Definition:Direct Sum of Bounded Linear Operators on Hilbert Space

Definition

Let $\GF \in \set {\R, \C}$.

Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\GF$.

For each $i \in I$, let $T_i : \HH_i \to \HH_i$ be a bounded linear operator.

Suppose that:

$\ds \sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } < \infty$

where $\norm {\, \cdot \,}_{\map B {\HH_i} }$ is the norm of a bounded linear operator on $\HH_i$.

Let:

$\ds \HH = \bigoplus_{i \mathop \in I} \HH_i$

be the Hilbert space direct sum of $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ with norm $\norm {\, \cdot \,}$.

Define $T : \HH \to \HH$ by:

$\map {\paren {T f} } i = \map {T_i} {\map f i}$

for each $f \in \HH$ and $i \in I$.

We say that $T$ is the direct sum of $\family {T_i}_{i \mathop \in I}$ and write:

$\ds T = \bigoplus_{i \mathop \in I} T_i$