TY - JOUR
AU - Mohammad M.H Rashid
PY - 2020/06/25
Y2 - 2020/05/28
TI - Property $(w)$ of upper triangular operator Matrices
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 51
IS - 2
SE - Papers
DO - 10.5556/j.tkjm.51.2020.2256
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/2256
AB - 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.
ER -