Bounded Linear 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 $H = \ds \bigoplus_{n \mathop = 1}^\infty H_n$ 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 $\ds \sup_{n \mathop \in \N} \norm {T_n} < \infty$, where $\norm {\, \cdot \,}$ signifies the norm on bounded linear operators.

Define $T: H \to H$ by:

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


Then $T \in \map B H$ is a bounded linear operator.


Proof

Let $c = \ds \sup_{n \mathop \in \N} \, \norm {T_n}$.

By assumption, $c < \infty$.


Let $h = \sequence {h_n}_{n \mathop \in \N} \in H$ be arbitrary.

Then:

\(\ds \norm {T h}_H^2\) \(=\) \(\ds \norm {\sequence {T_n h_n}_{n \mathop \in \N} }_H^2\) Definition of $T$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \norm {T_n h_n}_{H_n}^2\) Definition of $\norm {\, \cdot \,}_H$
\(\ds \) \(\le\) \(\ds \sum_{n \mathop = 1}^\infty \norm {T_n}^2 \norm {h_n}_{H_n}^2\)
\(\ds \) \(\le\) \(\ds \sum_{n \mathop = 1}^\infty c^2 \norm {h_n}_{H_n}^2\)
\(\ds \) \(=\) \(\ds c^2 \sum_{n \mathop = 1}^\infty \norm {h_n}_{H_n}^2\)
\(\ds \) \(=\) \(\ds c^2 \norm h_H^2\)

In summary:

$\norm {T h}_H^2 \le c^2 \norm h_H^2$

It follows that $T$ is bounded.

$\blacksquare$