Compact Linear Transformations Composed with Bounded Linear Operator

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Theorem

Let $H, K$ be Hilbert spaces.

Let $T \in B_0 \left({H, K}\right)$ be a compact linear transformation.


Let $A \in B \left({H}\right), B \in B \left({K}\right)$ be bounded linear operators.


Then the compositions $TA$ and $BT$ are also compact linear transformations.


Proof


Sources