Linear Operator is Sum of Real and Imaginary Parts
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Theorem
Let $\HH$ be a Hilbert space over $\C$.
Let $A \in \map B \HH$ be a bounded linear operator.
Let $B$ and $C$ be the real and imaginary parts of $A$, respectively.
Then $A = B + i C$.
Proof
\(\ds B + i C\) | \(=\) | \(\ds \frac 1 2 \paren {A + A^*} + i \frac 1 {2i} \paren {A - A^*}\) | Definitions of $B, C$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 A + \frac 1 2 A^* + \frac 1 2 A - \frac 1 2 A^*\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds A\) |
$\blacksquare$
Sources
1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\S \text{II}.2$