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Essentially coercive forms and asympotically compact semigroups

W. Arendt, I. Chalendar

Abstract

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that dist(S(t),K(H)) tends to 0 as t tends to infinity, where K(H) denotes the space of all compact operators on the underlying Hilbert space.

Essentially coercive forms and asympotically compact semigroups

Abstract

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that dist(S(t),K(H)) tends to 0 as t tends to infinity, where K(H) denotes the space of all compact operators on the underlying Hilbert space.

Paper Structure

This paper contains 11 sections, 35 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Numerical range and essential coercivity
  3. Selfadjoint, m-accretive and m-quasi-sectorial operators
  4. Selfadjoint operators
  5. m-accretive operators
  6. m-sectorial operators
  7. Structure theorems for essential coerciveness
  8. Asymptotic compactness and compact perturbation of forms
  9. Essentially positive coercive-forms and asymptotic compactness
  10. Essentially coercive forms and holomorphic semigroups
  11. Selfadjoint operators revisited

Key Result

Theorem 1.2

Assume that $a$ is essentially positive-coercive. Then $-A$ generates a holomorphic $C_0$-semigroup $S$ which is asymptotically compact.

Theorems & Definitions (80)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Example 2.4
  • Remark 2.5
  • ...and 70 more