Compact Operator on Hilbert Space Direct Sum

Theorem

Let $\sequence {\HH_n}_{n \mathop \in \N}$ be a sequence of Hilbert spaces.

Denote by $\HH = \ds \bigoplus_{n \mathop = 1}^\infty \HH_n$ their Hilbert space direct sum.

For each $n \in \N$, let $T_n \in \map B {\HH_n}$ be a bounded linear operator.

Suppose that:

$\ds \sup_{n \mathop \in \N} \norm {T_n} < \infty$

where $\norm {\, \cdot \, }$ signifies the norm on bounded linear operators.

Define $T \in \map B \HH$ by:

$\forall h = \sequence {h_n}_{n \mathop \in \N}: T h = \sequence {T_n h_n}_{n \mathop \in \N} \in \HH$

(That $T$ is indeed bounded follows from Bounded Linear Operator on Hilbert Space Direct Sum.)

Then $T$ is compact if and only if the following conditions hold:

For each $n \in \N$, $T_n$ is compact

$\ds \lim_{n \mathop \to \infty} \norm {T_n} = 0$

Proof

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text {II}.4$ Exercise $13$