Composition of Direct Sums of Bounded Linear Operators on Hilbert Space

Theorem

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 $T_i : \HH_i \to \HH_i$ and $S_i : \HH_i \to \HH_i$ be bounded linear operators.

Suppose that:

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

and:

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

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 inner product $\innerprod \cdot \cdot$ and norm $\norm {\, \cdot \,}$.

Let:

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

$\ds S = \bigoplus_{i \mathop \in I} S_i$

$\ds \bigoplus_{i \mathop \in I} T_i S_i$

be the direct sum of $\family {T_i}_{i \mathop \in I}$, $\family {S_i}_{i \mathop \in I}$ and $\family {T_i S_i}_{i \mathop \in I}$ respectively.

Then we have:

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

That is:

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

Proof

From Norm on Bounded Linear Transformation is Submultiplicative we have:

\(\ds \norm {T_i S_i}_{\map B {\HH_i} }\)

\(\le\)

\(\ds \norm {T_i}_{\map B {\HH_i} } \norm {S_i}_{\map B {\HH_i} }\)

\(\ds \)

\(\le\)

\(\ds \paren {\sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } } \paren {\sup_{i \mathop \in I} \norm {S_i}_{\map B {\HH_i} } }\)

Hence we have:

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

Hence:

$\ds \bigoplus_{i \mathop \in I} T_i S_i$ is well-defined.

Let $f \in \HH$ and $i \in I$.

Then we have:

$\ds \map {\paren {\paren {\bigoplus_{i \mathop \in I} T_i} \paren {\bigoplus_{i \mathop \in I} S_i} f} } i = T_i \map {\paren {\paren {\bigoplus_{i \mathop \in I} S_i} f} } i = T_i S_i \map f i = \map {\paren {\paren {\bigoplus_{i \mathop \in I} T_i S_i} f} } i$

by the definition of the direct sum.

Hence we have:

$\ds \paren {\bigoplus_{i \mathop \in I} T_i} \paren {\bigoplus_{i \mathop \in I} S_i} f = \paren {\bigoplus_{i \mathop \in I} T_i S_i} f$

for each $f \in \HH$.

Hence:

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

$\blacksquare$