aligned} (K_{\infty , \textrm{reg}})^{\frac{1}{2}}= U (T_{\textrm{reg}})^{**} \quad \text{ or } \quad (T_{\textrm{reg}})^{**}=U^*(K_{\infty , \textrm{reg}})^{\frac{1}{2}}. \end{aligned}$$
(10.8)

Moreover, for the limit \(T \in {{\textbf{L}}}({{\mathfrak {H}}},{{\mathfrak {K}}})\) one has

  1. (a)

    T is closable if and only if \(T \subset U^*(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\);

  2. (b)

    T is closed if and only if \(T=U^*(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\);

  3. (c)

    T is singular if and only if \(K_\infty \) is singular.

Proof

Let \(T \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {K}}})\) be the limit operator in (10.5) and (10.6). Recall from (10.6) that \(T \prec _c T_n\). This leads to \(T^{**} \prec _c (T_n)^{**}=T_n\), which gives \(T^*T^{**} \le T_n^*T_n=K_n\) by Theorem 2.3. Since this holds for all \(n \in {{\mathbb {N}}}\) one obtains

$$\begin{aligned} T^*T^{**} \le K_\infty . \end{aligned}$$
(10.9)

Moreover, recall from (10.1) that \(K_\infty \le K_n\), so that \((K_\infty )^{\frac{1}{2}}\prec _c T_n\) by Theorem 2.3. In particular, it follows that \((K_{\infty , \textrm{reg}})^{\frac{1}{2}}\prec _c T_n\). Hence one has

$$\begin{aligned} \mathrm{dom\,}T_n \subset \mathrm{dom\,}(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\quad \text{ and } \quad \Vert (K_{\infty , \textrm{reg}})^{\frac{1}{2}}\varphi \Vert \le \Vert T_n \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T_n. \end{aligned}$$

Clearly, with (10.5) this now leads to

$$\begin{aligned} \mathrm{dom\,}T \subset \mathrm{dom\,}(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\quad \text{ and } \quad \Vert (K_{\infty , \textrm{reg}})^{\frac{1}{2}}\varphi \Vert \le \inf _{n \in {{\mathbb {N}}}} \Vert T_n \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T. \end{aligned}$$

Thanks to (10.6) this reads

$$\begin{aligned} \mathrm{dom\,}T \subset \mathrm{dom\,}(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\quad \text{ and } \quad \Vert (K_{\infty , \textrm{reg}})^{\frac{1}{2}}\varphi \Vert \le \Vert T \varphi \Vert , \quad \varphi \in \mathrm{dom\,}T, \end{aligned}$$

or equivalently, \((K_{\infty , \textrm{reg}})^{\frac{1}{2}}\prec _c T\). Since closures and regular parts are preserved under the inequality, this gives \((K_{\infty , \textrm{reg}})^{\frac{1}{2}}\prec _c (T^{**})_{\textrm{reg}}\) or \((K_\infty )^{\frac{1}{2}}\prec _c T^{**}\) by Lemma 2.2. Therefore, one obtains

$$\begin{aligned} K_\infty \le T^* T^{**}. \end{aligned}$$
(10.10)

Combining the inequalities (10.9) and (10.10) leads to the inequalities

$$\begin{aligned} T^* T^{**} \le K_\infty \le T^* T^{**}, \end{aligned}$$

or, equivalently,

$$\begin{aligned} (T^*T^{**}-\lambda )^{-1} \le (K_\infty -\lambda )^{-1} \le (T^*T^{**}-\lambda )^{-1}, \quad \lambda <0. \end{aligned}$$

This shows that (10.7) holds. Next (10.8) follows thanks to Lemma 11.2.

Finally, the last assertions concerning the relationship between T and \(K_\infty \) will be discussed.

  1. (a)

    If \(T \subset U^*(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\), then T is closable. Conversely, if T is closable, then \(T=T_{\textrm{reg}} \subset (T_{\textrm{reg}})^{**}=U^* (K_{\infty , \textrm{reg}})^{\frac{1}{2}}\).

  2. (b)

    If \(T = U^*(K_{\infty , \textrm{reg}})^{\frac{1}{2}}\), then T is closed. Conversely, if T is closed, then \(T=(T_{\textrm{reg}})^{**} =U^* (K_{\infty , \textrm{reg}})^{\frac{1}{2}}\).

  3. (c)

    If T is singular, then \(T^*={{\mathfrak {A}}}\times {{\mathfrak {B}}}\) where \({{\mathfrak {A}}}\) and \({{\mathfrak {B}}}\) are closed linear subspaces of \({{\mathfrak {K}}}\) and \({{\mathfrak {H}}}\), respectively. Hence \(T^{**}={{\mathfrak {B}}}^\perp \times {{\mathfrak {A}}}^\perp \), so that \(T^*T^{**}={{\mathfrak {B}}}^\perp \times {{\mathfrak {B}}}\) and \(K_\infty \) is singular. Conversely, let \(K_\infty =T^*T^{**}\) be singular. Then \(T^*T^{**}={{\mathfrak {B}}}^\perp \times {{\mathfrak {B}}}\) with a closed linear subspace \({{\mathfrak {B}}}\) in \({{\mathfrak {H}}}\). Hence it follows that

    $$\begin{aligned} \left\{ \begin{array}{l} \mathrm{mul\,}T^*=\mathrm{mul\,}T^*T^{**}={{\mathfrak {B}}}, \\ \mathrm{ker\,}T^{**}=\mathrm{ker\,}T^*T^{**}={{\mathfrak {B}}}^\perp . \end{array} \right. \end{aligned}$$

    Therefore \(\mathrm{{\overline{ran}}\,}T^{*}= (\mathrm{ker\,}T^{**})^\perp =\mathrm{mul\,}T^*\), i.e. \(T^*\) and, hence, also T is singular.

\(\square \)

In the present circumstances there is in general no preservation of closedness in Theorem 10.3. This will be shown in the following example; it is a simple adaptation of [3, Example 4.5] or [12, p. 374].

Example 10.5

Let \(T_n \in {{\textbf{L}}}({{\mathfrak {H}}}, {{\mathfrak {H}}}\oplus {{\mathbb {C}}})\) with \({{\mathfrak {H}}}=L^2(0,1)\) be given as a column operator by \(T_n=\mathrm{col\,}(T_n^1, T_n^2)\) (see [8]) with the operators \(T_n^1\) and \(T_n^2\) given by

$$\begin{aligned} T_n^1 f=\frac{1}{\sqrt{n}}\, i Df, \quad f(1)=0, \quad \text{ and } \quad T_n^2 f=f(0). \end{aligned}$$

Here D stands for the maximal differentiation operator in \(L^2(0,1)\). Then \(T_n^1\) is closed, \(T_n^2\) is singular, while the column \(T_n\) is closed. It is clear that \(T_n \prec _c T_m\), \(m \le n\), and the limit \(T \in {{\textbf{L}}}({{\mathfrak {H}}})\) is given by \(Tf=f(0) e\), where the function \(e \in {{\mathfrak {H}}}=L^2(0,1)\) is defined by \(e(x)=1\). Moreover, the corresponding nonnegative selfadjoint relation \(K_n=T_n^*T_n\) is the operator in \({{\mathfrak {H}}}=L^2(0,1)\) given by

$$\begin{aligned} K_n f =- \frac{1}{n} D^2 f, \quad f'(0)=n f(0), \,\, f(1)=0. \end{aligned}$$

The relations \(K_n\) form a sequence that is nonincreasing with the nonnegative selfadjoint limit \(K_\infty \) and, by Theorem 10.4, one has

$$\begin{aligned} K_\infty =T^*T^{**}. \end{aligned}$$

Now observe that \(T^*= (\mathrm{span\,}\{e\})^\perp \times \{0\}\) and \(T^{**}={{\mathfrak {H}}}\times \mathrm{span\,}\{e \}\), so that T is a singular operator and, in fact \(T^*T^{**}={{\mathfrak {H}}}\times \{0\}\). Hence it follows that

$$\begin{aligned} K_\infty =T^*T^{**}={{\mathfrak {H}}}\times \{0\}. \end{aligned}$$