M. H. M. Rashid

This paper introduces and investigates the class of $k$-quasi $n$-power posinormal operators in Hilbert spaces, generalizing both posinormal and $n$-power posinormal operators. We establish fundamental properties including matrix representations in $2 \times 2$ block form, tensor product preservation ($T\otimes S$ remains in the class when $T,S$ are), and complete characterizations for weighted conditional type operators $T_{w,u} := wE(uf)$ on $L^2(\Sigma)$. Key theoretical contributions include a structural decomposition theorem for operators with non-dense range, spectral properties, invariant subspace behavior, and interactions with isometric operators. For weighted operators, we derive explicit conditions for $k$-quasi $n$-power posinormality in terms of weight functions $w,u$ and their conditional expectations. The work bridges abstract operator theory with concrete applications, particularly in conditional expectation analysis, while significantly extending posinormal operator theory. The results provide new tools for operator analysis with potential applications in spectral theory, functional calculus and mathematical physics. Concrete examples throughout the paper illustrate the theory and the framework opens new research directions in operator theory and its applications, offering both theoretical insights and practical computational tools for analyzing this important class of operators in Hilbert spaces.

A Hilbert space operator $T$ is said to be $p$-$w$-hyponormal with $0 < p\leq 1$ if $|\widetilde{T}|^p\geq |T|^p\geq |\widetilde{T}^{*}|^p$, where $\widetilde{T}$ is the Aluthge transform. In this paper we prove basic properties of these operators. Using these results, we also prove that if $P$ is a Riesz idempotent for a non-zero isolated point $\lambda$ of the spectrum of $T$, then $P$ is self-adjoint. Among other things, we prove these operators are finitely ascensive and that, for non-zero $p$-$w$-hyponormal $T$ and $S$, $T\otimes S$ is $p$-$w$-hyponormal if and only if $T$ and $S$ are $p$-$w$-hyponormal. Moreover, it is shown that property $(gt)$ holds for $f(T)$, where $f\in H_{nc}(\sigma(T)).$

Оператор $T$ у гільбертовім просторі називається $p$-$w$-гіпонормальним, де $0 < p\leq 1$, якщо $ |\widetilde{T}|^p\geq |T|^p\geq |{\widetilde{T}}^{*}|^p$, де $\widetilde{T}$ --- перетворення Алутге. В цій роботі досліджені основні властивості таких операторів. Показано також, що якщо $P$ --- ідемпотент Рісса, який відповідає ненульовій ізольованій точці $\lambda$ спектру $T$, то оператор $P$ самоспряжений. Доведено, що ці оператори мають скінченний підйом і що для ненульових $p$-$w$-гіпонормальних $T$ і $S$, $T\otimes S$ є $p$-$w$-гіпонормальним тоді й тільки тоді, коли $T$ і $S$ $p$-$w$-гіпонормальні. Крім того, доведено, що властивість $(gt)$ має місце для $f(T)$, де $f\in H_{nc}(\sigma(T)).$

In this paper, we show that every $k$-quasi-class $A$ operator has a scalar extension and give some spectral properties of the scalar extensions of $k$-quasi-class $A$ operators. As a corollary, we get that such an operator with rich spectrum has a nontrivial invariant subspace.

In this paper we study the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$, a variant of Weyl's type theorems introduced by Berkani. We established for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which the properties $(\rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ hold. Among other things, we study the stability of the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ for a polaroid operator $T$ acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with $T$.