# 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