for linear Operators is it true that $(A^*)^{-1}=I$, whrere $I$ is the neutral element?

Question

If A is a linear operator between Hilbert spaces. Is the following statement true?

$(A^*)^{-1}A=I$, where $I$ is the neutral element so that $A^{-1}A=I$

Answer 1

Not true. A counter-example is $iA$ where $A$ is an invertible self-adjoint operator.

Answer 2

We have $(A^*)^{-1}A=I \iff (A^*)^{-1}=A^{-1} \iff A^*=A \iff A $ is self-adjoint.