Difference between revisions of "Potential operator"
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| − | + | A mapping $ A $ | |
| + | of a [[Banach space|Banach space]] $ X $ | ||
| + | into the dual space $ X ^ {*} $ | ||
| + | that is the gradient of some functional $ f \in X ^ {*} $, | ||
| + | i.e. is such that | ||
| − | + | $$ | |
| + | \langle A x , h \rangle = \lim\limits _ { t\rightarrow } 0 \ | ||
| + | |||
| + | \frac{f ( x + t h ) - f ( x) }{t} | ||
| + | . | ||
| + | $$ | ||
| + | |||
| + | For instance, any bounded [[Self-adjoint operator|self-adjoint operator]] $ A $ | ||
| + | defined on a Hilbert space $ H $ | ||
| + | is potential: | ||
| + | |||
| + | $$ | ||
| + | Ax = \mathop{\rm grad} \left \{ | ||
| + | \frac{1}{2} | ||
| + | \langle A x , x \rangle \right \} ,\ \ | ||
| + | x \in H . | ||
| + | $$ | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)</TD></TR></table> | ||
Latest revision as of 08:07, 6 June 2020
A mapping $ A $
of a Banach space $ X $
into the dual space $ X ^ {*} $
that is the gradient of some functional $ f \in X ^ {*} $,
i.e. is such that
$$ \langle A x , h \rangle = \lim\limits _ { t\rightarrow } 0 \ \frac{f ( x + t h ) - f ( x) }{t} . $$
For instance, any bounded self-adjoint operator $ A $ defined on a Hilbert space $ H $ is potential:
$$ Ax = \mathop{\rm grad} \left \{ \frac{1}{2} \langle A x , x \rangle \right \} ,\ \ x \in H . $$
References
| [1] | M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian) |
| [2] | H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974) |
How to Cite This Entry:
Potential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_operator&oldid=48265
Potential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_operator&oldid=48265
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article