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) |
Adjoint operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_operator&oldid=38989