In this paper, we introduce the generalized Cauchy dual <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$w(T) = T(T^{*}T)^{\dagger }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>T</mi> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mi>T</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow /> <mo>†</mo> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation> of a closed operator T with a closed range between Hilbert spaces and present intriguing findings that characterize the Cauchy dual of T. Additionally, we establish the result <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$w(T^{n}) = (w(T))^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, where T is a quasinormal EP operator.

On the generalized Cauchy dual of closed operators in Hilbert spaces

期刊论文
发表日期：

2025年9月5日

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Arup Majumdar,
P. Sam Johnson,
Ram N. Mohapatra

摘要

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