Invariant subspaces associated with class $p$-$wA(s,t)$ operators

Abstract

In this paper, we establish that when $\T\in\bh$ belongs to a class denoted as $p$-$wA(s,t)$ with $0<p\leq 1$ and $0<s,t, s+t\leq 1$, the quasi-nilpotent component $H_0(\T)$ of $\T$ is defined as follows:
$$\n(\T^p)=\{x\in\h:r_{\T}(x)=0\}=\bigcap_{\lambda\neq0}(\T-\lambda)^p\h.$$
\indent This characterization holds for sufficiently large integer values of $p$, where $r_{\T}(x)=\lim_{n\rightarrow \infty}\sup\norm{\T^nx}^{\frac{1}{n}}$.
Furthermore, when the spectrum $\s(\T)$ is finite and $\T$ belongs to the class $p$-$wA(s,t)$, we demonstrate that $\T$ is an algebraic operator. Moreover, in the case where $\T\in\bh$ is part of the class $p$-$wA(s,t)$ and possesses the decomposition property $(\delta)$, there exists a non-trivial invariant closed linear subspace of $\T$. Additionally, we uncover that an operator exhibiting such a diverse spectrum also possesses a nontrivial invariant subspace. The exploration of the existence of invariant and hyperinvariant subspaces is further elaborated upon in this study.