# Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation

@inproceedings{Chalmoukis2021HolomorphicSA, title={Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation}, author={Nikolaos Chalmoukis and Vassilis Daskalogiannis}, year={2021} }

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA), is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (φt) that can replace the semigroup of rotations in Sarason’s Theorem. We give a complete answer to this question, in terms of a…

## One Citation

Generators of $C_0$-semigroups of weighted composition operators

- Mathematics
- 2021

We prove that in a large class of Banach spaces of analytic functions in the unit disc D an (unbounded) operator Af = G · f ′ + g · f with G, g analytic in D generates a C0-semigroup of weighted…

## References

SHOWING 1-10 OF 25 REFERENCES

Semigroups of Composition Operators in BMOA and the Extension of a Theorem of Sarason

- Mathematics
- 2008

Abstract.In this paper we deal with the maximal subspace in BMOA where a general semigroup of analytic functions on the unit disk generates a strongly continuous semigroup of composition operators.…

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

- Mathematics
- 2013

We study the maximal spaces of strong continuity on BMOA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA…

On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point

- Mathematics
- 2020

Let {φt} be a semigroup of holomorphic self-maps of the unit disc D with Denjoy–Wolff point τ ∈ ∂D. We study the rate of convergence of the semigroup to τ , that is, given z ∈ D, we discuss the…

On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions

- Mathematics
- 2016

Let $$\{\phi _t\}_{t\ge 0}$${ϕt}t≥0 be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive…

Semigroups of composition operators and integral operators on mixed norm spaces

- MathematicsRevista Matemática Complutense
- 2019

We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces $$H(p,q,\alpha )$$H(p,q,α). First, we study the separable spaces $$H(p,q,\alpha…

Functions of vanishing mean oscillation

- Mathematics
- 1975

A function of bounded mean oscillation is said to have vanish- ing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of…

Semigroups of Composition Operators in Analytic Morrey Spaces

- Mathematics
- 2019

Analytic Morrey spaces belong to the class of function spaces which, like BMOA, are defined in terms of the degree of oscillation on the boundary of functions analytic in the unit disc. We consider…

A class of integral operators on spaces of analytic functions

- Mathematics
- 2014

Abstract We determine the spectrum and essential spectrum as well as resolvent estimates for a class of integral operators T μ , ν f ( z ) = z μ − 1 ( 1 − z ) − ν ∫ 0 z f ( ξ ) ξ − μ ( 1 − ξ ) ν − 1…

Essential norms and weak compactness of integration operators

- Mathematics
- 2011

Let g be an analytic function on the unit disc and consider the integration operator of the form $${T_g f(z) = \int_0^z fg'\,d\zeta}$$. We derive estimates for the essential and weak essential norms…

Volterra Operators on spaces of analytic functions-a survey

- Mathematics
- 2006

We give a short and selective account of results known about operators of the form Vg(f)(z) = 1 Z z 0 f(‡)g 0 (‡)d‡; where g is analytic on the disc and the operator Tg = zVg acts on spaces of…