Let be a Hilbert spaceMathworldPlanetmath and let A:𝒟(A) be a densely defined linear operatorMathworldPlanetmath. Suppose that for some y, there exists z such that (Ax,y)=(x,z) for all x𝒟(A). Then such z is unique, for if z is another element of satisfying that condition, we have (x,z-z)=0 for all x𝒟(A), which implies z-z=0 since 𝒟(A) is dense ( Hence we may define a new operator A*:𝒟(A*) by

𝒟(A*)= {y:there iszsuch that(Ax,y)=(x,z)},
A*(y)= z.

It is easy to see that A* is linear, and it is called the adjointPlanetmathPlanetmathPlanetmath of A.

Remark. The requirement for A to be densely defined is essential, for otherwise we cannot guarantee A* to be well defined.

Title adjoint
Canonical name Adjoint
Date of creation 2013-03-22 13:48:09
Last modified on 2013-03-22 13:48:09
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Definition
Classification msc 47A05
Synonym adjoint operator
Related topic TransposeOperator