The Bishop's property {\boldmath $(\beta)$} for class A operators

Abstract

We say that an operator $T$ on a Hilbert space $\mathcal H$ has the Bishop's property $(\beta)$ if for an arbitrary open set $\mathcal U \subset \mathbb C$ and

analytic functions $f_n: {\mathcal U} \to {\mathcal H}$ with

$\| (T-z)f_n(z) \| $ converges to $0$ uniformly on every compact subset of $\mathcal U$ as $n \to \infty$ then $\| f_n \| $ converges to $0$ uniformly on every compact subset of $\mathcal U$ as $n \to \infty$. An operator $T$ on $\mathcal H$ is called to be hyponormal if $T^*T \geq TT^*$, and $T$ is called to be class $A$ if $T^*T \leq |T^2|$.

In this papaer, we give an elementary proof of the assertion that every hyponormal operator has the Bishop's property $(\beta)$. And we show that every class $A$ operator has the Bishop's property $(\beta)$. Moreover, we also show a class $A$ operator $T$ is similar to a hyponormal operator if $T$ is invertible, and hence $T$ has the growth condition $(G_1)$.