*[Oops, this was a typo: all references to eigenvalues here have been replaced with singular values. There is no interlacing for eigenvalues for non-Hermitian matrices, as these eigenvalues need not even be real. -T.]*

For compact self-adjoint operators, basically all the inequalities for finite-dimensional operators extend to the infinite-dimensional setting by a limiting argument, see https://arxiv.org/abs/0709.1088 . For non-compact operators, in which the spectrum need not be pure point, the situation is significantly more complicated, and I am not sure what can cleanly be said (beyond the subadditivity of operator norms).

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