Bounded Linear Operator on Hilbert Space Direct Sum

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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:

\(\displaystyle \left\Vert{ T h }\right\Vert_H^2\) \(=\) \(\displaystyle \left\Vert{ \left({T_n h_n}\right)_{n \in \N} }\right\Vert_H^2\) $\quad$ Definition of $T$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty \left\Vert{ T_n h_n }\right\Vert_{H_n}^2\) $\quad$ Definition of $\left\Vert{\cdot}\right\Vert_H$ $\quad$
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{n \mathop = 1}^\infty \left\Vert{ T_n }\right\Vert^2 \left\Vert{ h_n }\right\Vert_{H_n}^2\) $\quad$ $\quad$
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{n \mathop = 1}^\infty c^2 \left\Vert{ h_n }\right\Vert_{H_n}^2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle c^2 \sum_{n \mathop = 1}^\infty \left\Vert{ h_n }\right\Vert_{H_n}^2\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle c^2 \left\Vert{ h }\right\Vert_H^2\) $\quad$ $\quad$

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.

$\blacksquare$