# Adjoint operator

A linear operator (where and are the strong duals of locally convex spaces and , respectively), constructed from a linear operator in the following way. Let the domain of definition of be everywhere dense in . If for all ,

(*) |

where , and , then is a uniquely defined operator from the set of elements satisfying (*) into . If and is continuous, then is also continuous. If, in addition, and are normed linear spaces, then . If is completely continuous, then so is . Adjoint operators are of particular interest in the case when and are Hilbert spaces.

#### References

[1] | K. Yosida, "Functional analysis" , Springer (1980) |

[2] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |

#### Comments

In Western literature the adjoint operator as defined above is usually called the dual or conjugate operator. The term adjoint operator is reserved for Hilbert spaces, in which case it is defined by

where denotes the Hilbert space inner product.

#### References

[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |

**How to Cite This Entry:**

Adjoint operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Adjoint_operator&oldid=16759