I am working in book "A Hilbert Space Problem Book" by Paul R. Halmos.
When I try to show each problem in detail, I got in stuck with problem 132 , page 72. It states that:
"If $A$ is a contraction (that means $\|A\| \le 1$), then $1-AA*$ is positive (A* is the adjoint operator of A); call it A'. Assertion: the operator matrix $$M(A) = \begin{pmatrix} A & A'\\ 0 & 0 \end{pmatrix}$$ is a partial isometry. Proof: Check that MM*M=M. "
I really confuse what the adjoint of $M$ (I mean $M*$) is. How to find it in general?
Moreover, How to apply to check that $MM*M=M$?
Could you please help me to understand it deeply? Thank you so much.