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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]].
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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|$m$-accretive operator]].

Latest revision as of 09:34, 18 July 2025

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=56034
This article was adapted from an original article by A.G. Kartsatos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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