scholarly journals Operator matrices and their Weyl type theorems

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3191-3204
Author(s):  
Ju An ◽  
Eungil Ko ◽  
Ji Lee
Keyword(s):  
Operator Matrix ◽  
Operator Matrices ◽  
Closed Range ◽  
Weyl Type Theorems ◽  
Drazin Spectrum

We denote the collection of the 2 x 2 operator matrices with (1,2)-entries having closed range by S. In this paper, we study the relations between the operator matrices in the class S and their component operators in terms of the Drazin spectrum and left Drazin spectrum, respectively. As some application of them, we investigate how the generalized Weyl?s theorem and the generalized a-Weyl?s theorem hold for operator matrices in S, respectively. In addition, we provide a simple example about an operator matrix in S satisfying such Weyl type theorems.

scholarly journals Local Spectral Properties Under Conjugations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Pietro Aiena ◽  
Fabio Burderi ◽  
Salvatore Triolo
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Analytic Core ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

scholarly journals Right and Left Weyl Operator Matrices in a Banach Space Setting

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Aichun Liu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Operator Matrix ◽  
Weyl Operator ◽  
Operator Matrices ◽  
Space Setting ◽  
Hamiltonian Operators ◽  
Equivalent Conditions

Let X i , Y i i = 1,2 be Banach spaces. The operator matrix of the form M C = A C 0 B acting between X 1 ⊕ X 2 and Y 1 ⊕ Y 2 is investigated. By using row and column operators, equivalent conditions are obtained for M C to be left Weyl, right Weyl, and Weyl for some C ∈ ℬ X 2 , Y 1 , respectively. Based on these results, some sufficient conditions are also presented. As applications, some discussions on Hamiltonian operators are given in the context of Hilbert spaces.

Generalized Drazin spectrum of operator matrices

2014 ◽  
Vol 29 (2) ◽  
pp. 162-170 ◽  
Author(s):  
Shi-fang Zhang ◽  
Huai-jie Zhong ◽  
Li-qiong Lin
Keyword(s):  
Operator Matrices ◽  
Drazin Spectrum

scholarly journals Drazin spectrum of operator matrices on the Banach space

2008 ◽  
Vol 429 (8-9) ◽  
pp. 2067-2075 ◽  
Author(s):  
Shifang Zhang ◽  
Huaijie Zhong ◽  
Qiaofen Jiang
Keyword(s):  
Banach Space ◽  
Operator Matrices ◽  
Drazin Spectrum

scholarly journals Local spectral property of 2 x 2 operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1845-1854
Author(s):  
Eungil Ko
Keyword(s):  
Spectral Property ◽  
Spectral Properties ◽  
Extension Property ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Single Valued Extension Property

In this paper we study the local spectral properties of 2 x 2 operator matrices. In particular, we show that every 2 x 2 operator matrix with three scalar entries has the single valued extension property. Moreover, we consider the spectral properties of such operator matrices. Finally, we show that some of such operator matrices are decomposable.

scholarly journals On the mixed-type generalized inverses of the products of two operators

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4361-4376
Author(s):  
Rufang Liu ◽  
Haiyan Zhang ◽  
Chunyuan Deng
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Operator Matrix ◽  
Closed Range ◽  
Matrix Methods ◽  
Block Operator Matrix ◽  
Inclusion Relations ◽  
Block Operator ◽  
Necessary And Sufficient

Let A, B and be closed range operators. The explicit matrix expressions for various generalized inverses are obtained by using block operator matrix methods. Some subtle relationships between the properties of sub-blocks in operator matrices A, B and their range relations are built. New necessary and sufficient conditions for the equivalent relations, inclusion relations and mixed-type generalized inverses relations are presented. Some recent mixed-type reverse-order laws results are covered and many new mixed-type generalized inverses relations are established by using this block-operator matrix technique.

Weyl type theorems for complex symmetric operator matrices

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2891-2900 ◽  
Author(s):  
Il An ◽  
Eungil Ko ◽  
Ji Lee
Keyword(s):  
Symmetric Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Complex Symmetric Operator ◽  
Weyl Type Theorems ◽  
Complex Symmetric ◽  
Necessary And Sufficient

In this paper, we study Weyl type theorems for complex symmetric operator matrices. In particular, we give a necessary and sufficient condition for complex symmetric operator matrices to satisfy a-Weyl?s theorem. Moreover, we also provide the conditions for such operator matrices to satisfy generalized a-Weyl?s theorem and generalized a-Browder?s theorem, respectively. As some applications, we give various examples of such operator matrices which satisfy Weyl type theorems.

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