Invariant subspaces associated with class p-w A(s,t) operators

In this paper, we establish that when T ∈ B(H) belongs to a class denoted as p-wA(s, t) with 0 < p ≤ 1 and 0 < s, t, s + t ≤ 1 , the quasi-nilpotent component H_0(T) of T is defined as follows: (T^p) = \{x ∈ H : r_T(x) = 0\} = λ ≠ 0 (T - λ)^p H. . This characterization holds for sufficiently large integer values of p , where r_T(x) = limn → ∞ \|T^n x\|1/n . Furthermore, when the spectrum σ(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 ∈ B(H) is part of the class p-wA(s, t) and possesses the decomposition property (8), 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.