On the generalized Cauchy dual of closed operators in Hilbert spaces

TL;DR

This paper introduces a generalized Cauchy dual for closed operators in Hilbert spaces, explores its properties, and proves a power relation for quasinormal EP operators, advancing operator theory understanding.

Contribution

It defines the generalized Cauchy dual for closed operators with closed range and proves a key power relation for specific classes of operators, extending existing theory.

Findings

Defined the generalized Cauchy dual $w(T)$ for closed operators.

Proved that $w(T^n) = (w(T))^n$ for quasinormal EP operators.

Characterized the properties of the Cauchy dual in Hilbert space operators.

Abstract

In this paper, we introduce the generalized Cauchy dual $w(T) = T(T^{*}T)^{\dagger}$ of a closed operator $T$ with the closed range between Hilbert spaces and present intriguing findings that characterize the Cauchy dual of $T$. Additionally, we establish the result $w(T^{n}) = (w(T))^{n}$, for all $n \in \mathbb{N}$, where $T$ is a quasinormal EP operator.