Compact Operator on Hilbert Space Direct Sum

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Theorem

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

Denote by $H = \displaystyle \bigoplus_{n \mathop = 1}^\infty H_n$ their Hilbert space direct sum.


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

Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \norm {T_n} < \infty$, where $\norm {\, \cdot \, }$ signifies the norm on bounded linear operators.

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

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

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


Proof


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