On measuring unboundedness of the H∞H^\infty-calculus for generators of analytic semigroups

We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield $b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$.Comment: Preprint of the final, published version. In comparison with previous version, Prop. 2.2 was added and Thm. 3.5 has been slightly adapted in order to point out the major assertio

Strongly Kreiss Bounded Operators in UMD Banach Spaces

In this paper we give growth estimates for $\|T^n\|$ for $n\to \infty$ in the case $T$ is a strongly Kreiss bounded operator on a UMD Banach space $X$. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and $L^p$-spaces, but also intermediate UMD spaces such as non-commutative $L^p$-spaces and variable Lebesgue spaces.Comment: 27 page

The stability of linear operators

In the approximation and solution of both ordinary and partial differential equations by finite difference equations, it is well-known that for different ratios of the time interval to the spatial intervals widely differing solutions are obtained. This problem was first attacked by John von Neumann using Fourier analysis. It has also been studied in the context of the theory of semi-groups of operators. It seemed that the problem could be studied with profit if set in a more abstract structure. The concepts of the stability of a linear operator on a (complex) Banach space and the stability of a Banach sub-algebra of operators were formed in an attempt to generalize the matrix 2 theorems of H.O. Kreiss as applied to the L² stability problem. Chapter 1 deals with the stability and strict stability of linear operators. The equivalence of stability and convergence is discussed in Chapter 2 and special cases of the Equivalence Theorem are considered in Chapters 3 and 4. In Chapter 5 a brief account of the theory of discretizations is given and used to predict instability in non-linear algorithms

Operator Theory and Harmonic Analysis

The major topics discussed in this workshop were the Feichtinger conjecture and related questions of harmonic analysis, the corona problem for the ball Bn, the weighted approximation problem, and questions related to the model spaces, to multipliers, (hyper-)cyclicity, differentiability, Bezout and Fermat equations, traces and Toeplitz operators in different function spaces. A list of open problems raised at this workshop is also included

Non-Normality In Scalar Delay Differential Equations

Thesis (M.S.) University of Alaska Fairbanks, 2006Analysis of stability for delay differential equations (DDEs) is a tool in a variety of fields such as nonlinear dynamics in physics, biology, and chemistry, engineering and pure mathematics. Stability analysis is based primarily on the eigenvalues of a discretized system. Situations exist in which practical and numerical results may not match expected stability inferred from such approaches. The reasons and mechanisms for this behavior can be related to the eigenvectors associated with the eigenvalues. When the operator associated to a linear (or linearized) DDE is significantly non-normal, the stability analysis must be adapted as demonstrated here. Example DDEs are shown to have solutions which exhibit transient growth not accounted for by eigenvalues alone. Pseudospectra are computed and related to transient growth

Fully discrete hyperbolic initial boundary value problems with nonzero initial data

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that the maximal stability estimates that are known for zero initial data and nonzero boundary source term extend to the case of nonzero initial data in \^a 2 . The main novelty of our approach is to cover finite difference schemes with an arbitrary number of time levels. As an easy corollary of our main trace estimate, we recover former stability results in the semigroup sense by Kreiss [Kre68] and Osher [Osh69b]

Resolvent conditions and growth of powers of operators

Following Berm\'udez et al. (ArXiv: 1706.03638v1), we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Ces\`aro boundedness assumptions. We show that $T$ is power-bounded if (and only if) both $T$ and $T^*$ are absolutely Ces\`aro bounded. In Hilbert spaces, we prove that if $T$ satisfies the Kreiss condition, $\|T^n\|=O(n/\sqrt {\log n})$; if $T$ is absolutely Ces\`aro bounded, $\|T^n\|=O(n^{1/2 -\varepsilon})$ for some $\varepsilon >0$ (which depends on $T$); if $T$ is strongly Kreiss bounded, then $\|T^n\|=O((\log n)^\kappa)$ for some $\kappa >0$. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Ces\`aro means of order $\alpha$ converge strongly when $\alpha >1$.Comment: Added references [35] and [38] and updated some remark