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 H₀ (T) of T is defined as follows: (Tᵖ) = \x H: rT (x) = 0\ = ₀ (T -) ᵖ H. . This characterization holds for sufficiently large integer values of p, where rT (x) = ₍ \|Tⁿ x\|^1{n}. Furthermore, when the spectrum (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.
M. H. M. Rashid (Wed,) studied this question.