Boundary triplets and maximal accretive extensions of sectorial operators

doi:10.1017/CBO9781139135061.004

3 - Boundary triplets and maximal accretive extensions of sectorial operators

Published online by Cambridge University Press: 05 November 2012

Y. Arlinskiĭ

Edited by

Seppo Hassi ,

Hendrik S. V. de Snoo and

Franciszek Hugon Szafraniec

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Y. Arlinskiĭ: Affiliation:
East Ukrainian National University
Seppo Hassi: Affiliation:
University of Vaasa, Finland
Hendrik S. V. de Snoo: Affiliation:
Rijksuniversiteit Groningen, The Netherlands
Franciszek Hugon Szafraniec: Affiliation:
Jagiellonian University, Krakow

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Summary

Abstract This chapter is a survey of results related to the problem of a description of all maximal accretive extensions for a densely defined sectorial operator; the problem in more generality was originally posed by R. Phillips. We also treat maximal sectorial extensions. Our approach uses the concepts of boundary pairs and boundary triplets.

Introduction

A linear operator S in a complex Hilbert space h is called accretive [Kato, 1995] if Re (Su, u) ≥ 0 for all u ∈ dom S. An accretive operator S is called maximal accretive (m-accretive) if one of the following equivalent conditions is satisfied [Kato, 1995; Lyantse, 1954; Phillips, 1959a, b, 1969]:

the operator S is closed and has no accretive extensions in h;
ρ(S) ∩ ≠ 0, where π_ denotes the open left half-plane;
the operator S is densely defined and closed, and S* is accretive;
the operator −S generates a one-parameter contractive semigroup T(t) = exp(−tS), t ≥ 0.

The resolvent set ρ(S) of an m-accretive operator contains the open left half-plane and

The class of m-accretive operators plays an essential role in differential equations, scattering theory, stochastic processes, passive linear systems and, for instance, hydrodynamics. It should be noted that Phillips calls an operator A dissipative if −A is accretive. Nowadays the term dissipative is used for operators A with Im (Af, f) ≥ 0.

The Phillips problem is to find all m-accretive extensions of a densely defined accretive operator; cf. [Phillips, 1959a, b, 1969].

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Type: Chapter
Information: Operator Methods for Boundary Value Problems , pp. 35 - 72

DOI: https://doi.org/10.1017/CBO9781139135061.004 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

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