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")
formula_1
A couple of equivalent definitions are given below. 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".
An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator.
The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups.
Properties.
A dissipative operator has the following properties:
formula_5
for all "z" in the range of "λI" − "A". This is the same inequality as that given at the beginning of this article, with formula_6 (We could equally well write these as formula_7 which must hold for any positive κ.)
Equivalent characterizations.
Define the duality set of "x" ∈ "X", a subset of the dual space "X"' of "X", by
formula_8
By the Hahn–Banach theorem this set is nonempty. In the Hilbert space case (using the canonical duality between a Hilbert space and its dual) it consists of the single element "x". More generally, if "X" is a Banach space with a strictly convex dual, then "J"("x") consists of a single element.
Using this notation, "A" is dissipative if and only if for all "x" ∈ "D"("A") there exists a "x"' ∈ "J"("x") such that
formula_9
In the case of Hilbert spaces, this becomes formula_10 for all "x" in "D"("A"). Since this is non-positive, we have
formula_11
formula_12
Since "I−A" has an inverse, this implies that formula_13 is a contraction, and more generally, formula_14 is a contraction for any positive λ. The utility of this formulation is that if this operator is a contraction for some positive λ then "A" is dissipative. It is not necessary to show that it is a contraction for all positive λ (though this is true), in contrast to (λI−A)−1 which must be proved to be a contraction for all positive values of λ.
formula_15
 so "A" is a dissipative operator.
formula_23
 Hence, "A" is a dissipative operator. Furthermore, since there is a solution (almost everywhere) in "D" to formula_24 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.
formula_25
 so the Laplacian is a dissipative operator.
Notes.
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