Search

This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that limn→∞ ‖Tn(I - T)‖ = 0 if T is a power-bounded operator on a Banach space and σ(T) ∩ T ⊆ {1}. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory. [ABSTRACT FROM AUTHOR]

We study the rate of energy decay for solutions of a coupled wave-heat system on a rectangular domain. Using techniques from the theory of C0-semigroups, and in particular a well-known result due to Borichev and Tomilov, we prove that the energy of classical solutions decays like t-2/3 as t → ∞. This rate is moreover shown to be sharp. Our result implies in particular that a general estimate in the literature, which predicts at least logarithmic decay and is known to be best possible in general, is suboptimal in the special case under consideration here. Our strategy of proof involves direct estimates based on separation of variables and a refined version of the technique developed in our earlier paper for a one-dimensional wave-heat system. [ABSTRACT FROM AUTHOR]

We obtain integral representations for the resolvent of ψ ( A ) , where ψ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and A is a sectorial operator on a Banach space. As a corollary, for a wide class of functions ψ , we show that the operator − ψ ( A ) generates a sectorially bounded holomorphic C 0 -semigroup on a Banach space whenever − A does, and the sectorial angle of A is preserved. When ψ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for A can be described, at least on Hilbert spaces, in terms of the existence of a bounded H ∞ -calculus for A . As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators. [ABSTRACT FROM AUTHOR]

Motivated by potential applications to partial differential equations, we develop a theory of fine scales of decay rates for operator semigroups. The theory contains, unifies, and extends several notable results in the literature on decay of operator semigroups and yields a number of new ones. Its core is a new operator-theoretical method of deriving rates of decay combining ingredients from functional calculus and complex, real and harmonic analysis. It also leads to several results of independent interest. [ABSTRACT FROM AUTHOR]

We prove L p -analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive L p -decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the L p -rates of decay obtained in this way are best possible under our assumptions. [ABSTRACT FROM AUTHOR]

This paper pursues the 'formal darwinism' project of Grafen, whose aim is to construct formal links between dynamics of gene frequencies and optimization programmes, in very abstract settings with general implications for biologically relevant situations. A major outcome is the definition, within wide assumptions, of the ubiquitous but problematic concept of 'fitness'. This paper is the first to present the project for mathematicians. Within the framework of overlapping generations in discrete time and no social interactions, the current model shows links between fitness maximization and gene frequency change in a class-structured population, with individual-level uncertainty but no uncertainty in the class projection operator, where individuals are permitted to observe and condition their behaviour on arbitrary parts of the uncertainty. The results hold with arbitrary numbers of loci and alleles, arbitrary dominance and epistasis, and make no assumptions about linkage, linkage disequilibrium or mating system. An explicit derivation is given of Fisher's Fundamental Theorem of Natural Selection in its full generality. [ABSTRACT FROM AUTHOR]

Abstract: We obtain new stability results for those properties of -semigroups which admit characterisation in terms of decay of resolvents of infinitesimal generators on vertical lines, e.g. analyticity, Crandall–Pazy differentiability or immediate norm continuity in the case of Hilbert spaces. As a consequence we get a generalisation of the Kato–Neuberger theorem on approximation of the identity. Finally, we present examples shedding a new light on resolvent characterisation of eventually differentiable -semigroups for which differentiability is stable under bounded perturbations. [Copyright &y& Elsevier]

Abstract: We introduce the notion of quasi-hyperbolic operators and -semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a -semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property. [Copyright &y& Elsevier]

Abstract: Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, generates a cosine function for each . If A is unbounded and , then we show that there exists a rank-1 operator such that does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If generates a distribution semigroup for each operator of rank-1, then A generates a holomorphic -semigroup. If generates a -semigroup for each operator of rank-1 where , then the semigroup T generated by A is differentiable and as for any . This is an approximate converse of a perturbation theorem for this class of semigroups. [Copyright &y& Elsevier]

We improve a perturbation theorem of C. Kaiser and L. Weis for semigroups of operators on Hilbert spaces by using a generation theorem of A.M. Gomilko, D.H. Shi and D.X. Feng. [ABSTRACT FROM AUTHOR]

Some new and quite general conditions are presented to ensure equality of the spectral bound and the growth bound for those strongly continuous semigroups governing second-order (in time) equations in a Banach (or Hilbert) space. A counterexample arising from higher-order PDEs is given to show that the growth bound could be arbitrarily large while the spectral bound remains zero. [Copyright &y& Elsevier]

We consider the question of eventual differentiability of the delay semigroups associated with the retarded equation , where is the history function, A generates an immediately norm-continuous semigroup and Φ is bounded. We show that this is determined by the rate of decay of the resolvent of A along vertical lines. [Copyright &y& Elsevier]

The non-analytic growth bound ζ(T) of a C0-semigroup T measures the extent to which T can be approximated by a holomorphic function, and it is related to spectral properties of the generator A in regions of C far from the real axis. We show that ζ(T) can be characterised by means of Fourier multiplier properties of the resolvent of A far from the real axis, and also by existence and uniqueness of mild solutions of inhomogeneous Cauchy problems of the form u′(t)=Au(t)+f(t) on R where the Carleman spectra of f and u are far from the origin. The corresponding results for the exponential growth bound ω0(T) have been established earlier by other authors. [Copyright &y& Elsevier]

We prove several characterizations of strong stability of uniformly bounded evolution families (U(t,s))t⩾s⩾0 of bounded operators on a Banach space X, i.e. we characterize the property limt→∞ ∥U(t,s)x∥=0 for all s⩾0 and all x∈X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problemu(t)=A(t)u(t),t⩾s⩾0,u(s)=x,x∈X.In the autonomous case, i.e. when U(t,s)=T(t−s) for some C0-semigroup (T(t))t⩾0, we present, in addition, a range condition on the generator A of (T(t))t⩾0 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A. [Copyright &y& Elsevier]

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra B of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of B and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory. [ABSTRACT FROM AUTHOR]

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R+: u′(t)+Au(t)+φ(t)(t≥0). Suppose that u has uniformly convergent means, σ(A)∩iR is countable, and φ is asymptotically almost periodic. Then u is asymptotically almost periodic. Related results have been obtained by Ruess and Vũ, and by Basit, using different methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Räbiger, and applications to Volterra equations are discussed. 1991 Mathematics Subject Classification 34C28, 44A10, 47D03. [ABSTRACT FROM AUTHOR]