Modern Arabic mathematical notation

2013-07-12T09:47:56Z

83.227.51.194: /* Some features of Arabic mathematical notation */

In [[mathematics]], a '''dissipative operator''' is a [[linear operator]] ''A'' defined on a [[linear subspace]] ''D''(''A'') of [[Banach space]] ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'')
:<math>\|(\lambda I-A)x\|\geq\lambda\|x\|.</math>
A dissipative operator is called '''maximally dissipative''' if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''.

The main importance of dissipative operators is their appearance in the [[Lumer–Phillips theorem]] which characterizes maximally dissipative operators as the generators of [[contraction semigroup]]s.

==Properties==
A dissipative operator has the following properties<ref>Engel and Nagel Proposition II.3.14</ref>
* ''λI'' − ''A'' is [[injective]] for all ''λ'' > 0 and
:::<math>\|(\lambda I-A)^{-1}z\|\leq\frac{1}{\lambda}\|z\|</math>
::for all ''z'' in the range of ''λI'' − ''A''.
* ''λI'' − ''A'' is [[surjective]] for some ''λ'' > 0 if and only if it is surjective for all ''λ'' > 0. (This is the aforementioned maximally dissipative case.) In that case one has (0, ∞) ⊂ ''ρ''(''A'') (the [[resolvent set]] of ''A'').
* ''A'' is a [[closed operator]] if and only if the range of ''λI'' − ''A'' is closed for some (equivalently: for all) ''λ'' > 0.

==Equivalent characterization==
Define the duality set of ''x'' ∈ ''X'', a subset of the [[dual space]] ''X''' of ''X'', by
:<math>J(x):=\left\{x'\in X':\|x'\|_{X'}^2=\|x\|_{X}^2=\langle x',x\rangle \right\}.</math>
By the [[Hahn–Banach theorem]] this set is nonempty. If ''X'' is reflexive, then ''J''(''x'') consists of a single element{{Citation needed|date=November 2011}}. In the [[Hilbert space]] case (using the canonical duality between a Hilbert space and its dual) it consists of the single element ''x''.<ref>Engel and Nagel Exercise II.3.25i</ref>
Using this notation, ''A'' is dissipative if and only if<ref>Engel and Nagel Proposition II.3.23</ref> for all ''x'' ∈ ''D''(''A'') there exists a ''x''<nowiki>'</nowiki> ∈ ''J''(''x'') such that
:<math>{\rm Re}\langle Ax,x'\rangle\leq0.</math>

==Examples==

* For a simple finite-dimensional example, consider ''n''-dimensional [[Euclidean space]] '''R'''''n'' with its usual [[dot product]]. If ''A'' denotes the negative of the [[identity operator]], defined on all of '''R'''''n'', then

::<math>x \cdot A x = x \cdot (-x) = - \| x \|^{2} \leq 0,</math>

: so ''A'' is a dissipative operator.

* So long as the domain of an operator ''A'' is the whole Euclidean space, then it is dissipative if and only if it does not have any eigenvalue with positive real part, and (consequently) all such operators are maximally dissipative. An equivalent condition is that for some (and hence any) positive <math>\lambda, \lambda-A</math> has an inverse and the operator <math>(\lambda+A)(\lambda-A)^{-1}</math> is a contraction (that is, it diminishes the norm of its operand). If the time derivative of a point ''x'' in the space is given by ''Ax'', then the time evolution is governed by a [[contraction semigroup]] that constantly decreases the norm. (Note however that if the domain of ''A'' is a proper subspace, then ''A'' cannot be maximally dissipative because the range will not have a high enough dimensionality.)

* Consider ''H'' = [[Square-integrable function|''L''2]]([0, 1]; '''R''') with its usual inner product, and let ''Au'' = ''u''′ with domain ''D''(''A'') equal to those functions ''u'' in the [[Sobolev space]] ''H''1([0, 1]; '''R''') with ''u''(1) = 0. ''D''(''A'') is dense in ''L''2([0, 1]; '''R'''). Moreover, for every ''u'' in ''D''(''A''), using [[integration by parts]],

::<math>\langle u, A u \rangle = \int_{0}^{1} u(x) u'(x) \, \mathrm{d} x = - \frac1{2} u(0)^{2} \leq 0.</math>

: Hence, ''A'' is a dissipative operator. Furthermore, since there is a solution in ''D'' to <math>u-\lambda u'=f</math> for any ''f'' in ''H'', the operator ''A'' is maximally dissipative. Note that in a case of infinite dimensionality like this, the range can be the whole Banach space even though the domain is only a proper subspace thereof.

* Consider ''H'' = ''H''02(Ω; '''R''') for an [[open set|open]] and [[connected space|connected]] domain Ω &sube; '''R'''''n'' and let ''A'' = Δ, the [[Laplace operator]], defined on the dense subspace of [[compactly supported]] smooth functions on Ω. Then, using integration by parts,

::<math>\langle u, \Delta u \rangle = \int_\Omega u(x) \Delta u(x) \, \mathrm{d} x = - \int_\Omega \big| \nabla u(x) \big|^{2} \, \mathrm{d} x = - \| \nabla u \|_{L^{2} (\Omega; \mathbf{R})} \leq 0,</math>

: so the Laplacian is a dissipative operator. However in this case it is not maximally dissipative because there is no compactly supported solution to <math>u-\lambda \Delta u=f</math> if ''f'' is not compactly supported.

==Notes==

{{reflist|2}}

==References==
*{{ Cite book | last1=Engel| first1=Klaus-Jochen| last2=Nagel| first2=Rainer | title=One-parameter semigroups for linear evolution equations | year=2000| publisher=Springer}}
* {{cite book
| author = Renardy, Michael and Rogers, Robert C.
| title = An introduction to partial differential equations
| series = Texts in Applied Mathematics 13
| edition = Second
|publisher = Springer-Verlag
| location = New York
| year = 2004
| pages = 356
| isbn = 0-387-00444-0
}} (Definition 12.25)

[[Category:Operator theory]]

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Modern Arabic mathematical notation