Upper Semi-Weyl and Upper Semi-Browder Spectra of Unbounded Upper Triangular Operator Matrices

Abstract

In this paper, we study the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which upper semi-Weyl spectrum (resp. upper semi-Browder spectrum) of such operator matrix is equal to the union of the upper semi-Weyl spectra (resp. the upper semi-Browder spectra) of its diagonal entries. As an application, the corresponding spectral properties of Hamiltonian operator matrix are obtained.

1. Introduction

Let be the infinite dimensional separable Hilbert spaces and be the set of all closed (closable) linear operators from into . We also write as . Let be a linear subspace in . Then and denote the closure and the orthogonal complement of , respectively. For a (linear) operator between Hilbert spaces, we use , and to denote the domain, the range, and the kernel of and write and for the dimensions of the kernel and the quotient space , respectively. According to [1], an operator with dense domain is Fredholm, which can be defined as follows. An operator with dense domain is said to be upper semi-Fredholm (resp., lower semi-Fredholm) if (resp., ) and is closed. If both and are finite, then is called Fredholm operator. We call that is upper semi-Weyl (resp., lower semi-Weyl) if it is upper semi-Fredholm (resp., lower semi-Fredholm) with the index (resp., ) and is upper semi-Browder (resp., lower semi-Browder) if it is upper semi-Fredholm (resp., lower semi-Fredholm) of finite ascent (resp.,finite descent ), where We call that is Weyl if it is Fredholm with . Then, the upper semi-Weyl spectrum and upper semi-Browder spectrum of are, respectively, defined by

Block operator matrices play a major role in coupled systems of partial differential equations, and their spectral properties are of concerned interest. In Particular, the study of upper triangular operator matrices and related subjects is one of the hottest areas in operator theory. Recently, a number of mathematicians have studied bounded upper triangular operator matrices (see, e.g., [2–4]). In [5–9], the authors, making use of the single-valued extension property, estimated the defect sets ( and obtained some sufficient conditions for where is a bounded operator matrix acting on Banach space and . In [10], the authors extend these results to unbounded case. The main aim of this paper is to get sufficient and necessary conditions for (3) of an unbounded operator with . One of the significant differences between unbounded and bounded operator matrices arises in their domains. In general, one could not get certain spectral properties of unbounded operator matrix using the factorization where is a closed operator matrix.

Applying different method—space decomposition technique—we present some sufficient and necessary conditions for (3) in this paper. More precisely, the defect sets with are actually described, and, in addition, these results are applied to a Hamiltonian operator matrix.

Definition 1 (see [11]). A closed (linear) operator with dense domain is called a Hamiltonian operator matrix, if is a closed operator with dense domain and are self-adjoint operators.

Definition 2 (see [12]). Let and be linear operators from to . We say that is -compact if(1)(2) is compact on where denotes endowed with the graph norm i.e., , for .

2. Some Properties of Upper Triangular Operator Matrix

Lemma 3 (see [12]). Suppose that is a Fredholm operator and is -compact. Then(1) is a Fredholm operator,(2).

Lemma 4. Let be closed operator matrix such that , with dense domains and let ; then there exists some such that is upper semi-Weyl operator if and only if is upper semi-Fredholm operator and

Proof. Suppose that is upper semi-Weyl operator for some . Then is upper semi-Fredholm operator.
If , then is Fredholm operator. Also since is upper semi-Fredholm operator, we have is upper semi-Fredholm operator. In fact, can be written as follows: is a bijection, then there exists operator such that Since , then are compact operators; therefore is upper semi-Fredholm operator from Lemma 3. Hence ; i.e., . So when is upper semi-Weyl operator for some , we have is upper semi-Fredholm operator and Again if and only if Conversely, if are upper semi-Fredholm operators and , then is upper semi-Weyl operator for every , from [10, Lemma 2.2].
If is upper semi-Fredholm operator and , then there exist two infinite dimensional subspaces of such that . We define an operator where is an unitary operator. Therefore is upper semi-Weyl operator and . In fact, if , then and . Since , then and . And we obtain by the definition of . Therefore . Moreover . The following proof is that is a closed set. Since is a closed set, then we have to prove that is a closed set. Let , then , , and . By the definition of and the closeness of , we have and ; i.e., , so is a closed set. Since , then ; therefore is upper semi-Weyl operator.

Lemma 5. Let be closed operator matrix such that with dense domains and let ; then(1)if and are upper semi-Weyl, then is upper semi-Weyl(2)if and are upper semi-Browder, then is upper semi-Browder

Proof. It is easily obtained by Lemma 2.2 of [10, 13].

3. Main Results

Theorem 6. Let be closed operator matrix such that with dense domains and let ; then where and .

Proof. Let ; then is upper semi-Fredholm and .
(1) If , then from Theorem 3.2 of [10].
(2) If , then is upper semi-Fredholm operator and by Lemma 4.
If , then , so . From Lemma 4, if is closed, we have is upper semi-Fredholm and or (but it is impossible), so . If is not closed, then ; it is impossible. Therefore .
If , then is not upper semi-Fredholm or .
Let be not upper semi-Fredholm; then is not closed or ; moreover, by Lemma 4. Thus .
Let be upper semi-Fredholm and ; then , so .
Conversely, from Lemma 5, we have and .
The proof is complete.

Corollary 7. Let be closed operator matrix such that with dense domains and let ; then if and only if In particular, if and , , then .

Theorem 8. Let be closed operator matrix such that with dense domains and let ; then where .

Proof. Let ; then is upper semi-Fredholm and ; therefore, is upper semi-Fredholm and , from [13]. That is, does not belong to , so ; i.e., is not upper semi-Fredholm or . If is not closed or , then by Lemma 4. Therefore . If is upper semi-Fredholm but , then . So .
Conversely, we have by Lemma 5, and is easily obtained.

Corollary 9. Let be closed operator matrix such that with dense domains and let ; then if and only if In particular, if and , then .

4. Applications

In this section, we obtained some properties of Hamiltonian operator matrix.

Proposition 10. Let be a Hamiltonian operator matrix. Then if and only if In particular, if and , , then .

Proposition 11. Let be a Hamiltonian operator matrix. Then if and only if In particular, if and , then .

Example 12. Consider the plate bending equation in domain with boundary conditions Setting then the equation can be written as the following Hamiltonian system [14]: and the corresponding Hamiltonian operator matrix is given by with domain being , where , , and Through a simple calculation, we know , and . Then from Propositions 10 and 11, we have where .

Data Availability

No data were used to support this study.

Conflicts of Interest

No potential conflicts of interest were reported by the authors.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (grant number 11561053) (grant number 11761029), Natural Science Foundation of Inner Mongolia (grant number 2018BS01001), Research Program of Sciences at Universities of Inner Mongolia Autonomous Region (grant number NJZZ18018) (grant number NJZY18021), and Subject of Research Foundation of Inner Mongolia Normal University of China (grant number 2016ZRYB001).