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|Let A : D(A)→X be a densely defined dissipative operator. Then A is m-|
dissipative if and only if A is closed and A∗ is dissipative. Proof. Suppose that A
is m-dissipative. According to Proposition 3.1.9, A is closed and according to
|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 C,(H) such that G(A) = (|
A — K). " for any X > 0. K is called the infinitesimal generator of (P). It is clearly m-
dissipative” in C,(H) since, O-K)'''.< ||f|, A-0 recoil). It is an interesting problem ...
|Perturbation results for m-dissipative operators In many cases the operator under |
consideration can be viewed as a perturbation of an m-dissipative operator.
Therefore it is important to know whether the sum of an m-dissipative operator
|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 ...
|2.2. Definition. and. main. properties. of. m-dissipative. operators. Definition 2.2.1. |
An operator A in X is dissipative if ||u-Aj4u|| > H|, for all u e D(A) and all A > 0.
Definition 2.2.2. An operator A in X is m-dissipative if (i) A is dissipative; (ii) for all
|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 ...
|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