Definition:Hermitian Operator
(Redirected from Definition:Self-Adjoint Operator)
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Definition
Let $\HH$ be a Hilbert space.
Let $\mathbf T: \HH \to \HH$ be a bounded linear operator.
Then $\mathbf T$ is said to be Hermitian if and only if:
- $\mathbf T = \mathbf T^*$
That is, if and only if it equals its adjoint $\mathbf T^*$.
Also known as
A Hermitian operator is also known as a self-adjoint operator.
Also see
- Definition:Hermitian Matrix
- Definition:Unitary Operator
- Definition:Self-Adjoint Densely-Defined Linear Operator
- Results about Hermitian operators can be found here.
Source of Name
This entry was named for Charles Hermite.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text {II}.2.11 \ \text {(a)}$
- 2013: Brian C. Hall: Quantum Theory for Mathematicians: Definition $7.1$
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next): $13.1$: Existence of Hilbert Adjoint