Compact Linear Transformations Composed with Bounded Linear Operator

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Theorem

Let $H, K$ be Hilbert spaces.

Let $T \in \map {B_0} {H, K}$ be a compact linear transformation.


Let $A \in \map B H, B \in \map B K$ be bounded linear operators.


Then the compositions $T A$ and $B T$ are also compact linear transformations.


Proof


Sources