Definition:Real Part (Linear Operator)
Jump to navigation
Jump to search
Definition
Let $\HH$ be a Hilbert space over $\C$.
Let $A \in \map B \HH$ be a bounded linear operator.
Then the real part of $A$ is the Hermitian operator:
- $\Re A := \dfrac 1 2 \paren {A + A^*}$
Also denoted as
The real part of $A$ may be denoted by $\map \Re A$, $\map {\mathrm {re} } A$ or $\map {\mathrm {Re} } A$.
This resembles the notation for the real part of a complex number.
Also see
- Imaginary Part
- Linear Operator is Sum of Real and Imaginary Parts
- Definition:Real Part of a Definition:Complex Number
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\S \text {II}.2$