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direct sum of bounded operators on Hilbert spaces

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Definition

Let $\{ H_i\}_{i \in I}$ be a family of Hilbert spaces indexed by a set

. For each $i \in I$ let $T_i:H_i \longrightarrow H_i$ be a bounded linear operator on

such that the family $\{T_i\}_{i \in I}$ is uniformly bounded, i.e. $\sup\,\{\Vert T_i\Vert: i \in I\} < \infty$ .

Definition - The direct sum of the family $\{T_i\}_{i \in I}$ is the operator

$\displaystyle \bigoplus_{i \in I} T_i : \bigoplus_{i \in I} H_i \longrightarrow \bigoplus_{i \in I} H_i$

on the direct sum of Hilbert spaces $\bigoplus_{i \in I} H_i$ defined by

$\displaystyle \left( \bigoplus_{i \in I} T_i \;(x)\right)_i := T_i x_i$

It can be seen that $\bigoplus_{i \in I} T_i$ is well-defined and is in fact a bounded linear operator, whose norm is

$\displaystyle \left\Vert\bigoplus_{i \in I} T_i \right\Vert = \sup\,\{\Vert T_i\Vert : i \in I\}$

$\displaystyle \bigoplus_{i \in I} (aT_i + b S_i) = a \bigoplus_{i \in I} T_i + b\bigoplus_{i \in I} S_i$ , where $a, b \in \mathbb{C}$ .

$\displaystyle \left(\bigoplus_{i \in I} T_i\right)^* = \bigoplus_{i \in I} T_i^*$ .

$\displaystyle \left(\bigoplus_{i \in I} T_i\right) \left(\bigoplus_{i \in I} S_i\right) = \bigoplus_{i \in I} T_iS_i$ .

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AMS MSC:	46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )
	47A05 (Operator theory :: General theory of linear operators :: General )

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