I was reading on Hilbert–Schmidt operators, so if I have a bounded linear operator $F \in B(H)$ on a Hilbert space $H$ that is also Hilbert–Schmidt, where I am using the wiki definition.
My question is if I find an inverse in the sense of bounded linear operators, I am thinking it is given to me or I use generalization of Sherman-Morrison-Woodbury formula to find an inverse, is that inverse going to be necessarily Hilbert–Schmidt?
A Hilbert-Schmidt operator on an infinite-dimensional Hilbert space is never invertible. This follows, for instance, from the fact that the product of a Hilbert-Schmidt operator and any other bounded operator is Hilbert-Schmidt, and the identity operator is not Hilbert-Schmidt.
(Of course, on a finite-dimensional Hilbert space, every operator is Hilbert-Schmidt.)