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 $T_i:H_i⟶H_i$ 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. .

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

${\displaystyle \underset{i\in I}{\oplus }}{T}_i:{\displaystyle \underset{i\in I}{\oplus }}{H}_i⟶{\displaystyle \underset{i\in I}{\oplus }}{H}_i$

on the direct sum of Hilbert spaces ${\oplus }_{i\in I}{H}_i$ defined by

${\left({\displaystyle \underset{i\in I}{\oplus }}{T}_i(x)\right)}_i:=T_i{x}_i$

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

$\parallel {\displaystyle \underset{i\in I}{\oplus }}{T}_i\parallel =sup\{\parallel T_i\parallel :i\in I\}$

0.2 Properties

• ${\displaystyle \underset{i\in I}{\oplus }}(a{T}_i+b{S}_i)=a{\displaystyle \underset{i\in I}{\oplus }}{T}_i+b{\displaystyle \underset{i\in I}{\oplus }}{S}_i$, where $a,b\in ℂ$.

• ${\left({\displaystyle \underset{i\in I}{\oplus }}{T}_i\right)}^{*}={\displaystyle \underset{i\in I}{\oplus }}{T}_i^{*}$.

• $\left({\displaystyle \underset{i\in I}{\oplus }}{T}_i\right)\left({\displaystyle \underset{i\in I}{\oplus }}{S}_i\right)={\displaystyle \underset{i\in I}{\oplus }}{T}_i{S}_i$.