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) $\S \text{II}.2$