A functional model for a family of operators induced by Laguerre operator

Hatamleh Ra'ed

Displaying similar documents to “A functional model for a family of operators induced by Laguerre operator”

Is $A^{- 1}$ an infinitesimal generator?

Hans Zwart (2007)

Banach Center Publications

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In this paper we study the question whether $A^{- 1}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{- 1}$ . However, we construct a contraction semigroup with growth bound minus infinity for which $A^{- 1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its...

Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter Trebels, Ursula Westphal (2010)

Banach Center Publications

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Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to $(X, D ({(- A)}^{α}))$ , α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, ${| | λ R (λ; A) f | |}_{Y} {\leq c φ (1 / λ) | | f | |}_{X}$ , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has...

On the theory of remediability

Hassan Emamirad (2003)

Banach Center Publications

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Suppose ${G ₁ (t)}_{t \geq 0}$ and ${G ₂ (t)}_{t \geq 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators...

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Sameer Chavan (2011)

Studia Mathematica

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The Cauchy dual operator T’, given by $T {(T * T)}^{- 1}$ , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1...

Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski (2002)

Studia Mathematica

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We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (S) of complex-valued functions on the free semigroup S = FS(z,w) on two generators. First, to given probability measures μ, ν with all moments finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (ℬ,ε,Δ) on two self-adjoint generators X,X’ and a projection P. Then we introduce and study cumulants which are additive under the convolution μ̂* ν̂ = μ̂ ⊗...

Linear modulus of a multidimensional semigroup of $L_{1}$ -contractions and a differentiation theorem

Dogan Çömez (1989)

Colloquium Mathematicae

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Complete Pick positivity and unitary invariance

Angshuman Bhattacharya, Tirthankar Bhattacharyya (2010)

Studia Mathematica

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The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_{S} (z, w) = {(1 - z w ̅)}^{- 1}$ for |z|,|w| < 1, by means of $(1 / k_{S}) (T, T *) \geq 0$ , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense...

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

Ralph deLaubenfels (1992)

Studia Mathematica

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Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup ${e^{- s A}}_{s \leq 0}$ such that ${(1 / s^{2}) e^{- s A}}_{s > 0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{- s A}}_{s \geq 0}$ and ∃ M < ∞ such that $∥ H_{n} (s) ∥ \equiv ∥ (\sum_{k = 0}^{n} (s^{k} A^{k}) / k!) e^{- s A} ∥ \leq M$ , ∀s > 0, n ∈ ℕ ∪ 0. (4) -A generates a strongly continuous holomorphic semigroup ${e^{- z A}}_{R e (z) > 0}$ that is O(|z|)...

A $C^{*}$ -algebraic Schoenberg theorem

Ola Bratteli, Palle E. T. Jorgensen, Akitaka Kishimoto, Donald W. Robinson (1984)

Annales de l'institut Fourier

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Let $𝔄$ be a $C^{*}$ -algebra, $G$ a compact abelian group, $τ$ an action of $G$ by $*$ -automorphisms of $𝔄, 𝔄^{τ}$ the fixed point algebra of $τ$ and $𝔄_{F}$ the dense sub-algebra of $G$ -finite elements in $𝔄$ . Further let $H$ be a linear operator from $𝔄_{F}$ into $𝔄$ which commutes with $τ$ and vanishes on $𝔄^{τ}$ . We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_{0}$ -semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative...

Locally adequate semigroup algebras

Yingdan Ji, Yanfeng Luo (2016)

Open Mathematics

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We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant [...] 0-J* $0 - 𝒥 *$ -simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the [...] ℛ* $ℛ *$ -classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we...

Some model theory of SL(2,ℝ)

Jakub Gismatullin, Davide Penazzi, Anand Pillay (2015)

Fundamenta Mathematicae

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We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space $S_{G} (M)$ . We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on $S_{G} (M)$ ). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

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...

On semigroups with an infinitesimal operator

Jolanta Olko (2005)

Annales Polonici Mathematici

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Let $F^{t} : t \geq 0$ be an iteration semigroup of linear continuous set-valued functions. If the semigroup has an infinitesimal operator then it is a uniformly continuous semigroup majorized by an exponential semigroup. Moreover, for sufficiently small t every linear selection of $F^{t}$ is invertible and there exists an exponential semigroup $f^{t} : t \geq 0$ of linear continuous selections $f^{t}$ of $F^{t}$ .

A Class of Contractions in Hilbert Space and Applications

Nick Dungey (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${(T ⁿ)}_{n = 1, 2, . . .}$ by the continuous semigroup ${(e^{- t (I - T)})}_{t \geq 0}$ . Moreover, we give a stronger quadratic form inequality which ensures that $s u p n ∥ T ⁿ - T^{n + 1} ∥ : n = 1, 2, . . . < \infty$ . The results apply to large classes of Markov operators on countable spaces or on locally compact groups.