In mathematics, a self-adjoint operator on a complex vector space V with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is a linear...

specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle...

perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The...

\mathbb {N} } . If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral...

the adjoint on a dense subset of L2: P* is a densely defined operator. The Sturm–Liouville operator is a well-known example of a formal self-adjoint operator...

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element...

consider properties of an operator up to compact perturbation. A bounded operator T on a Hilbert space H is said to be self-adjoint if T = T*, or equivalently...

authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex...

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a ∗ {\displaystyle a=a^{*}} ). Let A {\displaystyle...

for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators...

{\displaystyle X} be a Hilbert space and let T {\displaystyle T} be a self-adjoint operator on X {\displaystyle X} . The essential spectrum of T {\displaystyle...

zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of...

, self-adjoint operators): N ∗ = N {\displaystyle N^{\ast }=N} skew-Hermitian operators: N ∗ = − N {\displaystyle N^{\ast }=-N} positive operators: N...

von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable".: 17  These observables...

that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. These operators are often contrasted...

everywhere-defined operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded...

of as skew-adjoint (since they are like 1 × 1 {\displaystyle 1\times 1} matrices), whereas real numbers correspond to self-adjoint operators. For example...

essentially self-adjoint if and only if it has one and only one self-adjoint extension. A symmetric operator may have more than one self-adjoint extension...

satisfy the above regular boundary conditions. Moreover, L is a self-adjoint operator: ⟨ L f , g ⟩ = ⟨ f , L g ⟩ . {\displaystyle \langle Lf,g\rangle...

Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. If T is a self-adjoint operator on a finite-dimensional...

defined operator. The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded...

space in question. In quantum mechanics, observables manifest as self-adjoint operators on a separable complex Hilbert space representing the quantum state...

the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H)...

perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The...

Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician...

functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter...

{\displaystyle V^{*}} is the adjoint of V. If T is a self-adjoint operator, then the compression T W {\displaystyle T_{W}} is also self-adjoint. When V is replaced...

symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions...

context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann...

square-integrable functions on the real line. The position operator is defined as the self-adjoint operator Q : D Q → L 2 ( R , C ) : ψ ↦ q ψ , {\displaystyle...