Compact Linear Transformations Composed with Bounded Linear Operator

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

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.4.2 \text {(c)}$