Property $(w)$ of upper triangular operator Matrices

  • Mohammad M.H Rashid University
Keywords: Property $(w)$, Matrix theory, Weyl's theorem, Weyl spectrum

Abstract

Let $M_C=\begin{pmatrix}
A & C \\
0 & B \\
\end{pmatrix}\in\LB(\x,\y)
$ be be an upper triangulate Banach space
operator. The relationship between the spectra of $M_C$ and $M_0,$ and their
various distinguished parts, has been studied by a large number of authors in
the 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.

Author Biography

Mohammad M.H Rashid, University
Mathematics/Prof.Dr

References

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%==============================================================

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%===============================================================

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%=========================================================

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%==================================================================

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%=======================================================================

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%=====================================================================================

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%=======================================================

bibitem{aiena2004}

P. Aiena, Fredholm and local spectral theory with applications to

multipliers, Kluwer, 2004.

%==================================================================

bibitem{AiCoGo2002}

P. Aiena, M.L. Colasantce and M. Gonzealez, Operators which have

a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002) 2701--2710

%================================================================

bibitem{AiPe2006}

P. Aiena and P. Pe~na, Variations on Weyl’s theorem, J. Math.

Anal. Appl. textbf{324}~(1)(2006), 566–-579.

%==============================================================================

bibitem{AiBiVill2009}

P. Aiena, Maria T. Biondi, F. Villafa~ne, Property (w) and perturbations III, J. Math. Anal. Appl.

{bf353} (2009), 205–-214.

%=====================================================================

bibitem{AiCaRo2005}

P. Aiena, C. Carpintero, E. Rosas, Some characterization of

operators satisfying a-Browder theorem, J. Math. Anal. Appl. 311

(2005) 530–-544.

%=============================================================================================

bibitem{Cao2005}

X. Cao and B. Meng, Essential approximate point spectra and Weyl’s theorem for

operator matrices, J. Math. Anal. Appl. 304 (2005), 759--771.

%=============================================================================

bibitem{CaoGuoMeng2005}

X. Cao, M. Guo, B. Meng, Weyl's theorem for upper triangular operator matrices, Linear Alg. Appl. 402 (2005), 61--73.

%=============================================================================================

bibitem{coburn66}

L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan

Math. J textbf{13}(1966), 285--288.

%==============================================================

bibitem{Djordjevi1999}

D.S. Djordjevi'c, Operators obeying a-Weyl’s theorem, Publ. Math.

Debrecen textbf{55} (3) (1999), 283–-298.

%================================================================

bibitem{Djordjevi2003}

S. V. Djordjevi'c and Y. M. Han, A note on Weyl's theorem for operator matrices, Proc.

Amer. Math. Soc. 131 (2003), 2543--2547.

%===============================================================

bibitem{Djordjevi2008}

S. V. Djordjevi'c and H. Zguitti, Essential point spectra of

operator matrices through local spectral theory, J. Math. Anal.

Appl. 338 (2008), 285--291.

%=========================================================

bibitem{Duggal2005}

B.P. Duggal, Hereditarily normaloid operators, Extracta Math. 20 (2005), 203--217.

%============================================================

bibitem{Duggal2008}

B. P. Duggal, Hereditarily polaroid operators, SVEP and Weyl’s theorem,

J. Math. Anal. Appl. textbf{340}(2008), 366--373.

%======================================================================

bibitem{B.P.Duggal2008}

B.P. Duggal, Upper triangular operator matrices, SVEP and Browder, Weyl Theorems, Integral Equations

and Operator Theory 63 (2009), 17–-28.

%================================================================

bibitem{B.P.Duggal2010}

B. P. Duggal, Browder and Weyl spectra of upper

triangular operator matrices, Filomat 24:2 (2010), 111--130.

%===============================================================

bibitem{harte1997}

R. Harte, W.Y. Lee, Another note on Weyl’s theorem, Trans. Amer.

Math. Soc. textbf{349} (1997), 2115-–2124.

%=======================================================================

bibitem{Heuser82}

H. Heuser, Functional analysis, Marcel Dekker, New York, 1982.

%==========================================================================

bibitem{Houimdi2000}

M. Houimdi and H. Zguitti, Proporties spectrales locales d'une matrice carree des

operateurs, Acta Math. Vietnam. 25 (2000), 137–-144.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

bibitem{LaNe2000}

K.~B. Laursen, M.~M. Neumann, An introduction to local spectral theory, Oxford. Clarendon, 2000.

%==============================================================================

bibitem{Lee1998}

W. Y. Lee, Weyl’s theorem for operator matrices, Integr. Equat.

Op. Th. 32 (1998), 319--331

%=====================================================================

bibitem{rako89}

V. Rako~cevi'c, Operators obeying a-Weyl's theorem, Rev.

Roumaine Math. Pures Appl. textbf{10}(1986), 915--919.

%====================================================================

bibitem{rakoce1985}

V. Rako~cevi'c, On a class of operators, Math. Vesnik {bf 37} (1985), 423-–426.

%==================================================================

bibitem{rako1996}

V. Rako~cevi'c, Semi-Browder operators and perturbations, Studia Math.

textbf{122}(1997), 131--137.

%=======================================================================

bibitem{Taylor1966}

A.E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163

(1966) 18–-49.

%=====================================================================================

bibitem{Taylor1980}

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, John

Wiley and Sons (1980).

%=======================================================

Published
2020-06-25
How to Cite
Rashid, M. M. (2020). Property $(w)$ of upper triangular operator Matrices. Tamkang Journal of Mathematics, 51(2), 81-99. https://doi.org/10.5556/j.tkjm.51.2020.2256
Section
Papers