Definition:Imaginary Part (Linear Operator)

Definition

Let $\HH$ be a Hilbert space over $\C$.

Let $A \in \map B \HH$ be a bounded linear operator.

Then the imaginary part of $A$ is the Hermitian operator:

$\Im A := \dfrac 1 {2 i} \paren {A - A^*}$

Also denoted as

The imaginary part of $A$ may be denoted by $\map \Im A$, $\map {\mathrm {im} } A$ or $\map {\mathrm {Im} } A$.

This resembles the notation for the imaginary part of a complex number.

Also see

• Definition:Real Part (Linear Operator)
• Linear Operator is Sum of Real and Imaginary Parts
• Definition:Imaginary Part of a complex number

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\S \text {II}.2$