Property $(w)$ of upper triangular operator Matrices

Authors

  • Mohammad M.H Rashid University

DOI:

https://doi.org/10.5556/j.tkjm.51.2020.2256

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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V. Rako~cevi'c, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl. textbf{10}(1986), 915--919.

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

V. Rako~cevi'c, Semi-Browder operators and perturbations, Studia Math. textbf{122}(1997), 131--137.

A.E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163 (1966) 18–-49.

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons (1980).

P. Aiena, Fredholm and local spectral theory with applications to multipliers, Kluwer, 2004.

P. Aiena, M.L. Colasantce and M. Gonzealez, Operators which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002) 2701--2710

P. Aiena and P. Pe~na, Variations on Weyl’s theorem, J. Math. Anal. Appl. textbf{324}~(1)(2006), 566–-579.

P. Aiena, Maria T. Biondi, F. Villafa~ne, Property (w) and perturbations III, J. Math. Anal. Appl. {bf353} (2009), 205–-214.

P. Aiena, C. Carpintero, E. Rosas, Some characterization of operators satisfying a-Browder theorem, J. Math. Anal. Appl. 311 (2005) 530–-544.

X. Cao and B. Meng, Essential approximate point spectra and Weyl’s theorem for operator matrices, J. Math. Anal. Appl. 304 (2005), 759--771.

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

L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J textbf{13}(1966), 285--288.

D.S. Djordjevi'c, Operators obeying a-Weyl’s theorem, Publ. Math. Debrecen textbf{55} (3) (1999), 283–-298.

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.

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.

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

B. P. Duggal, Hereditarily polaroid operators, SVEP and Weyl’s theorem, J. Math. Anal. Appl. textbf{340}(2008), 366--373.

B.P. Duggal, Upper triangular operator matrices, SVEP and Browder, Weyl Theorems, Integral Equations and Operator Theory 63 (2009), 17–-28.

B. P. Duggal, Browder and Weyl spectra of upper triangular operator matrices, Filomat 24:2 (2010), 111--130.

R. Harte, W.Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. textbf{349} (1997), 2115-–2124.

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

M. Houimdi and H. Zguitti, Proporties spectrales locales d'une matrice carree des operateurs, Acta Math. Vietnam. 25 (2000), 137–-144.

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

W. Y. Lee, Weyl’s theorem for operator matrices, Integr. Equat. Op. Th. 32 (1998), 319--331

V. Rako~cevi'c, Operators obeying a-Weyl's theorem, Rev. Roumaine Math. Pures Appl. textbf{10}(1986), 915--919.

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

V. Rako~cevi'c, Semi-Browder operators and perturbations, Studia Math. textbf{122}(1997), 131--137.

A.E. Taylor, Theorems on ascent, descent, nullity and defect of linear operators, Math. Ann. 163 (1966) 18–-49.

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons (1980).

Additional Files

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

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Section

Papers