For self-adjoint automorphism $T$, to what extent is $U \circ T \circ S$ equivalent to $U \circ S$?

Question

I would like to know to what extent can composition with self-adjoint isomorphism affect composition of linear operators. More specifically, for linear operators of Hilbert spaces as $H \xrightarrow{S} H' \xrightarrow{T} H' \xrightarrow{U} H''$ so that $T$ is self-adjoint vector-space-automorphism, is there an equivalence between $U \circ S$ and $U \circ T \circ S$ so that would exist vector-space-automorphisms $I \colon H,J \colon H''$ so that $J \circ (U \circ S) \circ I = U \circ T \circ S$. Does this apply if the spaces are finite-dimensional?