Difference between revisions of "M-dissipative-operator"
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| + | Let $X$ be a real [[Banach space|Banach space]] with dual space $X ^ { * }$ and normalized duality mapping $J$ (cf. also [[Duality|Duality]]; [[Adjoint space|Adjoint space]]). An operator $T : X \supset D ( T ) \rightarrow 2 ^ { X }$ is called dissipative if for every $x , y \in D ( T )$ and every $u \in T x , v \in T y$ there exists a $j \in J ( x - y )$ such that $\langle u - v , j \rangle \leq 0$ (cf. also [[Dissipative operator|Dissipative operator]]). A dissipative operator $T$ is called $m$-dissipative if $\lambda I - T$ is surjective for all $\lambda > 0$. Thus, an operator $T$ is dissipative (respectively, $m$-dissipative) if and only if the operator $- T$ is accretive (respectively, $m$-accretive). For more information, see [[Accretive mapping|Accretive mapping]] and [[M-accretive-operator|$m$-accretive operator]]. | ||
Latest revision as of 16:58, 1 July 2020
Let $X$ be a real Banach space with dual space $X ^ { * }$ and normalized duality mapping $J$ (cf. also Duality; Adjoint space). An operator $T : X \supset D ( T ) \rightarrow 2 ^ { X }$ is called dissipative if for every $x , y \in D ( T )$ and every $u \in T x , v \in T y$ there exists a $j \in J ( x - y )$ such that $\langle u - v , j \rangle \leq 0$ (cf. also Dissipative operator). A dissipative operator $T$ is called $m$-dissipative if $\lambda I - T$ is surjective for all $\lambda > 0$. Thus, an operator $T$ is dissipative (respectively, $m$-dissipative) if and only if the operator $- T$ is accretive (respectively, $m$-accretive). For more information, see Accretive mapping and $m$-accretive operator.
How to Cite This Entry:
M-dissipative-operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M-dissipative-operator&oldid=50267
M-dissipative-operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M-dissipative-operator&oldid=50267
This article was adapted from an original article by A.G. Kartsatos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article