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  2. Self-adjoint operator - Wikipedia

    en.wikipedia.org/wiki/Self-adjoint_operator

    In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.

  3. Friedrichs extension - Wikipedia

    en.wikipedia.org/wiki/Friedrichs_extension

    Example.Multiplication by a non-negative function on an L 2 space is a non-negative self-adjoint operator.. Example.Let U be an open set in R n.On L 2 (U) we consider differential operators of the form

  4. Extensions of symmetric operators - Wikipedia

    en.wikipedia.org/wiki/Extensions_of_symmetric...

    An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of ) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all.

  5. Unbounded operator - Wikipedia

    en.wikipedia.org/wiki/Unbounded_operator

    An operator is called essentially self-adjoint if its closure is self-adjoint. [40] An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension. [24] A symmetric operator may have more than one self-adjoint extension, and even a continuum of them. [26] A densely defined, symmetric operator T is ...

  6. Lagrange's identity (boundary value problem) - Wikipedia

    en.wikipedia.org/wiki/Lagrange's_identity...

    In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory.

  7. Operator theory - Wikipedia

    en.wikipedia.org/wiki/Operator_theory

    In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.

  8. Hermitian matrix - Wikipedia

    en.wikipedia.org/wiki/Hermitian_matrix

    Self-adjoint operatorLinear operator equal to its own adjoint; Skew-Hermitian matrix – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix) Unitary matrix – Complex matrix whose conjugate transpose equals its inverse; Vector space – Algebraic structure in linear algebra

  9. Self-adjoint - Wikipedia

    en.wikipedia.org/wiki/Self-adjoint

    The set of self-adjoint elements is a real linear subspace of . From the previous property, it follows that A {\displaystyle {\mathcal {A}}} is the direct sum of two real linear subspaces, i.e. A = A s a ⊕ i A s a {\displaystyle {\mathcal {A}}={\mathcal {A}}_{sa}\oplus \mathrm {i} {\mathcal {A}}_{sa}} .