Denote by $H_1$ and $H_2$ some Hilbert spaces and $S: H_1 \rightarrow H_2$ is some linear bounded operator, such that $SS^*: H_1\rightarrow H_1$ is some compact lineare operator, where $S^*$ denotes the adjoint operator.
My question is how do $SS^*$ and $S^*S$ correlate? Do both operators have the same eigenvalues?