scholarly journals The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Shifang Zhang ◽  
Huaijie Zhong ◽  
Long Long
Keyword(s):  
Separable Hilbert Space ◽  
Invertible Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Infinite Dimensional ◽  
Lower Semi ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

WhenA∈B(H)andB∈B(K)are given, we denote byMCthe operator acting on the infinite-dimensional separable Hilbert spaceH⊕Kof the formMC=(AC0B). In this paper, it is proved that there exists some operatorC∈B(K,H)such thatMCis upper semi-Browder if and only if there exists some left invertible operatorC∈B(K,H)such thatMCis upper semi-Browder. Moreover, a necessary and sufficient condition forMCto be upper semi-Browder for someC∈G(K,H)is given, whereG(K,H)denotes the subset of all of the invertible operators ofB(K,H).

scholarly journals The point spectrum and residual spectrum of upper triangular operator matrices

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1759-1771
Author(s):  
Xiufeng Wu ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Hilbert Spaces ◽  
Point Spectrum ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Residual Spectrum ◽  
Infinite Dimensional ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient

The point and residual spectra of an operator are, respectively, split into 1,2-point spectrum and 1,2-residual spectrum, based on the denseness and closedness of its range. Let H,K be infinite dimensional complex separable Hilbert spaces and write MX = (AX0B) ? B(H?K). For given operators A ? B(H) and B ? B(K), the sets ? X?B(K,H) ?+,i(MX)(+ = p,r;i = 1,2), are characterized. Moreover, we obtain some necessary and sufficient condition such that ?*,i(MX) = ?*,i(A) ?*,i(B) (* = p,r;i = 1,2) for every X ? B(K,H).

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.

scholarly journals Characterizations of majorization on summable sequences

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2193-2202
Author(s):  
Kosuru Raju ◽  
Subhajit Saha
Keyword(s):  
Sequence Space ◽  
Dimensional Space ◽  
Sufficient Condition ◽  
Infinite Dimensional Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Summable Sequence ◽  
Necessary And Sufficient

In this paper, we prove a necessary and sufficient condition for majorization on the summable sequence space. For this we redefine the notion of majorization on an infinite dimensional space and therein investigate properties of the majorization. We also prove the infinite dimensional Schur-Horn type and Hardy-Littlewood-P?lya type theorems.

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

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.

MULTIPLICATIVE RENORMALIZABILITY AND THE LAPLACE-BELTRAMI OPERATOR

1991 ◽  
Vol 06 (06) ◽  
pp. 955-976
Author(s):  
D. OLIVIER ◽  
G. VALENT
Keyword(s):  
Irreducible Representation ◽  
Isometry Group ◽  
Zero Mode ◽  
Beltrami Operator ◽  
Sufficient Condition ◽  
Representation Space ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Multiplicative Renormalizability

For some rank 1 non-linear σ models we prove that a necessary and sufficient condition of multiplicative renormalizability for composite fields is that they should be eigenfunctions of the coset Laplace-Beltrami operator. These eigenfunctions span the irreducible representation space of the isometry group and may be finite- or infinite-dimensional. The zero mode of the Laplace-Beltrami operator plays a particular role since it is not renormalized at all.

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.

Generalized left and right Weyl spectra of upper triangular operator matrices

2017 ◽  
Vol 32 ◽  
pp. 41-50
Author(s):  
Guojun Hai ◽  
Dragana Cvetkovic-Ilic
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Operator ◽  
Operator Matrices ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Left And Right ◽  
Necessary And Sufficient

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

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