# Compact Linear Transformations Composed with Bounded Linear Operator

## 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

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.4.2(c)$