Perturbation of operators and eigenvalues Suppose $P\in\mathcal{B}(\mathcal{H})$ is a self-adjoint compact operator. Lets perturb $P$ by multiplying it by a bounded operator $S$ and set $T=PS.$ Then what can be said for the spectrum of $T?$ reference suggestions is greatly appreciated.

The new operator $T$ is going to be also compact, as the product of a bounded and a compact and a bounded operator is also compact, and hence its spectrum, which could now contain even complex numbers, could only have as a limit point the $0$.

The fact that $P$ is also self-adjoint does not change anything.