Operator norm of $TT^*$

Question

Let $T:H \to H$ be a bounded operator

Prove that:$||TT^* ||=||T^*T || =||T||^2$ Any indication please

Answer 1

Let $h∈H$ such that $‖h‖_H≤1$

Then:

$‖Ah‖^2_H = ⟨Ah,Ah⟩_H = ⟨A^∗Ah,h⟩_H ≤ ‖‖A^∗Ah‖‖_H‖h‖_H ≤ ‖‖A^∗A‖‖‖h‖^2_H ≤ ‖‖A^∗A‖‖ ≤ ‖‖A^∗‖‖‖A‖ $
Therefore:

$‖Ah‖‖‖Ah′‖‖ ≤ ‖‖A^∗‖‖‖A‖$
it follows that: $ ‖A‖2≤‖A^∗A‖≤‖A^∗‖‖A‖$

That is, $‖A‖≤‖A^∗‖$

By substituting $A^∗$ for $A$, and using $A^{∗∗}=A$ from Double Adjoint is Itself, the reverse inequality is obtained.

Hence $ ‖A‖^2=‖A^∗A‖=‖A^∗‖^2$