The adjoint operator of a bounded operator between Hilbert spaces is the unique bounded operator satisfying

for all and , where the inner products are taken to be conjugate-linear in their first arguments.

In orthonormal bases, the matrix of is the conjugate transpose of the matrix of , so

For arbitrary bases having Gram matrices and , the corresponding formula is

Thus the identification of an adjoint with a conjugate transpose depends on the choice of orthonormal bases.

For bounded operators and for which the product is defined,

For densely defined unbounded linear operators, the domain of the adjoint must also be specified, and product formulas require additional domain hypotheses.

See also

Banach Adjoint, Conjugate Transpose, Hermitian Operator, Inner Product, Self-Adjoint

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References

Conway, J. B. A Course in Functional Analysis. New York: Springer-Verlag, 1990.

Cite this as:

Weisstein, Eric W. "Adjoint Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AdjointOperator.html

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