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In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule A x , y = x , A ∗ y , {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}
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.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian. [1]
The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits).
In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices. On the vector space C of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form e iθ for θ ∈ R, is a unitary operator.
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian adjoint N*, that is: NN* = N*N. [1] Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood.
Specifically, adjoint or adjunction may mean: Adjoint of a linear map, also called its transpose in case of matrices; Hermitian adjoint (adjoint of a linear operator) in functional analysis; Adjoint endomorphism of a Lie algebra; Adjoint representation of a Lie group; Adjoint functors in category theory; Adjunction (field theory)
Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let T ∗ : dom ( T ∗ ) ⊆ H → G {\displaystyle T^{*}:\operatorname {dom} \left(T^{*}\right)\subseteq H\to G} denote the adjoint of T . {\displaystyle T.}
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