I have a linear transformation $T: V \to W$ where $V$ and $W$ are finitely generated real inner product spaces and their inner product is not necessarily standard. I also have $K: V \to V$. My goal is to study $TKT^*$ where $T^*$ is the adjoint with respect to my non-standard inner-products.
The SVD theorem says that $T=UDV^*$. So apparently $$TKT^*=UDV^*K(UDV^*)^*=UDV^*KVD^*U^*$$ and $V^*KV=V^{-1}KV$ so this is just a change of basis for $K$, and similarly with $U$ and $U^*$. This gives me some insights.
The problem is that this is not true since in the SVD theorem as I know it, $V^*$ actually means conjugate transpose which does not respect my inner-product. SVD says that $U$ and $V$ are unitary which again does not respect my inner-product. Is there any version of SVD that overcomes this issue?
Thank you!