Semigroups of analytic functions and composition operators.

Berkson, Earl; Porta, Horácio

doi:10.1307/mmj/1029002009

Cited by 219 publications

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“…This connection between composition operators and operator semigroups opens the possibility of studying spectral properties, operator ideal properties or dynamical properties of the semigroup of operators (C t ) in terms of the theory of functions. The paper [3] can be considered as the starting point in this direction.…”

Section: F ∈ H(d)mentioning

confidence: 99%

Semigroups of composition operators and integral operators in spaces of analytic functions

Contreras¹,

Díaz-Madrigal²,

Martínez³

et al. 2013

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study the maximal spaces of strong continuity on BM OA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V M OA or the little Bloch B 0 . These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator T g . For the second characterization we prove and use an independent result, namely that the operators T g are weakly compact on the above mentioned spaces if and only if they are compact.

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Section: F ∈ H(d)mentioning

confidence: 99%

Semigroups of composition operators and integral operators in spaces of analytic functions

Contreras¹,

Díaz-Madrigal²,

Martínez³

et al. 2013

Ann. Acad. Sci. Fenn. Math.

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“…A parametric representation of such generators can be found in [5]. Earlier, in the one-dimensional case, similar facts were presented by E. Berkson and H. Porta in their study [14] of linear C 0 -semigroups of composition operators on Hardy spaces. E. Vesentini investigated semigroups of those fractional-linear transformations which are isometries with respect to the infinitesimal hyperbolic metric on the unit ball of a Banach space (see [83][84][85]).…”

Section: Introductionmentioning

confidence: 56%

“…In other words, a semigroup of holomorphic self-mappings of D is right differentiable at t = 0 + if and only if it is locally uniformly right continuous at t = 0 + (see [14], [3], [70]). Note also that in this case g ∈ Hol(D, X) defined by (3.1.3) is bounded on each subset strictly inside D. Finally, we observe that if…”

Section: Semicomplete Vector Fields and Semigroups Of Holomorphic Mapmentioning

confidence: 99%

Complex dynamical systems and the geometry of domains in Banach spaces

Elin

Reich

Shoikhet

2004

Dissertationes Math.

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consisting of all the holomorphic self-mappings of D, is a semigroup with respect to composition. Definition 1.1.2. A mapping f ∈ Hol(D, Ω) is said to be univalent on D if for each pair of distinct points x 1 and x 2 in D we have f (x 1 ) = f (x 2 ).In this case one can define the inverse mappingIt is well known (see, for example, [39]) that if X = C n and Y = C m are finite dimensional complex spaces, then f :However, this fact is no longer true in the infinite dimensional case (see counterexamples in [79,4]). Therefore, in the general case a univalent mappingIt is also known (see, for example, [29,49]is a linear isomorphism between X and Y . In this situation we will say that D and Ω are biholomorphically equivalent.As we have already mentioned, in this case X and Y must be linearly isomorphic Banach spaces. Generally speaking, the converse is not true. That is, even if X and Y are isomorphic Banach spaces and f ∈ Hol(D, Y ) has at each point x in D a continuously invertible Fréchet derivative, the mapping f need not be univalent on D.Nevertheless, in this case the mapping f is biholomorphic on a neighborhood of each x ∈ D by the inverse function theorem. In this situation we will say that f ∈ Hol(D, Ω) is locally biholomorphic.The set of all univalent mappings from a domain D ⊂ X into X will be denoted by Univ(D). For the special case when D is the open unit ball of X, the subset of Univ(D) normalized by the conditions f (0) = 0 and f (0) = I will be denoted by S(D). This notation conforms to the one used in the classical onedimensional case, when D = ∆ = {z ∈ C : |z| < 1}. In this case we simply writeThat is, S consists of all the mappings f ∈ Univ(∆) such that f has the following Taylor series at the origin: Convex, star-shaped and spiral-shaped domainsDefinition 1.2.1. A set M in X is called convex if for each pair of points w 1 and w 2 in M , the line segment joining w 1 and w 2 is contained in M , i.e., for each t ∈ [0, 1], the point w = (1 − t)w 1 + tw 2 is also in M .If D is a domain in X, then f ∈ Univ(D) is said to be a convex mapping on D if its image f (D) = Ω is a convex domain in X.Geometry of domains in Banach spaces 9 Definition 1.2.2. A set M in X is called star-shaped (with respect to the origin) if given any w ∈ M , the point w t = tw also belongs to M for every t with 0 ≤ t ≤ 1. That is, if M contains w, then it also contains the entire line segment joining w to the origin. In this definition the origin is in Ω. If, in particular, the origin belongs to Ω, then we will say that f is star-like, to make our definition consistent with the classical one. In this case the mapping f has a null point τ in D. The set of all biholomorphic mappings on D which are star-shaped on D will be denoted by Star(D).If, in addition, there is τ ∈ D such that f (τ ) = 0, (1.2.1) then we will write f ∈ S * τ (D). Of course, in this case such a point τ is unique because. Again, in the one-dimensional case, when X = C, we will simply write S * to denote the family of all biholomorphic (univalent) star-like functions...

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“…The theory of semigroups of weighted composition operators on Hardy and Dirichlet spaces of the unit disc D is well understood (see [1,[3][4][5][6]12,15,16]), and these have generators that are first order differential operators of the form A : f → G 0 f + G 1 f .…”

Section: Introductionmentioning

confidence: 99%

A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces

Chalendar¹,

Partington

2016

Semigroup Forum

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This paper studies semigroups of operators on Hardy and Dirichlet spaces whose generators are differential operators of order greater than one. The theory of forms is used to provide conditions for the generation of semigroups by second order differential operators. Finally, a class of more general weighted Hardy spaces is considered and necessary and sufficient conditions are given for an operator of the form f → G f (n 0 ) (for holomorphic G and arbitrary n 0 ) to generate a semigroup of quasicontractions.

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Semigroups of analytic functions and composition operators.

Cited by 219 publications

References 0 publications

Semigroups of composition operators and integral operators in spaces of analytic functions

Semigroups of composition operators and integral operators in spaces of analytic functions

Complex dynamical systems and the geometry of domains in Banach spaces

A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces

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