Semigroups with nonquasianalytic growth

Phóng Vũ

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Spectral inclusions and stability results for strongly continuous semigroups.

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This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators $T^{n} : n = 0, 1, . . .$ on a Banach space X (discrete one-parameter semigroups), one-parameter $C_{0}$ -semigroups $T (t) : t \geq 0$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family...

Functional calculi, regularized semigroups and integrated semigroups

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We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of $A^{n}$ , for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the right half-plane. We construct a semigroup analogue of a spectral distribution for such operators, and an extended functional calculus: When...

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We study a class of closed linear operators on a Banach space whose nonzero spectrum lies in the open left half plane, and for which $0$ is at most a simple pole of the operator resolvent. Our spectral theory based methods enable us to give a simple proof of the characterization of $C_{0}$ -semigroups of bounded linear operators with asynchronous exponential growth, and recover results of Thieme, Webb and van Neerven. The results are applied to the study of the asymptotic behavior of the solutions...

On the spectral bound of the generator of a $C_{0}$ -semigroup

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We give several conditions implying that the spectral bound of the generator of a $C_{0}$ -semigroup is negative. Applications to stability theory are considered.

Corrigendum to “On the spectral bound of the generator of a $C_{0}$ -semigroup” (Studia Math. 125 (1997), 23-33)

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Some statements of the paper [4] are corrected.

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Norm continuity of $c_{0}$ -semigroups

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We show that a positive semigroup $T_{t}$ on $L_{p} (Ω, ν)$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_{p}$ -scale, which may be of independent interest.

On the growth of analytic semigroups along vertical lines

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We construct two Banach algebras, one which contains analytic semigroups ${(a^{z})}_{R e z > 0}$ such that $| a^{1 + i y} | \to \infty$ arbitrarily slowly as $| y | \to \infty$ , the other which contains ones such that $| a^{1 + i y} | \to \infty$ arbitrarily fast

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On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiable $C_{0}$ -semigroups.

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A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

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Let T be a semigroup of linear contractions on a Banach space X, and let $X_{s} (T) = x \in X : l i m_{s \to \infty} ∥ T (s) x ∥ = 0$ . Then $X_{s} (T)$ is the annihilator of the bounded trajectories of T*. If the unitary spectrum of T is countable, then $X_{s} (T)$ is the annihilator of the unitary eigenvectors of T*, and $l i m_{s} ∥ T (s) x ∥ = i n f ∥ x - y ∥ : y \in X_{s} (T)$ for each x in X.