On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.


Introduction
Spectral properties of unbounded operator matrices are of major interest in operator theory and its applications [24]. Important examples are systems of partial differential equations with λ-dependent coefficients or boundary conditions [1,9,10,19,23]. A concept of equivalence can be used to compare spectral properties of different operator functions and the problem of classifying bounded analytic operator functions modulo equivalence has been studied intensely [6,7,11,15]. The properties preserved by equivalences include the spectrum and for holomorphic operator functions there is a one-to-one correspondence between their Jordan chains, [14,Prop. 1.2]. Our aim is to generalize some of the results in those articles and study a concept of equivalence for classes of operator functions whose values are unbounded linear operators. A prominent result in this direction is the equivalence between an operator matrix and its Schur complements [2,21,24].
In this paper, we consider systems described by n × n operator matrix functions and study a concept of equivalence when some of the entries are Schur complements, polynomials, or can be written as a product of operator IEOT Gohberg et al. [11] and Bart et al. [5] studied a generalization of equivalence called equivalence after extension. Here, we introduce a more general definition of equivalent after extension, which we for clarity call equivalence after operator function extension. The definition of equivalent after extension in [5] correspond in Definition 2.2 to the case W S (λ) = IȞ S and W T (λ) = IȞ T for all λ ∈ Ω. We allow W S and W T to be unbounded operator functions and can therefore study a concept of equivalence for a larger class of unbounded operator function pairs S and T .
In particular, the equivalence results for Schur complements and polynomial problems presented in Sect. 3.1 respectively Sect. 3.3, can not be described by an equivalence after extension with the identity operator. In the equivalence results for multiplication operators in Sect. 3.2 the operator function W is bounded (actually W (λ) = I for all λ ∈ C). Thus, in that case the standard definition of equivalence after extension is sufficient as well.
Proposition 2.1 shows that two equivalent unbounded operator functions have the same spectral properties and it provides the correspondence between the domains. In the following proposition, those results are extended to include operator functions that are equivalent after operator function extension.
and we have then where S Ω and T Ω denote the restrictions of S and T to Ω.
Proof. From Definition 2.2 it follows that for λ ∈ Ω the following relations hold The result then follows from Proposition 2.1 and that the closure of a block diagonal operator coincides with the closures of the blocks.
Below we show how an equivalence for an entry in an operator matrix function can be used to find an equivalence for the full operator matrix function. A general operator matrix function S : Proof. Under the assumption (2.4), the lemma follows immediately by verifying S(λ) = E(λ)T (λ)F(λ).
Remark 2.5. The condition (2.4) is satisfied in the trivial case E = 0, F = 0, and for the problems we study in Sect. 3. A similar result holds also when (2.4) is not satisfied, but then the (2, 2)-entry in T (λ) will not be of the same form.

Equivalences for Classes of Operator Matrix Functions
In this section, we study Schur complements, operator functions consisting of multiplications of operator functions, and operator polynomials. Each type will be studied similarly: First an equivalence after operator function extension is shown, which then together with Lemma 2.4 is utilized in an operator matrix function.
Remark 3.1. Assume that S(λ) ⊕ W (λ) is equivalent to T (λ) for λ ∈ Ω and let S be defined as (2.3). For the equivalence relation between T and S we want the block S(λ) ⊕ W (λ) intact to be able to apply Lemma 2.4 directly. Therefore, an equivalence after W -extension of S(λ) is given as

Schur Complements
Then S is after D-extension equivalent to T on Ω , where the operator matrix functions E and F in the equivalence relation (2.1) are

The operator T (λ) is closable if and only if S(λ) is closable, and
Proof. The operators matrices E(λ) and F (λ) are bounded on D(C(λ)) and The result then follows from the factorization and Proposition 2.3. The domain and the closure are not explicitly stated in the equivalences in the remaining part of the article but they can be derived using the relations in Proposition 2.3.

Define the operator matrix function T : Ω → L(H ⊕ H ⊕ H, H ⊕ H ⊕ H) by
Then, S is after D-extension with respect to structure (

Products of Operator Functions
Assume that for some n ∈ N the operator M : Ω → B(H n , H 0 ) can be written as The following lemma is a straightforward generalization of a result in [11].
Proof. For n = 2 the equivalence result is used in the proof of [11, Theorem 4.1] and the claims in the lemma follows by applying that equivalence iteratively.
Then M is after I H -extension, with respect to the structure (3.1), equivalent to T :

The operator matrix functions
Proof. The claims follow by combining the extension in Lemma 3.5 with Lemma 2.4 for the case E(λ) = 0, F (λ) = 0. This derivation is similar to the proof of Theorem 3.4.

Operator Polynomials
Let l ∈ {0, . . . , d} and consider the operator polynomial P : C → L(H), where P i ∈ B(H) for i = l. A linear equivalence is for l = 0 in principal given by [11, p. 112]. Only bounded operator coefficients are considered in that paper but the operator matrix functions E and F in the equivalence relation (2.1) are independent of P 0 . Hence they remain bounded also when P 0 is unbounded. However, the method in [11] can not be used directly if P i is unbounded for some i > 0. The following example illustrates the problem for a quadratic polynomial.
Example 3.8. Consider the operator polynomial P : C → L(H) defined as where A ∈ L(H) is an unbounded operator and B ∈ B(H). Then the method in [11] is not applicable to find an equivalent linear problem after extension as E(λ) and E(λ) −1 would be unbounded for all λ as can be seen below: However for all λ = 0, an equivalent spectral problem is S(λ) : . By extending S(λ) by −λI H an equivalent problem is given by Lemma 3.2 as and as a consequence Using this method, the obtained T has the same entries as the operator given in [11, p. 112], but the functions E(λ), F (λ) are bounded for λ = 0. Inspired by the previous example, we show how an equivalence can be found independent of which operator P i in Lemma 3.9 that is unbounded. Note that Lemma 3.9 is the standard companion block linearization for operator polynomials formulated as an equivalence after extension.
Further, define the operator matrix function W : Ω → L(H max(d−1,l) ) as Then, the following equivalence results hold: The operator matrix functions in the equivalence relation (2.1) are for λ ∈ Ω defined in the following steps: For l < d, define the operator matrix Vol. 89 (2017) On Equivalence of Operator Matrix Functions 475 Then, for all λ ∈ Ω the operator matrix functions E and F in the equivalence relation (2.1) are given by Proof. For l = 0, the result follows in principle from [11, p. 112]. Hence, we show the claim for l > 0 and Ω = C \ {0}. Define for all λ ∈ Ω the operator function S by Assume l < d, then apart from the sum Since, the following identity holds, By multiplying the first column in S(λ) ⊕ W (λ) with λ l the same result is obtained for P (λ). The operators E(λ), F (λ) are obtained by multiplying the corresponding operator matrix functions for the different equivalences.
Since T − λ can be written in the form where A is invertible. If A is bounded, P (λ) is equivalent to T − λ, T = −A −1 B but this equivalence do not hold if A is unbounded. However, these operator functions are equivalent on C \{0} after operator function extension as can be seen from Lemma 3.9 where the lemma for λ ∈ C \ {0} gives that Then, with respect to (3.1), the following equivalence results hold: The operator matrix functions in the equivalence relation (2.1) are for λ ∈ Ω defined in the following steps: If l < d, define the operator matrix function E α : Ω → L(H d−l , H) as The operator matrices E : Ω → B (H max(d,l+1) , H) and F : Ω → B ( H, H max(d,l+1) ) are then defined as Proof. Similar to the proof of Theorem 3.4, where Lemma 3.9 with (3.5) is used in Lemma 2.4. Note that D(X(λ)). Remark 3.12. Theorem 3.11 requires Q to be an operator polynomial. For a general Q an equivalence is obtained by using the equivalence given in Lemma 3.9 together with Lemma 2.4 with E := 0 and F := 0.

Linearization of Classes of Operator Matrix Functions
In Sect. 3 we considered three types of operator functions. One vital property differs between operator functions of the forms (3.2)  3) are operator polynomials, Lemma 3.2 respective Lemma 3.5 can be used to find an equivalence after operator function extension to an operator matrix polynomial. Hence, if the entries in a n × n operator matrix function are either multiplications of polynomials or Schur complements, then Theorem 3.4 and Theorem 3.7 can be used iteratively to find an equivalence to a operator matrix polynomial. An example of this form is considered in Sect. 4.3.  where

Linearization of Operator Matrix Polynomials
There are different ways to formulate (4.1) that highlight different methods to linearize the operator matrix polynomial. By using the notation: P In the formulation (4.2), the problem is written as a single operator function, which makes it possible to utilize Lemma 3.9, provided certain conditions hold. This is the most commonly used formulation, see e.g., [3]. For the original formulation (4.1), Theorem 3.11 can be applied iteratively for each Then the following results hold: is defined on its natural domain.
In the case L = ∅ the operator matrix functions in the equivalence relation (2.1) with respect to the structure (3.1) are defined in the following 480 C. Engström, A. Torshage IEOT

steps: Let the operator matrix functions E
For i = j define the operators matrices:

Then the operator matrices E(λ) and F(λ) in the equivalence relation
Proof. The claims follows from applying Theorem 3.11 to each column in (4.1). However, for columns 2, . . . , n reordering of the diagonal blocks as in (2.3) is needed to be able to apply Theorem 3.11 directly. is that then E(λ) and F(λ) depend on the order of which Theorem 3.11 is applied to the columns and are very complicated albeit possible to determine.
Remark 4.3. For operator polynomials it is common to consider equivalence after extension to a non-monic linear operator pencil, T −λS, [11]. In Theorem 4.1 the condition that P i,i is invertible for i = 1, . . . , n can be dropped if the matrix block in the equivalence is non-monic. However, the reduction of a non-monic pencil to an operator is as pointed out by Kato [12, VII, Section 6.1] non-trivial; see also Example 3.10.
There are both advantages and disadvantages of using Theorem 4.1 instead of Lemma 3.9 for operator matrix polynomials. One advantage is that P d does not have to be invertible. Furthermore, for unbounded operators functions Theorem 4.1 can handle more cases since it allows l i = l j while in Lemma 3.9, P l is unbounded for at most one l ∈ {0, . . . , d}. However, a disadvantage of this method is that the highest degree in each column has to be in the diagonal. Importantly, if both methods are applicable for P, then the obtained linearization using Theorem 4.1 and Lemma 3.9 is the same up to ordering of the spaces. Even if the conditions on P in Lemma 3.9 and/or Theorem 4.1 are not satisfied an equivalent operator matrix function P that satisfies these conditions can in many cases still be found. For example, Lemma 3.9 cannot be applied if the highest degree in the columns, d i , are not the same. However, for λ ∈ Ω \{0} an equivalent operator matrix function is obtained as where in P, the highest degree is the same in each column, unless one column is identically 0. However, the coefficient to the highest order, P d , might still be non-invertible and the boundedness condition might not be satisfied. Even if all conditions are satisfied the method increases the size of the linearization and introduces false solutions at 0. This is connected to the column reduction concept for matrix polynomials discussed for example in [20]. Due to these common problems that restrict use of Lemma 3.9 and the problems that can occur when trying to find a suitable equivalent problem, we prefer to use the results in Theorem 4.1. Therefore we develop a method that for a given operator matrix polynomial P provides an equivalent operator matrix polynomial P for which the conditions in Theorem 4.1 are satisfied.

Column Reduction of Operator Matrix Polynomials
Theorem 4.1 is only applicable when the diagonal entries in (4.1) are of strictly higher degree than the degrees of the rest of the entries in the same column. The aim of this subsection is to find for given operator matrix polynomial P a sequence of transformations that yields an equivalent operator matrix polynomial, where the diagonal entries have the highest degrees.

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C. Engström, A. Torshage IEOT One type of column reduction algorithms of polynomial matrices was considered in [20], but the column reduction algorithms presented in this section are different also in the finite dimensional case. Naturally, new challenges emerge in the infinite dimensional case and when some of the operators are unbounded. This can be seen in the following example, which also illustrates that it is not necessary to have an equivalence in each step.
Example 4.4. Consider the operator matrix function P : on its natural domain. P does not have the highest degrees in the diagonal entries. However, under the assumptions stated at the end of the example, an equivalent operator matrix polynomial can be found, where the highest degrees are on the diagonal. In the following, we will apply particular transformations that for the general case are defined in (4.4). Let K 1 denote the operator matrix The operator matrix function K 1 P is then which for the first two columns has the highest degree in the diagonal but not in the last column. Let K 3 denote the operator matrix function defined by Then (4.3) Hence, for K 3 K 1 P the third column has the highest degree in the diagonal. However, in the first column the entry in the diagonal is not of strictly higher degree than the rest of the column. We will therefore apply the operator matrix Let E : Define P : The operator matrix polynomial P has the highest degrees in the diagonal. Furthermore, since E(λ) is bounded and invertible for λ ∈ C it follows that P and P are equivalent on C.
Example 4.4 indicates that in the general case it is not feasible to obtain a closed formula for the final equivalent operator matrix polynomial. However, algorithms that follow the steps in Example 4.4 will below be developed for bounded operator matrix polynomials. These algorithms also work for classes of operator matrix functions with unbounded entries, as in Example 4.4, and it is in each case possible to check if one of the algorithms is applicable.
Let P denote the operator matrix polynomial (4.1) and assume that for i = j there exists operator polynomials K j,i (P) and R j,i (P) such that P j,i = K j,i (P)P i,i + R j,i (P), where deg R j,i (P) < deg P i,i (P). A sufficient condition for the existence of these operators is that P The dependence on P : C → B(H) is written out explicitly since we want to use K j,i (P) : C → B(H i , H j ) in the algorithms. Define K j,i (P) : C → B(H) as (4.4) Multiplying an operator matrix polynomial P from the left with K j,i (P) will be called reduction of the i-th column in the j-th row. Additionally a column in P is said to be reduced if the highest degree is in the diagonal of P in that column. When we in the algorithms presented below reduce the (i, j)-entry in P the condition that P j,i = K j,i (P)P i,i + R j,i (P) has a solution with deg R j,i (P) < deg P i,i (P) is not stated explicitly. Moreover, the notation K l:k,i (P) := K l,i (P) . . . K k,i (P) is used and it is clear that K j,i (P) 484 C. Engström, A. Torshage IEOT commutes so K l:k,i (P) is independent of the ordering in the multiplication. For convenience, the notation K i (P) := K 1:n,i (P) is used. For example, the first column in the operator function P defined by is reduced. The entries in P satisfy the conditions deg P 1,1 > deg R j,1 (P) and P j,i := P j,i − K j,1 (P)P 1,i . With the notation above the operator functions defined in Example 4.4 reads E := (K 1 • K 3 • K 1 )(P) and P := (K 1 • K 3 • K 1 )(P)P.  Define the functions and Lemma 4.6. The following properties hold for (4.7) : ii) f 0 is non-decreasing in the first and second argument.
Proof. i) Follows from the inequalities f (0, w, z) ≥ 0 and f (x, y, z) ≤ max(x, y + z). ii.) The function f (x, y, z) is non-decreasing in x and y, which implies the same properties for f 0 .
Proof. The result holds trivially for k = 1 and the proof for k > 1 is by induction. In the inductive step we show that P k = E k P and Δ(P k ) j,i < Δ(P k ) i,i for all j ∈ {1, . . . , n}, i ∈ {1, . . . , k − 1}, and j = i. Assume that induction hypothesis holds for k ≥ 1. By applying step 2 it follows that P k = E k P. Further since Δ(J k,i P k ) k,k ≥ Δ(J k,i P k ) l,k , the condition Δ(J k,i P k ) j,i < 0 for j > k and i ≤ k implies the condition Δ(P k ) j,i < 0 for j > k and i ≤ k. After step 3 we have P k = E k PJ 1,k and the inequality Δ( P k ) j,i < Δ( P k ) i,i holds for all j ∈ {1, . . . , n} and i ∈ {2, . . . , k}, since the k-th column is swapped with column one. The existence of J in step 4 is obvious and from the definitions P k = E k PJ 1,k J −1 and Δ( P k ) j,i < Δ( P k ) i,i for all j ∈ {1, . . . , n} and i ∈ {2, . . . , k}. By construction P k satisfies the assumptions of Lemma 4.8. This lemma then implies that P k = E k PJ 1,k J −1 and Δ( P k ) j,i < Δ( P k ) i,i for all j ∈ {1, . . . , n} and i ∈ {1, . . . , k}.
Hence, P k satisfies the desired condition for P k+1 , but the equivalence is P k = E k PJ 1,k J −1 .
Step 6 finds an equivalence of the desired type, P k+1 = E k+1 P and since J 1,k J −1 is a permutation operator matrix of first k rows the condition Δ( P k ) j,i < Δ( P k ) i,i for all j ∈ {1, . . . , n}, i ∈ {1, . . . , k} and i = j implies the same conditions for P k+1 . Hence, the result follows by induction.  P(λ) has the form assumed in Theorem 4.1, but the highest order in the (2, 2)-th entry, CB, might be degenerate for all operators C and B regardless if D is invertible or not.
By combining the results in Theorems 3.4, 3.7, 4.1, and Proposition 4.10 (or Proposition 4.9) we obtain a method of linearizing a class of operator matrix functions. This class consists of operator matrices where, each entry is a product and/or Schur complement of polynomials and the method extends the applicability of linearization to a larger class compared with a method based on the results in Sect. 3 alone. An illustrative example is presented in the following subsection.
In each step the operator matrix function is defined on its natural domain. Consider the operator matrix function S : Ω → L(H ⊕ H), Qλ .
This function can be linearized by the following steps: Theorem 3.7 states that after I H -extension S is equivalent to S : Ω → L(H 2 ⊕ H),