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|H. Isolated Singularities of the Resolvent Let T be a densely defined closed linear |
operator in a complex Banach space X and '10 an .... If X is a Hilbert space, then
conversely a maximal dissipative operator with dense domain is m-dissipative.
|Consequently, there exists a unique closed operator K in Cb(H) such that G()\) I (|
A — K)'1 for any X > 0. ... An m-dissipative operator is always maximal dissipative
(that is, it cannot be extended to a dissipative operator on a larger domain), ...
|We say that A is accretive if — A is dissipative. We shall compare this definition |
with Definition 1 (the case where X is a Hilbert space). Definition 6. m-dissipative;
m-strongly dissipative; m-accretive operators We say that an operator A with ...
|In particular, a maximal dissipative operator on the whole space X is called |
simply a maximal dissipative operator. (ii) A dissipative operator A with R(I~XA) =
X (2.19) for all X > 0 is called an m-dissipative operator. Lemma 2.12. (i) Let Sx c
|A linear operator is dissipative if and only if, for all u S D(A), and A > 0 we have ||(|
A/-A)u|U>||u|U. (2.6.11) This shows that a dissipative operator has a closed range
. Dissipative operators satisfying R(I - A) = X (2.6.12) are called m-dissipative.
|Recall that the operator T in the Hilbert space H is called dissipative if Im (Tf,f) ≥ |
0 for all f ∈ Dom(T). A dissipative operator T is called maximal dissipative (m-
dissipative) if one of the equivalent conditions is satisfied: • T has no dissipative ...
|By Proposition 3.1.3, A has a closed and dissipative extension A . By (3) we have |
A = A, so that A is closed. ... A dissipative operator is called maximal dissipative (
for brevity, m- dissipative) if it has one (hence, all) the properties listed in ...
|Any dissipative operator is closable. The dissipative operator ^f is called m- |
dissipative if the range of A — ,e/ coincides with JV for some (and consequently
for any) A > 0. An operator stf with dense domain is m-dissipative if and only if it is
|of a J —dissipative operator. In this definition it is required that Im [Au, u] 2 0 for |
all u G D(A). 3. If A is a maximal uniformly dissipative (J —dissipative) operator
then iR G p(A) (see [8, Chap. 2, Sect. 2, Prop. 232]). 4. IfA is m-dissipative ...
|To state such a theorem, we now introduce the concept of dissipative operator |
which is borrowed from the case where X is a ... A linear operator A : X D D(A) —
> R(A) C X is m-dissipative if A is dissipative and R(XI - A) = X for all X > 0 that is,