Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform

Ralph deLaubenfels

Displaying similar documents to “Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform”

Functional calculi, regularized semigroups and integrated semigroups

Ralph deLaubenfels, Mustapha Jazar (1999)

Studia Mathematica

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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...

Spectral projections, semigroups of operators, and the Laplace transform

Ralph deLaubenfels (1997)

Banach Center Publications

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Around Widder’s characterization of the Laplace transform of an element of $L^{\infty} (ℝ^{+})$

Jan Kisyński (2000)

Annales Polonici Mathematici

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Let ϰ be a positive, continuous, submultiplicative function on $ℝ^{+}$ such that $l i m_{t \to \infty} e^{- ω t} t^{- α} ϰ (t) = a$ for some ω ∈ ℝ, α ∈ $\bar{ℝ^{+}}$ and $a \in ℝ^{+}$ . For every λ ∈ (ω,∞) let $ϕ_{λ} (t) = e^{- λ t}$ for $t \in ℝ^{+}$ . Let $L_{ϰ}^{1} (ℝ^{+})$ be the space of functions Lebesgue integrable on $ℝ^{+}$ with weight $ϰ$ , and let E be a Banach space. Consider the map $ϕ_{•} : (ω, \infty) ∋ λ \to ϕ_{λ} \in L_{ϰ}^{1} (ℝ^{+})$ . Theorem 5.1 of the present paper characterizes the range of the linear map $T \to T ϕ_{•}$ defined on $L (L_{ϰ}^{1} (ℝ^{+}); E)$ , generalizing a result established by B. Hennig and F. Neubrander for $ϰ (t) = e^{ω t}$ . If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...

On analytic semigroups and cosine functions in Banach spaces

V. Keyantuo, P. Vieten (1998)

Studia Mathematica

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If A generates a bounded cosine function on a Banach space X then the negative square root B of A generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by B. The characterization relies on new results on the inversion of the vector-valued conjugate potential transform. ...

Norm continuity of $c_{0}$ -semigroups

V. Goersmeyer, L. Weis (1999)

Studia Mathematica

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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.

Joint subnormality of n-tuples and C₀-semigroups of composition operators on L²-spaces

Piotr Budzyński, Jan Stochel (2007)

Studia Mathematica

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Joint subnormality of a family of composition operators on L²-space is characterized by means of positive definiteness of appropriate Radon-Nikodym derivatives. Next, simplified positive definiteness conditions guaranteeing joint subnormality of a C₀-semigroup of composition operators are supplied. Finally, the Radon-Nikodym derivatives associated to a jointly subnormal C₀-semigroup of composition operators are shown to be the Laplace transforms of probability measures (modulo a C₀-group...

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...

Four characterizations of scalar-type operators with spectrum in a half-line

Peter Vieten (1997)

Studia Mathematica

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$C^{0}$ -scalar-type spectrality criterions for operators A whose resolvent set contains the negative reals are provided. The criterions are given in terms of growth conditions on the resolvent of A and the semigroup generated by A. These criterions characterize scalar-type operators on the Banach space X if and only if X has no subspace isomorphic to the space of complex null-sequences.

On α-times integrated C-semigroups and the abstract Cauchy problem

Chung-Cheng Kuo, Sen-Yen Shaw (2000)

Studia Mathematica

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This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{'} (t) = A u (t) + \frac{t^{α - 1}}{Γ (α)} x$ , t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x...

Semiflows and semigroups

Edoardo Vesentini (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Given a compact Hausdorff space $K$ and a strongly continuous semigroup $T$ of linear isometries of the Banach space of all complex-valued, continuous functions on $K$ , the semiflow induced by $T$ on $K$ is investigated. In the particular case in which $K$ is a compact, connected, differentiable manifold, a class of semigroups $T$ preserving the differentiable structure of $K$ is characterized.