Difference between revisions of "Hermitian operator"
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A [[linear operator]] $A$ on a [[Hilbert space]] $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a [[self-adjoint operator]]. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a [[Hermitian matrix]]. | A [[linear operator]] $A$ on a [[Hilbert space]] $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a [[self-adjoint operator]]. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a [[Hermitian matrix]]. | ||
Latest revision as of 18:36, 26 April 2016
See also: symmetric operator.
A linear operator $A$ on a Hilbert space $H$ with a dense domain of definition $D(A)$ and such that $\langle Ax,y\rangle = \langle x,Ay\rangle$ for any $x,y\in D(A)$. This condition is equivalent to: 1) $D(A)\subset D(A^*)$; and 2) $Ax = A^*x$ for all $x\in D(A)$, where $A^*$ is the adjoint of $A$, that is, to $A\subset A^*$. A bounded Hermitian operator is either defined on the whole of $H$ or can be so extended by continuity, and then $A=A^*$, that is, $A$ is a self-adjoint operator. An unbounded Hermitian operator may or may not have self-adjoint extensions. Sometimes any self-adjoint operator is called Hermitian, preserving the name symmetric for an operator that is Hermitian in the sense explained above. On a finite-dimensional space a Hermitian operator can be described by a Hermitian matrix.
References
| [1] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) |
| [2] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |
Hermitian operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_operator&oldid=38658