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Definition. A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator M f of multiplication by the function f (λ) = λ on. where Hn is a Hilbert space of dimension n. The domain of M f consists of vector-valued functions ψ on R such that.
The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators. Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum.
This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When is defined according to this formula, it is called the formal adjoint of T. A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.
Extensions of symmetric operators. In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for ...
The set of self-adjoint elements is referred to as . A subset that is closed under the involution *, i.e. , is called self-adjoint.[2] A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity ( ), which is called a C*-algebra.
Norm-coercive mappings. A mapping between two normed vector spaces and is called norm-coercive if and only if. More generally, a function between two topological spaces and is called coercive if for every compact subset of there exists a compact subset of such that. The composition of a bijective proper map followed by a coercive map is coercive.
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 in the i -th row and j -th column is equal to the complex conjugate of the element in the j -th row and i -th column, for all indices i and j: is Hermitian {\displaystyle A {\text { is ...
For many differential operators, such as , we can show that they are self-adjoint by integration by parts. While these operators may not be compact, their inverses (when they exist) may be, as in the case of the wave equation, and these inverses have the same eigenfunctions and eigenvalues as the original operator (with the possible exception ...