Definition:Imaginary Part (Linear Operator)
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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) $\S \text {II}.2$