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

Mehdi Mohammadzadeh Karizaki, Mahmoud Hassani, Maryam Amyari

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 $(TS)^\dag=S^dag T^\dagger$, where $S$ and $T$ have closed ranges and $(\ker(T))^\bot=\operatorname{ran}(S)$. Moreover, we investigate some results related to the polar decomposition of $T$. We also obtain the inverse of $1-T^\dag T+T$, when $T$ is a self-adjoint operator.