287 © 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License Perturbation of Weyl’s Theorems for Unbounded Upper Triangular Operator Matrices Dalia S. Ali 1* and Buthainah A.A. Ahmed 2 1,2 Mathematics Department, College of Science, University of Baghdad, Baghdad, Iraq *Corresponding Author. Received: 8 May 2023 Accepted: 11 October 2023 Published: 20 October 2025 doi.org/10.30526/38.4.3473 Abstract Let , - be an upper triangular operator matrix which is unbounded and defined on , where is infinite dimensional Hilbert space. This paper is concerned with new spectral properties which defined to other bounded operators. Some sufficient and necessary conditions are given in which these properties are equivalent. We further investigate the relations among Weyl’s type theorems and Brodwe’s theorems for this type of operator under some conditions. As an application the paper define the plate pending problem equation with henge end, fixed end and free end, after transform it to Hamitonian matrix then calculate the spectrum sets for this matrix which leads to if A has eigenvalues of finite multiplicity, so is M. Inaddition if has finite ascent this implies that the Hamiltonian operator M has finite ascent. Keywords: Browder’s Spectrum, Spectral Properties, Upper Triangular Operator Matrices,Weyl’s Spectrum, Weyl’s Theorems. 1. Introduction The conception of unbounded operator delivers a non-figurative background for allocating with differential operators, unbounded perceptible in quantum mechanics, and other circumstances. The Weyl’s Theorem for bounded hermitian operators was established by Weyl (1). Weyl’s Theorem has since been expanded to encompass the class of bounded normal, hyponormal, and Toeplitz operators (2) as well as a number of other non-normal categories of bounded operators. The familiar Weyl’s theorem is generalized in such a way. Furthermore, he established this modified version of the traditional Weyl’s theorem for limited hyponormal operators in (3). The works in this direction recently been expanded to include the classes of unbounded posinormal operators and unbounded hyponormal operators (4). For unbounded operators on different spaces such as the space of Banach with non-empty resolvent, the authors introduced the B-Fredholm theory in (5, 15). Weyl’s Theorem for the category of paranormal operators on Banach spaces was established by Ramanujan(6), and it was further developed by Aiena and Guillen to include the investigation of Weyl’s. Theorem for the perturbation of paranormal operators by algebraic operators and for unbounded compact operators defined on a Banach space are investigated by the authors (7,16,19) , including those by Browder and Weyl. The theory is demonstrated in the final section using https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:dalia.sami1103a@sc.uobaghdad.edu.iq https://orcid.org/0000-0003-4147-8134 mailto:buthaina.a@sc.uobaghdad.edu.iq IHJPAS. 2025,38(4) 288 examples involving isometrics, analytically Toeplitz operators, semi-shift operators, and weighted right shifts. Both generalized Weyl’s theorem and generalized Browder’s theorem are susceptible to failure for matrices with two by two operators. In this study, we also investigate the survival of generalized Weyl’s, generalized Browder’s, generalized a- Weyl’s, and generalized a-theorems Browder’s for 2D- upper triangular operator matrices on the Banach space (8, 20, 21). Operator matrices are important to determine the solvability and stability of the underlying systems and are found in various areas of pure and applied mathematics. If is a bounded linear operator on a Hilbert space , one always has the following block representation. ( ) In addition, if , then is an upper triangular operator matrix. There are many publications looking at the spectral properties of upper triangular operator matrices. It's worth mentioning that some authors estimate the set ( ( ) ( )) ( ) and obtain some sufficient conditions of ( ) ( ) ( ) where the upper triangular operator matrix is acting in a Banach space, and * , +. Block operator matrices play a significant role in coupled systems of partial differential equations of mixed order. The study of upper triangular operator matrices and related topics is one of the hottest areas in operator theory. A number of mathematicians have studied upper triangular operator matrices in the past. 2. Preliminaries In this section, we recall the following concepts, which are used later. All through this work, denotes to infinite dimensional complex Hilbert space, ( ) is the set of all closed linear operators defined on . For an operator ( ), we define ( ) as the kernel of , while ( ) represents the domain, and ( ) denotes the range of . The upper semi Fredholm operator is define if ( ) is closed and ( ) ( ) is finite while we say that is lower semi Fredholm operator if ( ) ( ) is finite. A Fredholm operator is upper and lower semi Fredholm operator. ( ) * ( ) + ( ) * ( ) +. The of is defined as ( ) ( ) ( ) An operator ( ) which is Fredholm operator of index is defined as Weyl operator, while ( ) * is not weyl} is used to define the Weyl spectrum of . In addition we can assign the following notations: ( ) * ( ) ( ), ( ) } ( ) * ( ) ( ), ( ) + In (4), Berkani generalized the concept of Fredholm operators to B-Fredholm operators as follows ( ) { ( ) ( ) ( ) ( )} The degree of stable iteration of is denoted by ( ) and defined by ( ) ( ) and ( ) when ( ) . Furthermore, for ( ) the operator is upper and lower semi operator, where is (resp., ) semi operator if IHJPAS. 2025,38(4) 289 ( ) * ( ) ( )+ is finite and ( ) closed and (resp., * ( ) ( )+ is finite), and the index of is ( ) * ( ) ( )+ * ( ) ( )+ We call ( ) as if it’s - operator with and ( ) is used to symbolize the - spectrum of and defined by ( ) * is not -weyl + Moreover, the ascent ( ) and descent ( ) for ( ) are defined as: ( ) { . / . /} ( ) { . / . /} An operator ( ) is called Browder if it’s both and semi Browder, where ( ) is semi- Browder if ( ) with is semi- Fredholm and it is semi-Browder if ( ) with is semi – Fredholm. Now, we can define the following spectrum for an operator as: ( ) * not upper semi Fredholm+, ( ) * not lower semi Fredholm+, ( ) * not Fredholm+, ( ) * ( )+, ( ) * not upper semi-Browder +, ( ) * not lower Semi-Browder+ and ( ) * not Browder +, respectively. Evidently ( ) ( ) ( ) ( ) ( ) where ( ) denotes the set of accumulation points of the spectrum ( ) of . approximate point spectrum of . Weyl claims that the Wey’l spectrum of a Hermition operator contains exactly all of the points in the spectrum of with the exception of those points, which are isolate eigenvalues of restricted pluralism, in (9), where he proved the Weyl’s theorems for bounded hermition operators. Weyl’s theorem has now been extended to other types of bounded operators (10). Recall that one says that obeys Weyl's theorem if ( ) ( ) ( ) where ( ) is the set of isolated points of ( ) which are eigenvalues of finite multiplicity, and that one says that obeys Browder's theorem if ( ) ( ). We say that obeys a-Weyl's theorem if ( ) ( ) ( ) where ( )is the set of isolated points of ( ) which are eigenvalues of finite multiplicity, and that obeys a-Browder's theorem if ( ) ( ). Let be an infinite dimensional Hilbert space. Hamiltonian operator can be defined as densely closed operator matrix ( ) ( ( ) ( )) ( ( ) ( )) where is a densely defined closed operator, and are self adjoint operators.(see (4)). For the proof of the main results in the next section, we need the following auxiliary lemmas. 2.1. Lemma 1. is upper semi- B-Fredholm and ( ) if and only if is left Fredholm. 2. is lower semi- B-Fredholm and ( ) if and only if is right Fredholm. Proof. The proof of this lemma is similar to the proof in bounded case. IHJPAS. 2025,38(4) 290 2.2. Lemma (see (13)) Let ( ) ( ) ( ) be a closed operator matrix such that are closed operators with dense domains and is a closable operator. Then (1) If and are right Fredholm, then is right Fredholm. (2) If and are left Fredholm, then is left Fredholm. (3) If (resp., D) and are Fredholm, then (resp., ) is Fredholm. (4) If A (resp., D) and are Weyl, then D (resp., A) is Weyl. 2.3. Lemma (see (14)) Suppose that either ( ) or ( ) is finite, and that ( ) is finite then ( ) ( ). 2.4. Lemma (see (14)) Suppose that either ( ) or ( ) is finite, and that ( ) is finite then ( ) ( ) ( ). In particular, ( ) ( ) if ( ) . 2.5. Lemma (see (14)) a) If ( ) and ( ) are finite, then ( ) ( ). If also ( ) , then ( ) ( ). b) Suppose that ( ) is finite and that ( ) ( ) . Then ( ) ( ). c) Suppose that ( ) , that ( ) is finite, and that ( ) ( ) . Then ( ) ( ). 2.6. Lemma (see (11)) If is linear operator on a vector space then the following hold: 1. If ( ) then ( ) ( ) 2. If ( ) then ( ) ( ) 3. Results In this part of paper, we define some spectral properties for unbounded upper triangular operator matrices, these properties are defined for operators in bounded case (see (4), (5), (12) and (13)). Furthermore, we effort some necessary and sufficient conditions to obtain the equivalence among them and among the Weyl type theorems such as Weyl’s, a-Weyl’s, Browder’s and a-Browder’s. Before we proceed, we need to define the following spectrums: ( ) { ( ) ( ) }; ( ) { ( )}; ( ) { ( ) ( ) ( )} for an operator ( ) and ( ) * ( ) is not closed +. 3.1. Theorem If ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) with ( ) ( ) then obeys Weyl’s theorem if and only if obeys -Weyl’s theorem. Proof. Since ( ) ( ) this would imply that ( ) ( ).To prove the equivalence, it sufficient to show that ( ) ( ). Let ( ), to prove ( ) i.e., is weyl operator. Since ( ) ( ) ( ) ( ) these conditions imply that and are Fredholm operators, by Lemma (1) and (2), we get is Fredholm operator. Clearly ( ) , ( ) Since ( ) ( ) and ( ) ( ) , thus ( ) ( ) ( ) , but ( ) then ( ) which leads to ( ). The proof is completed. 3.2. Theorem obeys a-Weyl's theorem if and only if obeys a-Browder's theorem, and ( ) ( ) ( ) ( ) IHJPAS. 2025,38(4) 291 Proof. The proof is similar to the proof in bounded case (see (16)). 3.3. Theorem M obeys Weyl's theorem if and only if obeys Browder's theorem, and ( ) ( ) Proof. The proof is similar to the proof in bounded case (see (16)). 3.4. Definition (6) For ( ) we say obeys 1. ( ) if ( ) ( ) ( ) 2. Property ( ) if ( ) ( ) ( ) 3. Property (b) if ( ) ( ) ( ) ( ) 4. Property ( ) if ( ) ( ) ( ) 3.5. Definition. For ( ), we say obeys 1. ( ) if ( ) ( ) ( ) 2. Property ( ) if ( ) ( ) ( ) 3. Property ( ) if ( ) ( ) ( ) 3.6. Theorem Let obeys property (gw) with ( ) ( ) and ( ) ( ) then obeys (sz) property. Proof. Let ( ) ( ), then ( ) and ( ) , by lemma 2.1.(1), we have is upper Semi B-Fredholm with ( ) , since by assumption ( ) ( ), then ( ( ), but obeys (gw) property, thus ( ). For the reverse inclusion, let ( ), then ( ), by assumption: ( ) ( ) and ( ) are right Fredholm with ( )+ ( ) , then by Lemma ( ), we have ( ) is right Fredholm with ( ) ( ) ( ) , then ( ) thus, ( ) ( ). Then obeys Property (sz). 3.7. Theorem Let M obeys property (gb) with ( ) ( ) and ( ) ( ) , then obeys property (asz). Proof. The proof of this theorem is similar to the Proof of theorem 3.6. 3.8. Theorem If ( ) ( ) and ( ) , then obeys (w) property with ( ) ( ) and ( ) ( ) if and only if obeys property ( ) with ( ) ( ) Proof. To prove ( ) obeys property ( ) with ( ) ( ) we need to proof ( ) ( ) ( ) ( ). Let ( ) ( ), then ( ) ( ) and ( ) is upper semi Fredholm with ( ) , to prove ( ), i.e., ( ) is Browder operator. Since by assumption ( ) ( ) and ( ) ( ) then ( ) and ( ) , by Lemma 2.4. we get ( ) ( ) , thus ( ) then ( ) ( ), thus ( ) ( ) ( ) ( ) but obeys property (w) , i.e., ( ) then ( ) ( ) . Now, Let ( ) ( ), then ( ) ( ) and is Browder i.e., ( ) is Fredholm operator with finite ascent and finite descent, then by definition of Fredholm operator: is upper and lower semi Fredholm operator. Since ( ) and ( ) , then by lemma 2.3 and lemma 2.4 we get ( ) ( ) thus is belong to ( ), thus ( ), from all of that we get ( ) ( ) then ( ) obeys property (b). IHJPAS. 2025,38(4) 292 To proof the reverse direction, i.e., to proof M obeys property (w) we need to show that ( ) ( ) ( ) let ( ) ( ), since obeys (w) property then ( ) ( ) ( ) * ( ) ( ) +, then iso ( ). Since is upper semi Fredholm operator with ( ) and ( ) and ( ) with ( ) . then ( ) ( ) , also ( ) ( ) must be larger than zero, if ( ) ( ) then is one-one mapping of ( ) onto all of . The inverse ( ) is then closed and hence bounded, thus ( ), which is Contradiction. Hence ( ) , then ( ). Now, assume ( ), To prove ( ) ( ) Since ( ) then iso ( ) which imply ( ) ( ). To prove is belong to ( ) Since obeys (b) property with ( ) ( ) and ( ), then ( ) and ( ) is closed with is Browder operator i.e., ( ) and ( ) ( ) and ( ) ( ) . The proof is completed. 3.9. Theorem If is upper triangular unbounded operator matrix with ( ) ( ) and ( ) ( ) , ( ) ( ) ( ) ( ) , then obeys property (am). Proof. Let ( ) ( ), to prove ( ) i.e., to prove ( ) * ( ) ( ) +. Since ( ) ( ) ( ) ( ( ) ( )) ( ) ( ) ( ), since ( ) then we get ( ) ( ) thus ( ). Let ( ), to prove ( ) ( ) Since ( ), then ( ), it remains to prove ( ), since ( ) ( ) then ( ) . From lemma 4 (see (9)) and ( ) ( ) we see that ( ) ( ) , and since ( ) ( ) ( ) ( ) , we have ( ) and ( ) Hence ( ) then by lemma ( ), we get ( ) ( ) , thus ( ). Then obeys (am) property. 3.10. Theorem If obeys Weyl’s theorem and ( ) with ( ) ( ) and ( ) ( ) then obeys (am) property. Proof. By using the same steps in theorem 3.9 one can show ( ) ( ) . It remains to proof ( ) ( ) ( ).Let ( ) then ( ) also ( ) but obeys Weyl's theorem then ( ) ( ) ( ) i.e., is Weyl theorem. By assumptions we have ( ) ( ) and ( ) ( ) these would imply that ( ) and ( ) are finite then ( ) is Browder operator. Then the proof is completed. 3.11. Example In this example, we tried to apply the results in (16) and (17) and some results in this paper to the Hamiltonian operator matrix by applying the plate bending problem. Assume the plate bending problem ( ) With from 0 to 1. For the y-direction: at we have (Hinge end) at we have =0 (Fixed end) at we have given function and (Free end) IHJPAS. 2025,38(4) 293 For the x-direction: are given function at to . the problem can be described by the following Hamiltonian system (18) ( ) ( ) ( ) and the corresponding Hamiltonian operator matrix is given by ( ) ( ) with domain is ( ) ( ) ( ) ( ) , -, and ( ) ( ) ( ) {( ) ( ) } With some simple calculation, we have ( ) ( ) ( ) ( ) ( ) , ( ) ( ) and ( ) . Then from Propositions 4.1, 4.2 and 4.3in (5), and from Propositions 10 and 11 in (10), we have ( ) ( ) ( ) where { }. now, by theorem 3.1 we found that ( ) ( ) if ( ) ( ). 4. Conclusion In this paper, other spectral properties are introduced and studied for the upper triangular operator matrices. Furthermore, Weyl’s type theorems and Browder’s theorems are also proved under certain conditions. 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