-Product of a Hilbert Schmidt and bounded operators is Hilbert-Schmidt

functional-analysis operator-theory

Hint: Use $\Vert BAe_i\Vert_H\leq \Vert B\Vert_H \Vert Ae_i\Vert_H$ to show $\Vert BA\Vert_{HS}\leq \Vert B\Vert_H \Vert A\Vert_{HS}$. For the other inequality it is useful to first show $\Vert B^*\Vert_H =\Vert B\Vert_H$ and $\Vert A^*\Vert_{HS}=\Vert A\Vert_{HS}.$

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