Dissipative operator

A linear operator $ A $ defined on a domain $ D _ {A} $ which is dense in a Hilbert space $ H $ such that

$$ \mathop{\rm Im} ( A x , x ) \geq 0 \ \textrm{ if } x \in D _ {A} . $$

This requirement is sometimes replaced by the condition $ \mathop{\rm Re} ( A x , x ) \leq 0 $ if $ x \in D _ {A} $, i.e. the dissipativeness of $ A $ in this sense is equivalent to that of the operator $ ( - iA ) $.

A dissipative operator is said to be maximal if it has no proper dissipative extensions. A dissipative operator always has a closure, which also is a dissipative operator; in particular, a maximal dissipative operator is a closed operator. Any dissipative operator can be extended to a maximal dissipative operator. For a dissipative operator all points $ \lambda $ with $ \mathop{\rm Im} \lambda < 0 $ belong to the resolvent set, and moreover

$$ \| A x - \lambda x \| \geq | \mathop{\rm Im} \lambda | \| x \| , \ x \in D _ {A} . $$

A dissipative operator is maximal if and only if $ ( A - \lambda I ) D _ {A} = H $ for all $ \lambda $ with $ \mathop{\rm Im} \lambda < 0 $. An equivalent condition for maximality of a dissipative operator is that it is closed and that

$$ \mathop{\rm Im} ( A ^ {*} y , y ) \leq 0 ,\ y \in D _ {A ^ {*} } . $$

If $ A _ {0} $ is a maximal symmetric operator, then either $ A _ {0} $ or $ ( - A _ {0} ) $ is a maximal dissipative operator. Dissipative and, in particular, maximal dissipative extensions may be considered for an arbitrary symmetric operator $ A _ {0} $; their description is equivalent to the description of all maximal dissipative extensions of the conservative operator $ B _ {0} = iA _ {0} $: $ \mathop{\rm Re} ( B _ {0} x , x ) = 0 $, $ x \in D _ {B} $.

Dissipative operators are closely connected with contractions (cf. Contraction) and with the so-called accretive operators, i.e. operators $ A $ for which $ iA $ is a dissipative operator. In particular, an accretive operator $ A $ is maximal if and only if $ ( - A ) $ is the generating operator (or generator) of a continuous one-parameter contraction semi-group $ \{ T _ {s} \} _ {s \geq 0 } $ on $ H $. The Cayley transform

$$ T = ( A - I ) ( A + I ) ^ {-} 1 ,\ \ A = ( I + T ) ( I - T ) ^ {-} 1 , $$

where $ A $ is a maximal accretive operator and $ T $ is a contraction not having $ \lambda = 1 $ as an eigen value, is used to construct the functional calculus and, in particular, the theory of fractional powers of maximal dissipative operators.

In the case of bounded linear operators $ A $ the definition of a dissipative operator is equivalent to the requirement $ A _ {J} \geq 0 $, where $ A _ {J} = ( A ^ {*} - A ) / 2 i $ is the imaginary part of the operator $ A $. For a completely-continuous dissipative operator $ A $ on a separable Hilbert space $ H $ with nuclear imaginary part $ A _ {J} $, several criteria (i.e. necessary and sufficient conditions) for the completeness of the system of its root vectors are available; for example,

$$ \sum _ {j = 1 } ^ { \nu ( A) } \mathop{\rm Im} \lambda _ {j} ( A) = \ \mathop{\rm tr} A _ {J} , $$

where $ \lambda _ {j} ( A) $ are all eigen values of the operator $ A $, $ j = 1 \dots \nu ( A) \leq \infty $, and $ \mathop{\rm tr} A _ {J} $ is the trace of the operator $ A _ {J} $( Livshits' criterion);

$$ \lim\limits _ {\rho \rightarrow \infty } \frac{n _ {+} ( \rho , A _ {R} ) } \rho = 0 \ \textrm{ or } \ \lim\limits _ {\rho \rightarrow \infty } \frac{n _ {-} ( \rho , A _ {R} ) } \rho = 0 , $$

where $ A _ {R} = ( A + A ^ {*} ) / 2 $ is the real part of $ A $, and $ n _ \pm $ is the number of characteristic numbers of the operator $ A _ {R} $ in the segment $ [ 0 , \rho ] $ and $ [ - \rho , 0 ] $( Krein's criterion). The system $ \{ \psi _ {j} \} $ of eigen vectors corresponding to different eigen values $ \lambda _ {j} $, $ j = 1 , 2 \dots $ of a dissipative operator forms a basis of its closed linear span and is equivalent to an orthonormal basis if

$$ \sum _ {\begin{array}{c} j , k = 1 , \\ j \neq k \end{array} } ^ \infty \frac{ \mathop{\rm Im} \lambda _ {j} \mathop{\rm Im} \lambda _ {k} }{| \lambda _ {j} - \overline \lambda \; _ {k} | } < \infty . $$

The concept of a dissipative operator was also introduced for non-linear and even for multi-valued operators $ A $. Such an operator on a Hilbert space is called dissipative if for any two of its values the inequality

$$ \mathop{\rm Re} ( A x _ {1} - A x _ {2} , x _ {1} - x _ {2} ) \leq 0 $$

holds. This concept also forms the base of the theory of one-parameter non-linear contraction semi-groups and the related differential equations. Another generalization of the concept of a dissipative operator concerns operators acting on a Banach space with a so-called semi-inner product. Another generalization concerns operators acting on a Hilbert space with an indefinite metric.

References

[1]	B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French)
[2]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)
[3]	M.S. Livshits, "On the spectral resolution of linear non-selfadjoint operators" Transl. Amer. Math. Soc. (2) , 5 (1957) pp. 67–114 Mat. Sb. , 34 : 1 (1954) pp. 145–199
[4]	R.S. Phillips, "Dissipative operators and hyperbolic systems of partial differential equations" Trans. Amer. Math. Soc. , 90 : 2 (1959) pp. 193–254
[5]	M. Crandall, A. Pazy, "Semi-groups of nonlinear contractions" J. Funct. Anal. , 3 (1969) pp. 376–418
[6]	G. Lumer, R. Phillips, "Dissipative operators in a Banach space" Pacific J. Math. , 11 (1961) pp. 679–698

Comments

A good reference for dissipative operators on more general spaces than Hilbert spaces is [a1]. For operators on Hilbert spaces see also [a2].

References

[a1]	H.O. Fattorini, "The Cauchy problem" , Addison-Wesley (1983) pp. 120–125; 154–159
[a2]	I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)