Moore-Penrose inverse of product operators in Hilbert C*- modules

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.