Adjoint pairs and unbounded normal operators

An adjoint pair is a pair of densely defined closed linear operators A,B on a Hilbert space such that ⟨Ax,y⟩=⟨x,By⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \langle Ax,y\rangle = \langle x,By\rangle$$\end{document} for x∈D(A),y∈D(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathcal{D} (A),y \in \mathcal{D} (B) $$\end{document}. We consider adjoint pairs for which 0 is a regular point for both operators and associate a boundary triplet to such an adjoint pair. Proper extensions of the operator B are in one-to-one correspondence Tc↔C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Tc\leftrightarrow \mathcal{C} $$\end{document} to closed subspaces C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C} $$\end{document} of N(A∗)⊕N(B∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}(A^{*}) \oplus\mathcal{N}(B^{*})$$\end{document}. In the case when B is formally normal andD(A)=D(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{D}(A) =\mathcal{D}(B) $$\end{document}, the normal operators Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Tc $$\end{document} are characterized. Next we assume that B has an extension to a normal operator with bounded inverse. Then the normal operators Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Tc$$\end{document} are described and the case when N(A∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}(A^{*}) $$\end{document} has dimension one is treated.


Introduction
This paper deals with various notions and constructions in unbounded operator theory on Hilbert space. The basic objects studied in this paper are adjoint pairs {A, B} (see Definition 2 below) of densely defined operators A, B on a Hilbert space H for which the number 0 is a regular point of A and B.
In Section 3, we use a result of M.I. Vishik [Vi] (stated as Theorem 5) and associate to such a pair a boundary triplet (see Definition 1) for the operator matrix acting as a symmetric operator with domain D (B)⊕D (A) on H ⊕H (Theorem 6). Then the theory of boundary triplets allows one to describe the proper extensions of the symmetric operator A in terms of closed relations. In the remaining Sections 4-6, we assume in addition that the operator B is formally normal and D (A) = D (B). Our aim is to study normal extensions of the operator B.

K. Schmüdgen
The proper extensions of B (that is, closed operators T satisfying B ⊆ T ⊆ A * ) can be described in terms of closed subspaces C of the Hilbert space N (A * ) ⊕ N (B * ). Let T C denote the corresponding operator.
In Section 4, the normality of the operator T C is characterized in terms of the subspace C (Theorems 10 and 12). In Sections 5 and 6, we assume that the formally normal operator B admits an extension to a normal operator R * with bounded inverse. Then there exists a unitary operator W satisfying R −1 W = (R * ) −1 which leads to simplifications of the normalcy criteria for the operator T C . The case when N (A * ) has dimension one is treated in Section 6; the corresponding result is Theorem 17.
Throughout the whole paper, {A, B} denotes an adjoint pair such that 0 is a regular point for A and B.
Let us add a few bibliographical comments and hints. Adjoint pairs are treated by Vishik [Vi] and in the monograph [EE,Section III.3]. Boundary triplets have been invented by Kochubei [Ko] and Bruk [Bk]; a fundamental paper on boundary triplets is [DM]. Boundary triplets associated with adjoint pairs were constructed and studied by Malamud and Mogilevskii [MM]. Pioneering work on formally normal operators and their extensions to normal operators was done by Biriuk and Coddington [BC], [Cd2]. The existence of formally normal operators which have no normal extension was discovered by Coddington [Cd1]; a very simple example can be found in [Sch86]. Unbounded normal operators have been extensively studied by Stochel and Szafraniec, see e.g. [SS1], [SS2], [SS3].
Concerning the theory of unbounded operators on Hilbert space we refer to the author's graduate text [Sch12] and also to the monograph [EE].

Some operator-theoretic notions
In this short section we collect a few concepts and notations from operator theory which are crucial in what follows.
Let T be a linear operator on a Hilbert space H . We denote by D (T ) its domain, by R(T ) its range und by N (T ) its kernel.
The symbol+ refers to the direct sum of vector spaces. The algebra of bounded operators on H is denoted by B(H ). A number λ ∈ C is called regular for T if there exists a constant γ > 0 (depending on λ in general) such that A densely defined operator A is called formally normal if D (A) ⊆ D (A * ) and By polarization, condition (3) implies that A formally normal operator A is called normal if D (A) = D (A * ).
Next we recall the notion of a boundary triplet.

Adjoint pairs and boundary triplets
The main concept of this paper is the following.
Definition 2. An adjoint pair is a pair {A, B} of densely defined closed linear operators A and B on a Hilbert space H such that Clearly, (5) is equivalent to the relations Thus, any pair of densely defined operators A, B satisfying (6) is an adjoint pair.
In the literature, "adjoint pairs" often appear as "dual pairs" (for instance, in [MM]). Since the latter notion is used in different context in other parts of mathematics, we prefer to speak about "adjoint pairs" (as in [EE]).
We mention two examples.
Example 3. Suppose A is a densely defined closed operator. Then {A, A * } and {A * , A} are adjoint pairs. Note that in both cases we have equalities in (6).
Example 4. Suppose T is a densely defined closed symmetric operator and α ∈ C. Then the operators A 0 := T + αI and B 0 := T * + αI form an adjoint pair. Likewise, A := T + αI and B := (T * + αI) D (T ) = T + αI are an adjoint pair. Set a = Re α and b = Im α. For x ∈ D (T ) we compute Hence A and B are formally normal operators. Further, if b = 0, it follows from (7) that 0 is a regular point for A and B. The considerations of this paper are essentially based on an important theorem of M. I. Vishik [Vi,Theorems 1 and 2]. It is an extension of a result of Calkin [Ck] for symmetric operators. We state this result as Theorem 5 and add an number of useful formulas.
A crucial part is the existence of the operator R with bounded inverse; a nice proof of this assertion can be found in [EE,Theorem 3.3].
Theorem 5. Suppose {A, B} is an adjoint pair and 0 is a regular number for the operators A and B, that is, exists a constant γ > 0 such that Then there exists a closed operator R on H such that R and R * have inverses and Let us adopt the following notational convention: Elements of N (A * ) are denoted by u, u , u 1 , u 2 , u 1 , u 2 , while symbols v, v , v 1 , v 2 , v 1 , v 2 always refer to vectors of N (B * ).
Recall that throughout this paper {A, B} denotes an adjoint pair such that 0 is a regular point for the operators A and B.
We define an operator A with domain D (A ) = D (B) ⊕ D (A) on the direct sum Hilbert space H ⊕ H by the operator matrix From (5) it follows at once that the operator A is symmetric. It is easily verified that the adjoint operator A * has the domain D (A * ) = D (A * ) ⊕ D (B * ) and is given by the matrix Let (x, y), (x , y ) ∈ D (A * ). Then x, x ∈ D (A * ) and y, y ∈ D (B * ). Therefore, by (10) and (11), x, y, x , y are of the form Using equation (9) we derive and similarly Replacing (9) by (10) the same reasoning yields Further, since u 1 , u 1 ∈ N (A * ) and v 2 , v 2 ∈ N (B * ), we have Now we apply the preceding formulas (5), (12), (17), (18), (19), (20) and compute Next we introduce an auxiliary Hilbert space and linear mappings Γ 0 : where x, y are of the form (15) and (16). Then This is condition (i) of Definition 1 for T = A . Condition (ii) of Definition 1 is obvious from the description of domains D (A * ) and D (B * ) given in Theorem 5. Summarizing the preceding we have proved the following Theorem 6. Suppose that {A, B} is an adjoint pair such that 0 is a regular point for A and B. Then the triplet (K, Γ 0 , Γ 1 ) of the Hilbert space K = N (A * ) ⊕ N (B * ) and the mappings Γ 0 and Γ 1 , defined by equation (23), is a boundary triplet for the operator A * . Next we restate some facts from the theory of boundary triplets adapted to the present situation (see e.g. [Sch12,Section 14.2]). Recall that a closed operator T on H 2 := H ⊕ H is called a proper extension of the symmetric Each proper extension of A is of this form. Further, the extension A C of A is self-adjoint if and only the relation C is self-adjoint.

Clearly, T is a proper extension of A if and only if there are closed operators S, T on H such that
Then, as discussed in the paragraph before last, self-adjoint extensions of A on H ⊕ H are in one-to-one correspondence to proper extensions T of the operator B on H , and equivalently, to proper extensions S of A on H . These operators T and S will be studied in the next section.
The passage to 2 × 2 operator matrices is an old and powerful trick in operator theory which was used in many papers and different contexts, see e.g.

Formally normal operators and normal operators
In this section we continue the considerations of the previous section and assume in addition that B is a closed formally normal operator and D (A) = D (B).
Recall that B is formally normal means that D (B) ⊆ D (B * ) and Bx = B * x for x ∈ D (B). Since (6) holds by assumption, we have A = B * D (B) and therefore In particular, the operator A is also formally normal and closed. In fact, the above assumption is symmetric in the operators B and A.
Let x ∈ D (A * ) and y ∈ D (B * ). As noted above (see (10) and (11)), x and y are of the form Then, setting x = 0, y = 0 and renaming y by y in formula (21) we obtain Now we suppose that C is a closed subspace of K = N (A * ) ⊕ N (B * ). We define linear operators T C and S C on the Hilbert space H by Further, let C denote the closed linear subspace of K given by Recall that for any relation Comparing (32) and (33) we conclude that Hence, since C * is a closed linear relation, C is a closed linear subspace of K.
Proof. By a general result on boundary triplets [Sch12,Lemma 14.13], the mappings Γ 0 , Γ 1 of D (A * ), endowed with the graph norm, into K are continuous.
Since the operators A, B are closed and the subspaces C , C of K are closed, it follows easily from this result that T C and S C are closed operators. The inclusions B ⊆ T C ⊆ A * are obvious from the definition of T C . Thus, T C is a proper extension of B. Now let T be an arbitrary proper extension of B. Then the matrix T, defined by (26), with S := T * , is a self-adjoint operator on H 2 . Hence, by Lemma 8, T = A C for some closed relation C on K 2 . Let C denote the set of vectors Γ 0 (x, y) for (Γ 0 (x, y), Γ 1 (x, y)) ∈ C. Then C is a closed linear subspace of K and from the definition of T = A C it follows that T = T C .
The following theorem characterizes the case when the operator T C is normal. Condition (i) ensures that the domains D (T C ) and D ((T C ) * ) coincide, while condition (ii) implies the equality of norms T C z and (T C ) * z .
Theorem 10. Suppose B is a closed formally normal operator and D (A) = D (B). Let C be a closed linear subspace of K satisfying the following two conditions: Proof. First we prove that the conditions (i) and (ii) imply that T C is a normal operator. Since (T C ) * = S C by Lemma 9, condition (i) ensures that Let z ∈ D (T C ) = D ((T C ) * ). Then, by (30) and (31), the vector z is of the form Similarly, by A * u 2 = 0, By assumption, B is formally normal and A = B * D (B). Hence Bx 0 2 = Ax 0 2 . Since v 1 = u 2 by condition (ii), comparing (38) and (39) The crucial step is to show that x 0 = y 0 . Recall that T C = A * z = Bx 0 +v 1 and (T C ) * z = S C z = B * z = Ay 0 + u 2 . Therefore, since T C is normal, using equation (44) we derive Using B * v 1 = 0 and A * z 2 = 0 we conclude that By assumption, the operator B is formally normal and A ⊆ B * . Hence, combining (41) and (44), we obtain Inserting the equality x 0 = y 0 into (40) we get (R * ) −1 v 1 +u 1 = R −1 u 2 +v 2 . Note that all vectors (R * ) −1 v 1 +u 1 with (u 1 , v 1 ) ∈ C and all vectors R −1 u 2 +v 2 with (u 2 , v 2 ) ∈ C are in D (T C ) = D ((T C ) * ). Thus, it follows that condition (i) is fulfilled.
By equations (38) and (39) and the normality of the operator T C , we have Recall that Bx 0 = Ax 0 , because B is formally normal. Hence v 1 = u 2 , which proves that condition (ii) holds.
Now we consider the special case where C is the the graph of a densely defined closed linear operator C of the Hilbert space N (B * ) into the Hilbert space N (A * ): Lemma 11. C = {(u 2 , C * u 2 ) : u 2 ∈ D (C * )}.
Proof. Let (u 2 , v 2 ) ∈ K. By the definitions (42) and (32) of C and C , we have (u 2 , v 2 ) ∈ C if and only if v 1 , v 2 = Cv 1 , u 2 for all v 1 ∈ D (C). The latter holds if and only if u 2 ∈ D (C * ) and v 2 = C * u 2 , which proves the assertion.
The following is the reformulation of Theorem 10 for subspaces of the form (42).

Theorem 12. Suppose B is a closed formally normal operator and D (A) = D (B). Let C be a closed linear subspace of K of the form (42).
Then the operator T C is normal if and only if there exists an isometric linear operator U of D (C * ) onto D (C) such that Proof. First we suppose that T C is normal. Let u 2 ∈ D (C * ). Then (u 2 , C * u 2 ) ∈ C , so by condition (i) there exists a vector v 1 ∈ D (C) such that (11) is a direct sum, v 1 is uniquely determined by u 2 . Clearly, the map u 2 → v 1 is linear. Since u 2 = v 1 by condition (ii) and equation (42), there is an isometric linear map U : D (C * ) → D (C) given by Uu 2 = v 1 . Inserting Uu 2 = v 1 , v 2 = C * u 2 , u 1 = Cv 1 = CU u 2 into condition (i), we obtain (43). Now we prove the converse implication. Let (u 2 , v 2 ) ∈ C . Then v 2 = C * u 2 by Lemma 11 and v 1 := Uu 2 ∈ D (C), so (Cv 1 , v 1 ) = (CU u 2 , Uv 1 ) ∈ C . Then (43) gives (42) and (43) yields R −1 u 2 + v 2 = (R * ) −1 v 1 + u 1 . This proves that condition (i) is satisfied. Since U is isometric by assumption, condition (ii) holds as well.

Normal operators
In this section, we assume in addition that the operator R is normal. Recall that we assumed throughout that R has a bounded inverse R −1 ∈ B(H ).
Note that R is normal if and only if R * is normal, or equivalently, R −1 (resp. (R * ) −1 = (R −1 ) * ) is normal. Again, the assumption is symmetric in the operators A and B.
Next we consider the polar decomposition of the operator R: Here |R| := [R * R) 1/2 is the modulus of R and U is the phase operator of R. Note that U is a partial isometry with initial space N (T ) ⊥ and final space N (T * ). Since R is normal with bounded inverse, U is a unitary which commutes with |R|. Basic properties of the polar decomposition combined with the normalcy of the operator R yield the following formulas: Therefore, U * z ∈ D (T * ) and T * U * z = 0. That is, we have U * z ∈ N (T * ) and Putting the preceding together, we have proved that U N (T * ) = N (A * ).
The proof of the second equality N (B * ) = U * N (T * ) is similar.
Next we introduce another unitary operator W . Since R is normal, the equation defines an isometric linear operator on H with dense domain and dense range.
Hence it extends to a unitary operator, denoted again by W , on H . Then Applying the adjoint to This implies that the unitary W commutes with R −1 and (R * ) −1 . The unitary operator W is the square of the phase operator U . Indeed, since W U * |R|x = U |R|x by (46) and the range of |R| is dense, we get W U * = U , so that W = U 2 . Combined with Lemma 13 we derive From (47) and (48) Further, by (45), and therefore by Lemma 13, , The formulas (49), (50), (51), (52) are useful descriptions of the domains D (A * ) and D (B * ).
Using the unitaries W and U we can reformulate and slightly simplify Theorems 10 and 12 under the assumption that R is normal. We do not carry out these restatements and mention only the corresponding changes in the case of W . Then condition (i) in Theorem 10 should be replaced by and in Theorem 12 equation (43) becomes Example 14. In this example we consider the special case , condition (i) of Theorem 10 is fullfilled. Condition (ii) holds trivially, so the operator T C is normal. From (30) and (31) we conclude easily that T C = R * and S C = R. Now we want to construct examples and reverse our considerations. We begin with a bounded normal operator Z on H with trivial kernel.
From now on suppose that U = {0} is a closed linear subspace of H such that Since D (R) = H , such spaces exists; one can even show that there are infinitedimensional closed subspaces U satisfying (53). We denote by P the orthogonal projection of H on U and by W the unitary operator defined by (46). Then equation (47) (53). Hence x = 0, which proves that D (A) is dense. The operators A and B are formally normal, because they are restrictions of the normal operators R and R * , respectively. Since R and R * have bounded inverses, 0 is a regular point for A and B. Clearly, A and B form an adjoint pair.
We prove that N (A * ) = P H and N (B * ) = W * P H . Let y ∈ H . Since AZ(I − P )x, y = (I − P )x, y for all x ∈ H , it follows that y ∈ N (A * ) if and only if y ∈ P H . Similarly, implies that y ∈ N (B * ) if and only if W y ∈ P H , that is, y ∈ W * P H .

The one-dimensional case
In this section we remain the setup and the assumptions of the preceding section. We shall treat the simplest case where U = P H = N (A * ) has dimension one. Throughout this section, we suppose that U = C · ξ, where ξ is a fixed vector of H such that ξ / ∈ D (R). Consider a linear subspace C of It is obvious that the operator T C is not normal if dim C = 0 or dim C = 2. If C = C · W * ξ, we know from Example (14) that T C = (R * ) −1 . Thus it remains to study the case Then, by (32), a vector (β 1 ξ, β 2 W * ξ) ∈ K with β 1 , β 2 ∈ C belongs to C if and only if αW * ξ, β 2 W * ξ = ξ, β 1 ξ , or equivalently, αβ 2 = β 1 . Therefore, Note that R −1 = Z and (R * ) −1 = Z * . Hence it follows from (56) and (57) that condition (i) of Theorem 10 holds if and only if there exists a number γ ∈ C, γ = 0, such that Clearly, condition (ii) is equivalent to αW * ξ = γ α ξ , that is, |γ| = 1. Recall that Z * W * = Z. Therefore, by the preceding, Theorem 10 yields the following: T C is a normal operator if and only if there is a number γ ∈ C such that Before we continue we illustrate this statement in a very special case.
Example 16. Suppose S is a densely defined symmetric operator with equal nonzero deficiency indices. Then S has a self-adjoint extension X on H . Clearly, A := S + iI and B := S − iI are formally normal operators with domain D (S) and R := X + iI is a normal extension of A with bounded inverse Z. We choose a vector ξ ∈ H such that ξ / ∈ D (X) and define C and C by (56) and (57), respectively. Then we are in the setup described above.
Therefore, since Zξ ∈ D (X) and ξ / ∈ D (X), (59) is fulfilled if and only if γ = 1 and α − α = −2i, or equivalently, γ = 1 and α = a − i with a real. Therefore, by the preceding, the operator T C is normal if and only if That is, the normal extensions of B are parametrized by the real number a. It can be shown that for C as in (61) the corresponding operator T C + iI is self-adjoint and hence a self-adjoint extension of the symmetric operator S = B + iI.
Now we return to the general case. The normal operator R = Z −1 can be written as R = X + iY , where X and Y are strongly commuting self-adjoint operators. Since R is unbounded by assumption, at least one of the operators X and Y has to be unbounded. Our aim is to reformulate both conditions (59) and (60) in terms of X and Y . For this we need some notation.

K. Schmüdgen
These formulas (62) and (63) express the real numbers t γ and s γ,α in terms of the parameters γ and α. Now we want to reverse these transformations. Let α = a + ib with a, b real. Then we obtain Thus, given t, s ∈ R, γ is uniquely determined by (65) and there is a oneparameter family of numbers α in (66), with b ∈ R as real parameter, satisfying the equations (62) and (63). Likewise, if t = ∞, s ∈ R, we have γ = 1 and α = a + is is a one-parameter family with real parameter a such that (64) holds. Further, we define operators Note that R ∞ and the closure of R t , t ∈ R, are self-adjoint operators.
Theorem 17. Suppose that C is a subspace of K of the form (56). Then the operator T C is a normal operator if and only if there exists a t ∈ R ∪ {∞} such that ξ is an eigenvector of the operator R t . In this case, if s denotes the eigenvalue of eigenvector ξ of R t , the pair (t, s) is uniquely determined by C . More precisely, if the above conditions (59) and (60) are satisfied, then t = t γ and s = s γ,α .
bounded inverses. We suppose that R is unbounded and choose a function ξ(z) ∈ L 2 (C; μ) such that zξ(z) / ∈ L 2 (C; μ). Then we are in the setup discussed above and R * is a normal extension of the operator B. By appropriate choices of μ and ξ we can construct interesting cases.
First fix s, t ∈ R and suppose that μ is supported on the line x − ty = s, where z = x + iy, x, y ∈ R. Then (X − tY )ξ = sξ. Define γ by (65), α by (66) with b ∈ R, and the vector space C by (56). Then, by Theorem 17, T C is normal and the operator B has, in addition to R * , precisely the one-parameter family of operators T C (with real parameter b) as normal extensions on the Hilbert space H .
A similar result is true for t = ∞, s ∈ R.
Next we choose μ and ξ such ξ is not an eigenvector of some operator R t with t ∈ R ∪ {∞}. (For instance, let μ be the Lebesgue measure outside of some ball around the origin and set ξ := |z| −3 .) Then the operator B has no other normal extension on H than the operator R * .