Definition:Imaginary Part (Linear Operator)

Definition
Let $H$ be a Hilbert space over $\C$.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Then the imaginary part of $A$ is the self-adjoint operator:


 * $\operatorname{Im} A := \dfrac 1 {2i} \left({A - A^*}\right)$

The imaginary part of $A$ may be denoted by $\operatorname{Im} \left({A}\right)$, $\operatorname{im} \left({A}\right)$ or $\Im \left({A}\right)$.

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