Operator theory

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Operator

An operator assigns to every function a function . It is therefore a mapping between two function spaces. If the range is on the real line or in the complex plane, the mapping is usually called a functional instead.

Hermitian operator

A second-order linear Hermitian operator is an operator that satisfies(1)where denotes a complex conjugate. As shown in Sturm-Liouville theory, if is self-adjoint and satisfies the boundary conditions(2)then it is automatically Hermitian.Hermitian operators have real eigenvalues, orthogonal eigenfunctions, and the corresponding eigenfunctions form a complete biorthogonal system when is second-order and linear.Note that the concept of Hermitian operator is somewhat extended in quantum mechanics to operators that need be neither second-order differential nor real. Simply assuming that the boundary conditions give sufficiently strongly vanishing near infinity or have periodic behavior allows an operator to be Hermitian in this extended sense if(3)which is identical to the previous definition except that quantities have been extended to be complex (Arfken 1985, p. 506).In order to prove that eigenvalues must be..

Operator extension

Let and be linear operators from domains and , respectively, into a Hilbert space . It is said that extends if and if for any vector .

Cyclic vector

A vector on a Hilbert space is said to be cyclic if there exists some bounded linear operator on so that the set of orbitsis dense in . In this case, the operator is said to be a cyclic operator.

Unitary

An operator satisfying (1)(2)where is the adjoint.

Cyclic operator

A bounded linear operator on a Hilbert space is said to be cyclic if there exists some vector for which the set of orbitsis dense in . In this case, the vector is said to be a cyclic vector.

Conjugate transpose

The conjugate transpose of an matrix is the matrix defined by(1)where denotes the transpose of the matrix and denotes the conjugate matrix. In all common spaces (i.e., separable Hilbert spaces), the conjugate and transpose operations commute, so(2)The symbol (where the "H" stands for "Hermitian") gives official recognition to the fact that for complex matrices, it is almost always the case that the combined operation of taking the transpose and complex conjugate arises in physical or computation contexts and virtually never the transpose in isolation (Strang 1988, pp. 220-221).The conjugate transpose of a matrix is implemented in the Wolfram Language as ConjugateTranspose[A].The conjugate transpose is also known as the adjoint matrix, adjugate matrix, Hermitian adjoint, or Hermitian transpose (Strang 1988, p. 221). Unfortunately, several different notations are in use as summarized in the..

Closed operator

A linear operator from its domain into a Hilbert space is closed if for any sequence of vectors such that and as , it follows that and .

Closable operator

A linear operator from its domain into a Hilbert space is closable if it has a closed extension where . Closable operators are sometimes called preclosed (Takesaki 1970), and the extension of is sometimes called the closure of .

Antiunitary

An operator is said to be antiunitary if it satisfies:(1)(2)(3)where is the inner product and is the complex conjugate of .

Separating vector

Given a subalgebra of the algebra of bounded linear transformations from a Hilbert space onto itself, the vector is a separating vector for if the only operator satisfying is the zero operator .

Antilinear

An antilinear operator satisfies the following two properties:(1)(2)where is the complex conjugate of .

Linear operator

An operator is said to be linear if, for every pair of functions and and scalar ,and

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