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...

for symmetric operators and A = A ∗ ∗ ⊆ A ∗ {\displaystyle A=A^{**}\subseteq A^{*}} for closed symmetric operators. The densely defined 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...

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...

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

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 )...

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...

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...

written in the coordinate system rotating and stretching with the fluid. The operator is specified by the following formula: A ▽ = D D t A − ( ∇ v ) T ⋅ A −...

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...

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...

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

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...

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...

bounded nets of operators. Positive normal functional are those that are non-negative on positive operators. For every non-zero operator, there is a positive...

mathematics, the symmetric closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest symmetric relation on X {\displaystyle...

\mathbf {N} =\mathbf {A} ^{\mathsf {T}}\mathbf {A} } is symmetric positive definite. Symmetric equation matrix may also come from an energy functional...

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

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

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...

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...

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...