Adjoint operator

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A linear operator $A^*\colon Y^* \rightarrow X^*$ (where $X^*$ and $Y^*$ are the strong duals of locally convex spaces $X$ and $Y$, respectively), constructed from a linear operator $A\colon X \rightarrow Y$ in the following way. Let the domain of definition $D_A$ of $A$ be everywhere dense in $X$. If for all $x \in D_A$, $$ (Ax, g) = (x, g^*) \label{eq:1} $$ where $Ax \in Y$, $g \in Y^*$ and $g^* \in X^*$, then $A^*g = g^*$ is a uniquely defined operator from the set $D_{A^*}$ of elements $g$ satisfying \eqref{eq:1} into $X^*$. If $D_A = X$ and $A$ is continuous, then $A^*$ is also continuous. If, in addition, $X$ and $Y$ are normed linear spaces, then $\Vert A^* \Vert = \Vert A \Vert$. If $A$ is completely continuous, then so is $A^*$. Adjoint operators are of particular interest in the case when $X$ and $Y$ are Hilbert spaces.


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 $$ (Ax,g) = (x,A^*g) $$ where $({\cdot},{\cdot})$ denotes the Hilbert space inner product.


[1] K. Yosida, "Functional analysis" , Springer (1980)
[2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[3] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
How to Cite This Entry:
Adjoint operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article