On the generalized Cauchy dual of closed operators in Hilbert spaces

In this paper, we introduce the generalized Cauchy dual $w(T)=T(T^{\ast }T{)}^{†}$ 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.