Moore-Penrose Inverse of Product Operators in Hilbert $C^*$-Modules

Abstract

Suppose $S$ and $T$ are adjointable linear operators‎

‎between Hilbert $C^*$-modules‎. ‎It is well known that an operator $T$ has closed range‎

‎if and only if its Moore-Penrose inverse $T^\dagger$ exists‎.

‎In this paper‎, ‎we show that $(TS)^\dagger=S^\dagger T^\dagger $‎,

‎where $S$ and $T$ have closed ranges and $(\ker(T))^\perp=\rm {ran}(S)$‎. ‎Moreover‎,

‎we investigate some results related to the polar decomposition of $T$‎. ‎We also obtain‎

‎the inverse of $1‎ - ‎T^{\dagger}T‎ + ‎T$‎, ‎when $T$ is a self-adjoint operator‎.