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For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation. An adjoint operator of the conjugate-linear operator A on a complex Hilbert space H is an conjugate-linear operator A ∗ : H → H with the property:
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [ 1 ] [ 2 ] It is occasionally known as adjunct matrix , [ 3 ] [ 4 ] or "adjoint", [ 5 ] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose .
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.
The last property given above shows that if one views as a linear transformation from Hilbert space to , then the matrix corresponds to the adjoint operator of . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.
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)
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.
For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.
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).