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 $\norm {\, \cdot \,}_H$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \norm {T_n}^2 \norm {h_n}_{H_n}^2\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty c^2 \norm {h_n}_{H_n}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c^2 \sum_{n \mathop = 1}^\infty \norm {h_n}_{H_n}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c^2 \norm h_H^2\) |
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In summary:
- $\norm {T h}_H^2 \le c^2 \norm h_H^2$
It follows that $T$ is bounded.
$\blacksquare$