$TT^*$ is unitary equivalent to $T^*T$.

Question

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ denote the bounded linear operators from $H$ to itself. An operator is said to be normal if $TT^* = T^*T$. I would like to know which properties of normal operators are preserved if we assume unitary equivalence between $TT^*$ and $T^*T$ instead of the equality.

Thank you very much!

Answer 1

I wouldn't expect much. In finite dimension, for instance, they are always unitarily equivalent: write $T=V|T|$ with $V$ a unitary (that's where one uses finite dimension), then $$ TT^*=V|T|^2V^*=V^*T^*TV. $$