# Hermitian Operator

## Hermitian operator

[er′mish·ən ′äp·ə‚rād·ər]
(mathematics)
A linear operator A on vectors in a Hilbert space, such that if x and y are in the range of A then the inner products (Ax,y) and (x,Ay) are equal.

## Hermitian Operator

an infinite-dimensional analogue of the Hermitian linear transformation. A bounded linear operator A in a complex Hilbert space H is said to be Hermitian if for any two vectors x and y in the space the relation (Ax, y) = (x, Ay) holds, where (x, y) is the scalar product of H. Examples of Hermitian operators are integral operators (seeINTEGRAL EQUATIONS) for which the kernel K(x, y) is given in a bounded region and is a continuous function such that ; in this case, K(x, y) is called a Hermitian kernel. The concept of Hermitian operators may be extended to unbounded linear operators in a Hilbert space.

Hermitian operators play an important role in quantum mechanics, providing a convenient means of describing mathematically the observable quantities that characterize a physical system.

References in periodicals archive ?
Given a Hermitian operator M on a vector space V with a subspace U [subset or equal to] V, let [pr.
Some other specific subjects which I found interesting were related to the iterative method for computing the square root for a positive Hermitian operator [6, p.
Taking into account that the Hamiltonian is a Hermitian operator, it is possible to show that for n [not equal to] 1:
Weyl [20] examined the spectra of all compact perturbations of a hermitian operator on Hilbert space and found in 1909 that their intersection consisted precisely of those points of the spectrum which were not isolated eigenvalues of finite multiplicity.
So, to sum it up: a semilocal pseudopotential is a general Hermitian operator in the spherically symmetric problem (i.
Since the Hamiltonian is a Hermitian operator, one concludes that if the Hilbert space basis yields a non-diagonal Hamiltonian matrix then the lowest eigenvalue "favors" eigenfunctions that are a linear combination of the Hilbert space basis functions.
Weyl [22] examined the spectra of all compact perturbations of a hermitian operator on Hilbert space and found in 1909 that their intersection consisted precisely of those points of the spectrum which were not isolated eigenvalues of finite multiplicity.
The difference in the resolvents of two self-adjoint extensions of the non-densely defined Hermitian operator A = -[Laplacian operator] (when restricted to an appropriate subspace of [L.
Lee, Maps preserving zero Jordan products on hermitian operators, Illinois J.
He explains vector spaces and bases, linear transformations and operators, eigenvalues, circles and ellipses, inner products, adjoints, Hermitian operators, unitary operators, the wave equation, continuous spectre and the Dirac delta function, Fourier transforms, Green's and functions, and includes an appendix on matrix operations (new to this edition) and a full chapter on crucial applications.
The way in which physical concepts are tied to the mathematical formalism is particularly helpful, for example in the treatment of Hermitian operators where orthogonality of non-degenerate eigenstates is tied to repeatability of measurements.
KREIN, Fundamental aspects of the representation theory of Hermitian operators with deficiency index (mrz, rn), Ukrain.

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