A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum

Charles Batty; Zdzisław Brzeźniak; David Greenfield

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On the invertibility of isometric semigroup representations

C. Batty, D. Greenfield (1994)

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Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $S p_{u} (T, S)$ . For $S = ℤ_{+}^{n}$ or $ℝ_{+}^{n}$ , we establish connections with polynomial convexity.

Almost periodic and strongly stable semigroups of operators

Vũ Phóng (1997)

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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...

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Yu Lyubich, Phóng Vũ (1988)

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Spectral inclusions and stability results for strongly continuous semigroups.

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Semigroups with nonquasianalytic growth

Phóng Vũ (1993)

Studia Mathematica

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We study asymptotic behavior of $C_{0}$ -semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then $l i m_{t \to \infty} 1 / α (t) ∥ T (t) x ∥ = 0$ , ∀x ∈ X. If, moreover, f is a function in $L_{α}^{1} (ℝ_{+})$ which is of spectral synthesis in a corresponding algebra $L_{α_{1}}^{1} (ℝ)$ with respect to (iσ(A)) ∩ ℝ, then $l i m_{t \to \infty} 1 / α (t) ∥ T (t) f ̂ (T) ∥ = 0$ , where $f ̂ (T) = ʃ_{0}^{\infty} f (t) T (t) d t$ . Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of...

A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

Sen Huang (1999)

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Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function $φ_{a} (t) : = φ (α_{t} a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_{w} * (φ_{a})$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_{φ}^{a}$ to be the...

Unitary dilations of multi-parameter semigroups of operators

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On the exponential stability and dichotomy of $C_{0}$ -semigroups

Phóng Vũ (1999)

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A characterization of exponentially dichotomic and exponentially stable $C_{0}$ -semigroups in terms of solutions of an operator equation of Lyapunov type is presented. As a corollary a new and shorter proof of van Neerven’s recent characterization of exponential stability in terms of boundedness of convolutions of a semigroup with almost periodic functions is given.

$C *$ algebras of operators on a half-space

L. A. Coburn, R. G. Douglas (1971)

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Closed semistable operators and singular differential equations

Jaromír J. Koliha, Trung Dinh Tran (2003)

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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...