Bounded Linear Operator on Hilbert Space Direct Sum
Theorem
Let $\left({H_n}\right)_{n \in \N}$ 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 $T_n \in B \left({H_n}\right)$ be a bounded linear operator.
Suppose that one has $\displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert < \infty$, where $\left\Vert{\cdot}\right\Vert$ signifies the norm on bounded linear operators.
Define $T: H \to H$ by:
- $\forall h = \left({h_n}\right)_{n \in \N}: T h = \left({T_n h_n}\right)_{n \in \N} \in H$
Then $T \in B \left({H}\right)$ is a bounded linear operator.
Proof
Let $c = \displaystyle \sup_{n \mathop \in \N} \, \left\Vert{T_n}\right\Vert$.
By assumption, $c < \infty$.
Let $h = \left({h_n}\right)_{n \in \N} \in H$ be arbitrary.
Then:
| \(\ds \left\Vert{ T h }\right\Vert_H^2\) | \(=\) | \(\ds \left\Vert{ \left({T_n h_n}\right)_{n \in \N} }\right\Vert_H^2\) | Definition of $T$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \left\Vert{ T_n h_n }\right\Vert_{H_n}^2\) | Definition of $\left\Vert{\cdot}\right\Vert_H$ | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \left\Vert{ T_n }\right\Vert^2 \left\Vert{ h_n }\right\Vert_{H_n}^2\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty c^2 \left\Vert{ h_n }\right\Vert_{H_n}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds c^2 \sum_{n \mathop = 1}^\infty \left\Vert{ h_n }\right\Vert_{H_n}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds c^2 \left\Vert{ h }\right\Vert_H^2\) |
In particular: results for some steps.
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In summary:
- $\left\Vert{ T h }\right\Vert_H^2 \le c^2 \left\Vert{ h }\right\Vert_H^2$
It follows that $T$ is bounded.
$\blacksquare$