# direct sum of bounded operators on Hilbert spaces

direct sum

## 0.1 Definition

Let $\{H_{i}\}_{i\in I}$ be a family of Hilbert spaces indexed by a set $I$. For each $i\in I$ let $T_{i}:H_{i}\longrightarrow H_{i}$ be a bounded linear operator on $H_{i}$ such that the family $\{T_{i}\}_{i\in I}$ of bounded linear operators is uniformly bounded, i.e. $\sup\,\{\|T_{i}\|:i\in I\}<\infty$.

Definition - The direct sum of the uniformly bounded 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

 $\displaystyle\left\|\bigoplus_{i\in I}T_{i}\right\|=\sup\,\{\|T_{i}\|:i\in I\}$

## 0.2 Properties

• $\displaystyle\bigoplus_{i\in I}(aT_{i}+bS_{i})=a\bigoplus_{i\in I}T_{i}+b% \bigoplus_{i\in I}S_{i}$, where $a,b\in\mathbb{C}$.

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

• $\displaystyle\left(\bigoplus_{i\in I}T_{i}\right)\left(\bigoplus_{i\in I}S_{i}% \right)=\bigoplus_{i\in I}T_{i}S_{i}$.

Title direct sum of bounded operators on Hilbert spaces DirectSumOfBoundedOperatorsOnHilbertSpaces 2013-03-22 18:00:32 2013-03-22 18:00:32 asteroid (17536) asteroid (17536) 7 asteroid (17536) Definition msc 46C05 msc 47A05