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

theorem on the representation of C*-algebras by bounded operators Naimark's dilation theorem on extensions of symmetric operators The Gelfand–Naimark–Segal...

T_{\mathrm {max} }.} Energetic extension Extensions of symmetric operators N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman...

is symmetric and Dom ⁡ A = Dom ⁡ A ∗ {\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}} . Equivalently, a closed symmetric operator A {\displaystyle...

the domain of U, dom U, is (A+iI) dom A. See self-adjoint operator for further details. Bilinear transform Extensions of symmetric operators Robert Everist...

closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators. A symmetric operator is often...

descriptions of redirect targets Category of matrices Quasilinearization "Linear transformations of V into V are often called linear operators on V." Rudin...

of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with...

quantum state then the operator is self-adjoint. In physics the term Hermitian often refers to both symmetric and self-adjoint operators. (In certain artificial...

The symmetrically normalized Laplacian matrix is symmetric if and only if the adjacency matrix is symmetric. For a non-symmetric adjacency matrix of a directed...

transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by F ( y )...

only real entries is symmetric if and only if it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix. Proof...

byproduct are Pauli operators. Note that a product of symmetric matrices is not symmetric in general. It is easy to check that the affine forms of H {\displaystyle...

rotors symmetric rotors oblate symmetric rotors prolate symmetric rotors asymmetric rotors This classification depends on the relative magnitudes of the...

M {\displaystyle M} is symmetric or Hermitian, and all its eigenvalues are real and positive. M {\displaystyle M} is symmetric or Hermitian, and all its...

important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such...

to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular...

operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm...

gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify...

eigenfunction Hermitian operator self-adjoint operator, Hermitian adjoint Hilbert matrix Shift operator Symmetric matrix Parseval's identity Rayleigh quotient...

by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types...

Hermitian symmetric space, a Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space Hermitian transpose, the transpose of a matrix...

consumers. The number of consumers serviced by a single laser is determined by the operator's engineering criteria; operators may opt to reduce the number...

gives an inner product invariant under H. The operators Ad p with p in P are positive symmetric operators. This new inner produst can be written as ( S...

(January 1944), "The identity of weak and strong extensions of differential operators", Transactions of the American Mathematical Society, 55 (1): 132–151...

matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if A T = A . {\displaystyle \mathbf {A} ^{\text{T}}=\mathbf...

length and perimeter. Spatial operators for determining geospatial set operations, like union, difference, symmetric difference and buffers (provided...

// Generalized inverse (F*F^(-1)F=F) of a : // symmetric positive semi-definite matrix : I [symmetric] 1 2 3 +-------------------------------------------+...

It is a symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X . {\displaystyle x\in X.} Every subadditive symmetric function...

linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues...