Adjoint operator

For a bounded linear map between inner-product spaces, its adjoint is the unique map that transfers the operator across the inner product: ⟨Tx,y⟩ = ⟨x,T*y⟩, with the convention adjusted to the inner product’s linearity.

Inner product: A positive-definite sesquilinear form that defines lengths and angles in a vector space. The adjoint is defined by moving an operator from one slot of this pairing to the other.
Bounded linear operator: A linear map between normed spaces whose output norms are bounded by a constant times input norms. Boundedness ensures a well-defined adjoint between Hilbert spaces.
Normal operator: An operator that commutes with its adjoint. Normality organizes operators whose action and adjoint action are compatible.
Transpose map: The linear map between algebraic dual spaces induced by a linear map. Unlike Hilbert-space adjoints, it needs no inner product and may act on functionals rather than vectors.
Dual space: The vector space of linear functionals on a given vector space. The adjoint is induced by the dual map after inner products identify vectors with functionals.
Hilbert space: A complete inner-product space. Completeness supports the representation theorem used to construct adjoints.
Self-adjoint operator: An operator equal to its adjoint on the same inner-product space. It is the central special case in spectral theory and mathematical physics.
Adjoint of an unbounded operator: The densely defined operator characterized by the inner-product identity for an unbounded operator. Its domain is a substantive part of the definition, unlike the bounded case.
Riesz representation theorem: A theorem identifying continuous linear functionals on a Hilbert space with inner products against unique vectors. It turns the dual map into an operator acting on vectors.
Kernel of a linear map: The subspace of inputs mapped to zero by a linear map. The adjoint’s kernel is orthogonal to the original operator’s range when the range is suitably closed.

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