Consistent in Invertibility Operators and Svep

Dragan S. Djordjević, Bhagwati P. Duggal, Robin E. Harte

If $S(\Cal X)\subset B(\Cal X)$, where $B(\Cal X)$ denotes the algebra of operators on a Banach space $\Cal X$, then $A\in B(\Cal X)$ is $S(\Cal X)$ consistent if $AB\in S(\Cal X)\Leftrightarrow BA\in S(\Cal X)$ for every $B\in B(\Cal X)$. SVEP is a powerful tool in determining the $S(\Cal X)$ consistency of operators $A$ for various choices of the subset $S(\Cal X)$