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Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates

Masashi Wakaiki

TL;DR

This work analyzes the Crandall–Pazy class of $C_0$-semigroups on Hilbert spaces, providing two sharp characterizations that relate immediate differentiability to resolvent growth and integral resolvent bounds. Using the $\mathcal{B}$-calculus and Lyapunov-equation techniques, the authors derive precise decay rates for Crank–Nicolson schemes under exponential stability in the CP class, and they extend these results to inverse generators, clarifying how bounded $-A^{-1}$ improves rates. A central contribution is bridging decay rates for Cayley-transform products with time-domain decay, yielding optimal or near-optimal rates of $n^{- rac{\alpha}{2-\beta}}$ (and corresponding $t$-domain rates) depending on whether $-A^{-1}$ is bounded. The results provide a cohesive framework linking continuous-time semigroup decay, discrete-time approximations, and inverse-generator dynamics, with implications for stability and convergence analyses in semigroup-based discretizations. Overall, the paper advances understanding of decay transfer from semigroup properties to numerical schemes via sophisticated functional-calculus tools.

Abstract

We investigate immediately differentiable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$ satisfying $\sup_{0 < t <1} t^{1/β}\|Ae^{-tA}\| < \infty$ for some $0 < β\leq 1$. Such $C_0$-semigroups are referred to as the Crandall--Pazy class of $C_0$-semigroups. In the Hilbert space setting, we present two characterizations of the Crandall--Pazy class. We then apply these characterizations to estimate decay rates for Crank--Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable $C_0$-semigroup in the Crandall--Pazy class. The first approach is based on a functional calculus called the $\mathcal{B}$-calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that $-A^{-1}$ generates a bounded $C_0$-semigroup.

Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates

TL;DR

This work analyzes the Crandall–Pazy class of -semigroups on Hilbert spaces, providing two sharp characterizations that relate immediate differentiability to resolvent growth and integral resolvent bounds. Using the -calculus and Lyapunov-equation techniques, the authors derive precise decay rates for Crank–Nicolson schemes under exponential stability in the CP class, and they extend these results to inverse generators, clarifying how bounded improves rates. A central contribution is bridging decay rates for Cayley-transform products with time-domain decay, yielding optimal or near-optimal rates of (and corresponding -domain rates) depending on whether is bounded. The results provide a cohesive framework linking continuous-time semigroup decay, discrete-time approximations, and inverse-generator dynamics, with implications for stability and convergence analyses in semigroup-based discretizations. Overall, the paper advances understanding of decay transfer from semigroup properties to numerical schemes via sophisticated functional-calculus tools.

Abstract

We investigate immediately differentiable -semigroups satisfying for some . Such -semigroups are referred to as the Crandall--Pazy class of -semigroups. In the Hilbert space setting, we present two characterizations of the Crandall--Pazy class. We then apply these characterizations to estimate decay rates for Crank--Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable -semigroup in the Crandall--Pazy class. The first approach is based on a functional calculus called the -calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that generates a bounded -semigroup.

Paper Structure

This paper contains 22 sections, 27 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Characterizations of Crandall--Pazy class of semigroups
  3. Moment inequality
  4. Crandall--Pazy class of semigroups
  5. Decay estimates via the $\mathcal{B}$-calculus
  6. Background material on the $\mathcal{B}$-calculus
  7. Norm estimate for the Crandall--Pazy class of semigroups
  8. Decay estimate for Crank--Nicolson schemes
  9. Holomorphic $C_0$-semigroups on Banach spaces
  10. Crandall--Pazy class of $C_0$-semigroups on Hilbert spaces
  11. Decay estimate for inverse generators
  12. Example
  13. Crank--Nicolson scheme
  14. Inverse generator
  15. Decay estimates when $-A^{-1}$ generates a bounded semigroup
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $-A$ be the generator of a bounded $C_0$-semigroup $(e^{-tA})_{t \geq 0}$ on a Hilbert space $H$ such that $\sigma(A) \cap i \mathbb{R} = \emptyset$. Then the following statements are equivalent for a fixed $\beta \in (0,1]$:

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['thm:decay_growth']}.
  • ...and 47 more