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

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Wurichaihu Bai ◽  
Qingmei Bai ◽  
Alatancang Chen
Keyword(s):  
Necessary Conditions ◽  
Hamiltonian Operator ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Weyl Spectrum ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Sufficient And Necessary Conditions ◽  
Upper Triangular Operator Matrix ◽  
Hamiltonian Operator Matrix

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.

scholarly journals The residual spectrum and the continuous spectrum of upper triangular operator matrices

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 65-71 ◽  
Author(s):  
Guojun Hai ◽  
Alatancang Chen
Keyword(s):  
Continuous Spectrum ◽  
Hilbert Spaces ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Upper Triangular Operator Matrix

Let H and K be separable infinite dimensional Hilbert spaces. We denote by MC the 2x2 upper triangular operator matrix acting on H ? K of the form MC = (A C/0 B ). For given operators A ? B(H) and B ? B(K), the sets C?B?(K,H) ?r(MC) and C?B?(K,H) ?c(MC) are characterized, where ?r(?) and ?c(?) denote the residual spectrum and the continuous spectrum, respectively

scholarly journals Spectra of 2 x 2 upper triangular operator matrices

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3587-3599
Author(s):  
Junjie Huang ◽  
Aichun Liu ◽  
Alatancang Chen
Keyword(s):  
Linear Operators ◽  
Operator Matrix ◽  
Operator Matrices ◽  
Bounded Linear ◽  
Spectrum Of An Operator ◽  
Residual Spectrum ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Upper Triangular Operator Matrix ◽  
Necessary And Sufficient

The spectra of the 2 x 2 upper triangular operator matrix MC = (A C 0 B ) acting on a Hilbert space H1 ? H2 are investigated. We obtain a necessary and sufficient condition of ?(MC) = ?(A)??(B) for every C ? B(H2,H1), in terms of the spectral properties of two diagonal elements A and B of MC. Also, the analogues for the point spectrum, residual spectrum and continuous spectrum are further presented. Moveover, we construct some examples illustrating our main results. In particular, it is shown that the inclusion ?r(MC) ? ?r(A) ? ?r(B) for every C ? B(H2,H1) is not correct in general. Note that ?(T) (resp. ?r(T)) denotes the spectrum (resp. residual spectrum) of an operator T, and B(H2,H1) is the set of all bounded linear operators from H2 to H1.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

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