# Dissipative operator

A linear operator defined on a domain which is dense in a Hilbert space such that

This requirement is sometimes replaced by the condition if , i.e. the dissipativeness of in this sense is equivalent to that of the operator .

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 with belong to the resolvent set, and moreover

A dissipative operator is maximal if and only if for all with . An equivalent condition for maximality of a dissipative operator is that it is closed and that

If is a maximal symmetric operator, then either or is a maximal dissipative operator. Dissipative and, in particular, maximal dissipative extensions may be considered for an arbitrary symmetric operator ; their description is equivalent to the description of all maximal dissipative extensions of the conservative operator : , .

Dissipative operators are closely connected with contractions (cf. Contraction) and with the so-called accretive operators, i.e. operators for which is a dissipative operator. In particular, an accretive operator is maximal if and only if is the generating operator (or generator) of a continuous one-parameter contraction semi-group on . The Cayley transform

where is a maximal accretive operator and is a contraction not having 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 the definition of a dissipative operator is equivalent to the requirement , where is the imaginary part of the operator . For a completely-continuous dissipative operator on a separable Hilbert space with nuclear imaginary part , several criteria (i.e. necessary and sufficient conditions) for the completeness of the system of its root vectors are available; for example,

where are all eigen values of the operator , , and is the trace of the operator (Livshits' criterion);

where is the real part of , and is the number of characteristic numbers of the operator in the segment and (Krein's criterion). The system of eigen vectors corresponding to different eigen values , of a dissipative operator forms a basis of its closed linear span and is equivalent to an orthonormal basis if

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

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) |

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Dissipative operator.

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