As I read the literature, I have noticed that the composition $T^*T$ of a linear operator $T:H\to H$ and its adjoint frequently turns up in all kind of places. I am aware that it is Hermitian (at least when $T$ is bounded), that $||Tx||^2=\langle Tx,Tx\rangle=\langle x,T^*Tx\rangle$ and other basic stuff like that. However, I don't quite "feel" what the notion $T^*T$ really is and why is it so ubiquitous.
I know that this might seem vague but can anyone give me a general idea of how I should view $T^*T$? What dose it do to a vector and what are its important properties?
I am particularly struck by the fact that $||A||_2^2=\rho(A^*A)$ when $A$ is a matrix representing a finite dimensional operator and that $I+T^*T$ is a bijection.
Considering it as an operator we have that its self adjont i.e $(T^{*}Tx,y)=(x,T^{*}Ty)$, and thus diagonalisable if its compact. One significant fact allowing this to happen is that the orthagonal complement of an eigenvector of the operator is invariant under the operator, atleast in Hilbert spaces. Hence once you prove that there is one eigenvector you inductively obtain a complete set of eigenvectors.
$I+T^{*}T$ has in the case when $T$ is compact index $0$ hence gives injectivity iff. surjectivity kind of resembling the fundamental theorem om linear algebra.