Compact Operator on Hilbert Space Direct Sum
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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 $13$