1,720,990 research outputs found
Strong stability of bounded evolution families and semigroups
No full textWe 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 ε X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem 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. © 2002 Elsevier Science (USA)
Rank-1 perturbations of cosine functions and semigroups
No full textLet A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A + B generates a cosine function for each B ∈ L (D ((ω - A)1 / 2), X). If A is unbounded and frac(1, 2) 1 / γ. This is an approximate converse of a perturbation theorem for this class of semigroups. © 2006 Elsevier Inc. All rights reserved
Bounded Laplace transforms, primitives and semigroup orbits
No full textLet f : ℝ+ → ℂ be an exponentially bounded, measurable function whose Laplace transform has a bounded holomorphic extension to the open right half-plane. It is known that there is a constant C such that |∫0t f (s) ds| ≦ C (1 + t) for all t ≧ 0. We show that this estimate is sharp. Furthermore, the corresponding estimates for orbits of C0-semigroups are also sharp
Laplace transforms, non-analytic growth bounds and -semigroups
Get PDFIn this thesis, we study a non-analytic growth bound associated with an exponentially bounded measurable function which measures the extent to which can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of far from the real axis. We study the properties of as well as two associated abscissas, namely the non-analytic abscissa of convergence, and the non-analytic abscissa of absolute convergence . These new bounds may be considered as non-analytic analogues of the exponential growth bound and the abscissas of convergence and absolute convergence of the Laplace transform of and . Analogues of several well known relations involving the growth bound and abscissas of convergence associated with and abscissas of holomorphy of the Laplace transform of are established. We examine the behaviour of under regularisation of by convolution and obtain, in particular, estimates for the non-analytic growth bound of the classical fractional integrals of . The definitions of and extend to the operator-valued case also. For a -semigroup of operators, is closely related to the critical growth bound of . We obtain a characterisation of the non-analytic growth bound of in terms of Fourier multiplier properties of the resolvent of the generator. Yet another characterisation of is obtained in terms of the existence of unique mild solutions of inhomogeneous Cauchy problems for which a non-resonance condition holds. We apply our theory of non-analytic growth bounds to prove some results in which does not appear explicitly; for example, we show that all the growth bounds of a -semigroup coincide with the spectral bound , provided the pseudo-spectrum is of a particular shape. Lastly, we shift our focus from non-analytic bounds to sun-reflexivity of a Banach space with respect to -semigroups. In particular, we study the relations between the existence of certain approximations of the identity on the Banach space \xspace and that of -semigroups on which make sun-reflexive
Spectral conditions for stability of one-parameter semigroups
Get PDFLet {S(t): t≥0} be a C0-semigroup on a Banach space Y with generator B and {T(t): t≥0} be a bounded C0-semigroup on a Banach space X with generator A. Suppose that σ(B) ∩ iR is countable, Pσ(A*) ∩ iR is empty and that there is a bounded linear operator C: Y → X with dense range which intertwines the two semigroups. Then ∥T(t)x∥X → 0 as t → ∞, for each x in X. This generalises results of W. Arendt and the author, Yu. I. Lyubich and Vũ Quôc Phóng, and Falun Huang. © 1996 Academic Press, Inc
On a perturbation theorem of Kaiser and Weis
No full textWe 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. © 2005 Springer
Differentiability of perturbed semigroups and delay semigroups
No full textSuppose that A generates a Co-semigroup T on a Banach space X. In 1953 R. S. Phillips showed that, for each bounded operator B on X, the perturbation A + B of A generates a Co-semigroup on X, and he considered whether certain classes of semigroups are stable under such perturbations. This study was extended in 1968 by A. Pazy who identified a condition on the resolvent of A which is sufficient for the perturbed semigroups to be immediately differentiable. However, M. Renardy showed in 1995 that immediate differentiability is not stable under bounded perturbations.We give a survey account of the partial answers already given to the question of differentiability of perturbed semigroups. Furthermore, we show that Pazy's condition is necessary, as well as sufficient, if one adds a natural requirement of uniformity for the differentiability of the perturbed semigroups. We also present an account of the corresponding theory for delay semigroups associated with A, based on an earlier paper of ours but with improved formulation. The necessary and sufficient condition for eventual differentiability of the delay semigroups is that the resolvent of A should have polynomial decay on vertical lines. We also give a brief account of the consequences for asymptotics of individual mild solutions of abstract Cauchy problems and delay differential equations
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