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

Authors: Wurichaihu Bai; Qingmei Bai; Alatancang Chen
Journal: Journal of Function Spaces
 (2018)

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## 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.


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## Body

## 1. Introduction

LetK,H be the infinite dimensional separable Hilbert spaces and C(H,K)(C+(H,K)) be the set of all closed (closable) linear operators from H into K. We also write C(H,H)(C+(H,H)) as C(H)(C+(H)). Let G be a linear subspace in H. Then G¯ and G⊥ denote the closure and the orthogonal complement of G, respectively. For a (linear) operator T between Hilbert spaces, we use D(T),R(T), and N(T) to denote the domain, the range, and the kernel of T and write α(T) and β(T) for the dimensions of the kernel N(T) and the quotient space H/R(T), respectively. According to [1], an operator T∈C(H,K) with dense domain is Fredholm, which can be defined as follows. An operator T∈C(H,K) with dense domain is said to be upper semi-Fredholm (resp., lower semi-Fredholm) if α(T)<∞ (resp., β(T)<∞) and R(T) is closed. If both α(T) and β(T) are finite, then T is called Fredholm operator. We call that T is upper semi-Weyl (resp., lower semi-Weyl) if it is upper semi-Fredholm (resp., lower semi-Fredholm) with the index ind(T)=α(T)-β(T)≤0 (resp., ind(T)≥0) and T is upper semi-Browder (resp., lower semi-Browder) if it is upper semi-Fredholm (resp., lower semi-Fredholm) of finite ascent asc(T) (resp.,finite descent dsc(T)), where (1)ascT=inf⁡n∣NTn=NTn+1,dscT=inf⁡n∣RTn=RTn+1withinf⁡∅=∞.We call that T is Weyl if it is Fredholm with ind(T)=0. Then, the upper semi-Weyl spectrum and upper semi-Browder spectrum of T are, respectively, defined by (2)σSF+-T=λ∈C:T-λis  not  upper  semi-Weyl,σlbT=λ∈C:T-λis  not  upper  semi-Browder.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 studied2×2 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 (σ∗(A)∪σ∗(D))\σ∗(T) and obtained some sufficient conditions for (3)σ∗T=σ∗A∪σ∗D,where (4)T=AB0Dis a bounded operator matrix acting on Banach space and σ∗∈{σe,σw,σb}. 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 σ∗∈{σSF+-,σlb}. 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 T using the factorization (5)T=I00DIB0IA00Iwhere (6)T=AB0D:DA⊕DD⊂H⊕H→H⊕His 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 (σ∗(A)∪σ∗(D))\σ∗(T) with σ∗∈{σSF+-,σlb} are actually described, and, in addition, these results are applied to a Hamiltonian operator matrix.Definition 1 (see [11]).
A closed (linear) operator(7)H=ABC-A∗:DH⊂H⊕H→H⊕Hwith dense domain (8)DH=DA∩DC⊕DB∩DA∗,is called a Hamiltonian operator matrix, if A is a closed operator with dense domain and B,C are self-adjoint operators.Definition 2 (see [12]).
LetT and B be linear operators from H to K. We say that B is T-compact if(1)
D(T)⊂D(B)(2)
B is compact on HT
whereHT denotes D(T) endowed with the graph norm i.e., xT=x+Tx, for x∈D(T).


## 2. Some Properties of Upper Triangular Operator Matrix

Lemma 3 (see [12]).
Suppose thatT∈C(H) is a Fredholm operator and B is T-compact. Then(1)
T+B is a Fredholm operator,(2)
ind(T+B)=ind(T).Lemma 4.
Let(9)T=AB0D:DA⊕DD⊂H⊕K→H⊕Kbe closed operator matrix such that A∈C(H), D∈C(K) with dense domains and let B∈CD+(K,H)={B∈C+(K,H):D(D)⊂D(B),D∈C(K)}; then there exists some B∈CD+(K,H) such that T is upper semi-Weyl operator if and only if A is upper semi-Fredholm operator and (10)αD<∞andαA+αD≤βA+βDorβA=αD=∞ifRDisclosedβA=∞ifRDisnotclosed.Proof.
Suppose thatT is upper semi-Weyl operator for some B∈CD+(K,H). Then A is upper semi-Fredholm operator.
Ifβ(A)<∞, then A is Fredholm operator. Also since T is upper semi-Fredholm operator, we have D is upper semi-Fredholm operator. In fact, T can be written as follows:(11)T=A10B1B300B2B400D100000:DA∩NA⊥NADD∩ND⊥ND→RARA⊥RD¯RD⊥,A1:D(A)∩N(A)⊥→R(A) is a bijection, then there exists operator (12)Q=I0-A1-1B1-A1-1B30I0000I0000I:RARA⊥RD¯RD⊥→RARA⊥RD¯RD⊥such that (13)TQ=A100000B2B400D100000=A1000000000D100000+000000B2B400000000.
Sinceβ(A)<∞, then B2,B4 are compact operators; therefore D is upper semi-Fredholm operator from Lemma 3. Hence ind(T)=ind(A)+ind(D)≤0; i.e., α(A)+α(D)≤β(A)+β(D). So when T is upper semi-Weyl operator for some B∈CD+(K,H), we have A is upper semi-Fredholm operator and (14)βA=∞orDisuppersemi-FredholmoperatorandαA+αD≤βA+βD.Again (15)βA=∞orDisuppersemi-FredholmoperatorandαA+αD≤βA+βD.if and only if (16)αD<∞andαA+αD≤βA+βDorβA=αD=∞ifRDis  closedβA=∞ifRDis  not  closed
Conversely, ifA,D are upper semi-Fredholm operators and α(A)+α(D)≤β(A)+β(D), then T is upper semi-Weyl operator for every B∈CD+(K,H), from [10, Lemma 2.2].
IfA is upper semi-Fredholm operator and β(A)=∞, then there exist two infinite dimensional subspaces M,N of R(A)⊥ such that R(A)⊥=M⊕N. We define an operator (17)B=0U0:K→RAMN,where U:K→M is an unitary operator. Therefore T is upper semi-Weyl operator and α(A)=α(T). In fact, if xy∈N(T), then Ax+By=0 and Dy=0. Since Ax=-By∈R(A)∩M={0}, then Ax=0 and By=0. And we obtain y=0 by the definition of U. Therefore N(T)⊂N(A)⊕{0}. Moreover α(T)=α(A)<∞. The following proof is that R(A) is a closed set. Since R(A) is a closed set, then we have to prove that RU0D is a closed set. Let U0Dxn→y1y2y3, then Uxn→y1, Dxn→y3, and y2=0. By the definition of U and the closeness of U, we have U-1y1∈D(D) and y3=DU-1y1; i.e., U0DU-1y1=y1y2y3∈RU0D, so RU0D is a closed set. Since dim⁡N=∞, then β(T)=∞; therefore T is upper semi-Weyl operator.Lemma 5.
LetT=AB0D:D(A)⊕D(D)⊂H⊕K→H⊕K be closed operator matrix such that A∈C(H),D∈C(K) with dense domains and let B∈CD+(K,H); then(1)
ifA and D are upper semi-Weyl, then T is upper semi-Weyl(2)
ifA and D are upper semi-Browder, then T is upper semi-BrowderProof.
It is easily obtained by Lemma 2.2 of [10, 13].


## 3. Main Results

Theorem 6.
LetT=AB0D:D(A)⊕D(D)⊂H⊕K→H⊕K be closed operator matrix such that A∈C(H),D∈C(K) with dense domains and let B∈CD+(K,H); then (18)σSF+-A∪σSF+-D=σSF+-T∪σp+D∩σp+A∗-∪σp+A∩σp+D∗-∪σp∞A∗-∩σp∞D,where σp+(·)={λ∈σp(·):α(·-λI)>β(·-λI)},σp∞(·)={λ∈σp(·):α(·-λI)=∞} and σp+(·)-={λ¯∈C:λ∈σp+(·)}.Proof.
Letλ∈(σSF+-(A)∪σSF+-(D))\σSF+-(T); then T-λI is upper semi-Fredholm and ind(T-λI)≤0.
(1) Ifind(T-λI)=0, then λ∈(σp+(D)∩σp+(A∗)-)∪(σp+(A)∩σp+(D∗)-) from Theorem 3.2 of [10].
(2) Ifind(T-λI)<0, then A-λI is upper semi-Fredholm operator and (19)αA-λI+αD-λI≤βA-λI+βD-λIandαD-λI<∞orβA-λI=αD-λI=∞ifRD-λIis  closedβA-λI=∞ifRD-λIis  not  closedby Lemma 4.
Ifλ∈σSF+-(A), then ind(A-λI)>0, so λ∈σp+(A). From Lemma 4, if R(D-λI) is closed, we have D-λI is upper semi-Fredholm and ind(D-λI)<0 or β(A-λI)=α(D-λI)=∞ (but it is impossible), so λ∈σp+(D∗)-. If R(D-λI) is not closed, then β(A-λI)=∞; it is impossible. Therefore λ∈σp+(A)∩σp+(D∗)-.
Ifλ∈σSF+-(D), then D-λI is not upper semi-Fredholm or ind(D-λI)>0.
LetD-λI be not upper semi-Fredholm; then R(D-λI) is not closed or α(D-λI)=∞; moreover, β(A-λI)=∞ by Lemma 4. Thus λ∈σp∞(A∗)-∩σp∞(D).
LetD-λI be upper semi-Fredholm and ind(D-λI)>0; then ind(A-λI)<0, so λ∈σp+(D)∩σp+(A∗)-.
Conversely, from Lemma5, we have σSF+-(A)∪σSF+-(D)⊇σSF+-(T) and (σp+(D)∩σp+(A∗)-)∪(σp+(A)∩σp+(D∗)-)∪(σp∞(A∗)-∩σp∞(D))⊆σSF+-(A)∪σSF+-(D).
The proof is complete.Corollary 7.
LetT=AB0D:D(A)⊕D(D)⊂H⊕K→H⊕K be closed operator matrix such that A∈C(H),D∈C(K) with dense domains and let B∈CD+(K,H); then (20)σSF+-A∪σSF+-D=σSF+-Tif and only if (21)σp+D∩σp+A∗-⊆σSF+-Tandσp+A∩σp+D∗-⊆σSF+-T,σp∞A∗-∩σp∞D⊆σSF+-T.In particular, if σp+(D)∩σp+(A∗)-=∅ and σp+(A)∩σp+(D∗)-=∅, σp∞(A∗)-∩σp∞(D)=∅, then σSF+-(A)∪σSF+-(D)=σSF+-(T).Theorem 8.
LetT=AB0D:D(A)⊕D(D)⊂H⊕K→H⊕K be closed operator matrix such that A∈C(H),D∈C(K) with dense domains and let B∈CD+(K,H); then (22)σlbA∪σlbD=σlbT∪σp∞A∗-∩σp∞D∪σascD,where σasc(·)={λ∈C:asc(·-λI)=∞}.Proof.
Letλ∈(σlb(A)∪σlb(D))\σlb(T); then T-λI is upper semi-Fredholm and asc(T-λI)<∞; therefore, A-λI is upper semi-Fredholm and asc(A-λI)<∞, from [13]. That is, λ does not belong to σlb(A), so λ∈σlb(D); i.e., D-λI is not upper semi-Fredholm or asc(D-λI)=∞. If R(D-λI) is not closed or α(D-λI)=∞, then β(A-λI)=∞ by Lemma 4. Therefore λ∈σp∞(A∗)-∩σp∞(D). If D-λI is upper semi-Fredholm but asc(D-λI)=∞, then λ∈σasc(D). So (σlb(A)∪σlb(D))\σlb(T)⊂σp∞(A∗)-∪σasc(D).
Conversely, we haveσlb(A)∪σlb(D)⊇σlb(T) by Lemma 5, and (σp∞(A∗)-∩σp∞(D))∪σasc(D)⊆σlb(A)∪σlb(D) is easily obtained.Corollary 9.
LetT=AB0D:D(A)⊕D(D)⊂H⊕K→H⊕K be closed operator matrix such that A∈C(H),D∈C(K) with dense domains and let B∈CD+(K,H); then (23)σlbA∪σlbD=σlbTif and only if (24)σascD⊆σlbTandσp∞A∗-∩σp∞D⊆σlbT.In particular, if σasc(D)=∅ and σp∞(A∗)-∩σp∞(D)=∅, then σlb(A)∪σlb(D)=σlb(T).


## 4. Applications

In this section, we obtained some properties of Hamiltonian operator matrix.Proposition 10.
Let(25)H=AB0-A∗:DA⊕DA∗⊂H⊕H→H⊕Hbe a Hamiltonian operator matrix. Then (26)σSF+-A∪σSF+--A∗=σSF+-Hif and only if (27)σp+-A∗∩σp+A∗-⊆σSF+-Handσp+A∩σp+-A∗∗-⊆σSF+-H,σp∞A∗-∩σp∞D⊆σSF+-H.In particular, if σp+(-A∗)∩σp+(A∗)-=∅ and σp+(A)∩σp+((-A∗)∗)-=∅, σp∞(A∗)-∩σp∞(D)=∅, then σSF+-(A)∪σSF+-(-A∗)=σSF+-(H).Proposition 11.
Let(28)H=AB0-A∗:DA⊕DA∗⊂H⊕H→H⊕Hbe a Hamiltonian operator matrix. Then (29)σlbA∪σlb-A∗=σlbHif and only if (30)σasc-A∗⊆σlbHandσp∞A∗-∩σp∞D⊆σlbH.In particular, if σasc(-A∗)=∅ and σp∞(A∗)-∩σp∞(D)=∅, then σlb(A)∪σlb(-A∗)=σlb(H).Example 12.
Consider the plate bending equation in domain{(x,y):0<x<1,0<y<1}(31)D∂2∂x2+∂2∂y22ω=0,with boundary conditions (32)ωx,0=ωx,1=0,∂2ω∂x2+∂2ω∂y2=0,y=0,1.Setting (33)θ=∂ω∂x,q=D∂3ω∂x3+∂3ω∂y3,m=-D∂2ω∂x2+∂2ω∂y2,then the equation can be written as the following Hamiltonian system [14]: (34)∂∂xωθqm=0100-∂2∂y200-1D000∂2∂y200-10ωθqmand the corresponding Hamiltonian operator matrix is given by (35)H=0100-d2dy200-1D000d2dy200-10≔AB0-A∗with domain being D(A)⊕D(A∗)⊂H⊕H, where H=L2(0,1)⊕L2(0,1), A=AC[0,1], and (36)A=01-d2dy20,B=000-1D,DA=ωθ∈H:ω0=ω1=0,ω′∈A,ω′′∈H.Through a simple calculation, we know σp∞(A∗)=∅,σp+(-A∗)∩σp+(A∗)-=∅,-σp+(A)-∩σp+(A)=∅, and σasc(A∗)=∅. Then from Propositions 10 and 11, we have (37)-σ∗A∗∪σ∗A=σ∗H,where σ∗∈{σSF+-,σlb}.



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