Q:

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

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 $A^{-1}A=I$

A:

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

A:

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