$k$-Quasi $n$-Power Posinormal Operators: Theory and Weighted Conditional Type Applications

TL;DR

This paper introduces and studies $k$-quasi $n$-power posinormal operators in Hilbert spaces, generalizing existing classes and providing structural, spectral, and application-oriented results, especially for weighted conditional operators.

Contribution

It extends posinormal operator theory by defining and analyzing $k$-quasi $n$-power posinormal operators, including their properties, decompositions, and applications to weighted conditional operators.

Findings

Matrix representations in $2 imes 2$ blocks

Tensor product preservation of the class

Explicit conditions for weighted operators

Abstract

This paper introduces and investigates the class of \textit{$k$-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 $\tTwu := wE(uf)$ 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.