Quasihyponormal operators and the continuity of the approximate point spectrum

Abstract

Let $H$ be a separable Hilbert space. We write $\sigma (A)$ for the spectrum of $AınB(H)$, ${\sigma }_a(A)$ and ${\sigma }_{ea}(A)$ for the approximate point and the essential approximate point spectrum of $A$. Operator $AınB(H)$ is quasihyponormal if $\Vert A^{\ast }Ax\Vert \le \Vert A^{2}x\Vert$ for all $xınH$.In this paper we show that the approximate point spectrum ${\sigma }_a$ and the essential approximate point spectrum ${\sigma }_{ea}$ are continuous in the set of all quasihyponormal operators.