Holomorphic subordinated semigroups (original) (raw)

Generation of generators of holomorphic semigroups

Journal of the Australian Mathematical Society, 1993

We construct a functional calculus, g H> g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (-oo, 0), with [\\r(r + A)' 1 1| | r > 0} bounded. For such functions g, we show that -g(A) generates a bounded holomorphic strongly continuous semigroup of angle 6, whenever -A does.

Generation of Subordinated Holomorphic Semigroups via Yosida’s Theorem

Operator Theory: Advances and Applications, 2015

Using functional calculi theory, we obtain several estimates for ψ(A)g(A) , where ψ is a Bernstein function, g is a bounded completely monotone function and −A is the generator of a holomorphic C0-semigroup on a Banach space, bounded on [0, ∞). Such estimates are of value, in particular, in approximation theory of operator semigroups. As a corollary, we obtain a new proof of the fact that −ψ(A) generates a holomorphic semigroup whenever −A does, established recently in [8] by a different approach.

On subordination of holomorphic semigroups

Advances in Mathematics, 2015

We prove that for any Bernstein function ψ the operator −ψ(A) generates a holomorphic C0-semigroup (e −tψ(A) ) t≥0 on a Banach space, whenever −A does. This answers a question posed by Kishimoto and Robinson. Moreover, giving a positive answer to a question by Berg, Boyadzhiev and de Laubenfels, we show that (e −tψ(A) ) t≥0 is holomorphic in the holomorphy sector of (e −tA ) t≥0 , and if (e −tA ) t≥0 is sectorially bounded in this sector then (e −tψ(A) ) t≥0 has the same property. We also obtain new sufficient conditions on ψ in order that, for every Banach space X, the semigroup (e −tψ(A) ) t≥0 on X is holomorphic whenever (e −tA ) t≥0 is a bounded C0-semigroup on X. These conditions improve and generalize well-known results by Carasso-Kato and Fujita.

Semigroups of Holomorphic Functions

Springer Monographs in Mathematics, 2020

In this chapter we introduce the primary subject of our study: continuous one-parameter semigroups of holomorphic self-maps of the unit disc. We establish their main basic properties and extend to this context the Denjoy-Wolff Theorem. Then we characterize groups of automorphisms and more generally of linear fractional self-maps of the unit disc. We also briefly consider continuous semigroups of holomorphic self-maps of C and C ∞ , proving that they reduce to groups of Möbius transformations, and we explain why a non-trivial theory of continuous semigroups of holomorphic maps only makes sense for self-maps of the unit disc. 8.1 Semigroups in the Unit Disc Definition 8.1.1 An algebraic semigroup (φ t) of holomorphic self-maps in the unit disc is a homomorphism between the additive semigroup of non-negative real numbers and the composition semigroup of all holomorphic self-maps of the unit disc. In other words: (1) φ t ∈ Hol(D, D) for all t ≥ 0; (2) φ 0 = id D , that is, φ 0 is the identity in D; (3) φ s+t = φ s • φ t , for all s, t ≥ 0. Every φ t is called an iterate of the semigroup. Moreover, the semigroup (φ t) is said to be continuous if, additionally, the map [0, +∞) t → φ t ∈ Hol(D, D) is continuous when [0, +∞) is endowed with the Euclidean topology and Hol(D, D) with the topology of uniform convergence on compacta.

Generators of semigroups on Banach spaces inducing holomorphic semiflows

Israel Journal of Mathematics

Let A be the generator of a C 0-semigroup T on a Banach space of analytic functions on the open unit disc. If T consists of composition operators, then there exists a holomorphic function G : D → C such that Af = Gf ′ with maximal domain. The aim of the paper is the study of the reciprocal implication. 2010 Mathematics Subject Classification. 30D05, 47D03, 47B33. Key words and phrases. semiflow of analytic functions, generator of C 0semigroup, semigroup of composition operators.

A study of Holomorphic Semigroups

arXiv: Dynamical Systems, 2019

In this paper, we investigate some characteristic features of holomorphic semigroups. In particular, we investigate nice examples of holomorphic semigroups whose every left or right ideal includes minimal ideal. These examples are compact topological holomorphic semigroups and examples of compact topological holomorphic semigroups are the spaces of ultrafilters of semigroups.

A Case Study of Holomorphic Semigroups

Journal of Nepal Mathematical Society

Modulus of continuity with respect to semigroups of analytic functions and applications

Journal of Mathematical Analysis and Applications, 2016

Given a complex Banach space E, a semigroup of analytic functions (ϕt) and an analytic function F : D → E we introduce the modulus wϕ(F, t) = sup |z|<1 F (ϕt(z)) − F (z). We show that if 0 < α ≤ 1 and F belongs to the vector-valued disc algebra A(D, E), the Lipschitz condition M∞(F , r) = O((1 − r) 1−α) as r → 1 is equivalent to wϕ(F, t) = O(t α) as t → 0 for any semigroup of analytic functions (ϕt), with ϕt(0) = 0 and infinitesimal generator G, satisfying that ϕ t and G belong to H ∞ (D), and in particular is equivalent to the condition F − Fr A(D,E) = O((1 − r) α) as r → 1. We apply this result to particular semigroups (ϕt) and particular spaces of analytic functions E, such as Hardy or Bergman spaces, to recover several known results about Lipschitz type functions.

Remarks on generators of analytic semigroups

Israel Journal of Mathematics, 1979

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex rn.mthers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups in L~, 1 < p < ~~, while the sufficient condition in the other characterization is meaningful in the case of non-j linear operators.

Smoothness and Shape Preserving Properties of Bernstein Semigroup

Mediterranean Journal of Mathematics, 2018

This paper is concerned with the strongly continuous semigroup (T (t)) t≥0 of operators on C[0, 1] which can be represented as a limit of suitable iterates of the Bernstein operators Bn. We present some new smoothness and shape preserving properties of the operators T (t) and Bn. The asymptotic behavior and simultaneous approximation results are also presented.

Holomorphic subordinated semigroups (original) (raw)

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