Displaying similar documents to “A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum”

On the invertibility of isometric semigroup representations

C. Batty, D. Greenfield (1994)

Studia Mathematica

Similarity:

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)

Banach Center Publications

Similarity:

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 ≥ 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...

Semigroups with nonquasianalytic growth

Phóng Vũ (1993)

Studia Mathematica

Similarity:

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 → ∞ 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 → ∞ 1 / α ( t ) ∥ T ( t ) f ̂ ( T ) ∥ = 0 , where f ̂ ( T ) = ʃ 0 ∞ 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)

Studia Mathematica

Similarity:

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...

On the exponential stability and dichotomy of C 0 -semigroups

Phóng Vũ (1999)

Studia Mathematica

Similarity:

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.