Semigroups of Composition Operators in Analytic Morrey Spaces

Galanopoulos, Petros; Merchán, Noel; Siskakis, Aristomenis G.

doi:10.1007/s00020-020-2568-5

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…For some K > 0 large enough ω K satisfies the hypothesis of Lemma 3.1. Then f ∈ B log,0 is equivalent to (14), hence it satisfies (15) which is equivalent to f ∈ V M OA log .…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…We believe that the techniques that we have employed can be used to prove similar characterizations in other kind of spaces. In particular recent studies investigated the maximal subspace of holomorphic semigroups in the analytic Morrey spaces [14] and in the setting of mixed norm spaces [4]. It would be interesting to know whether a similar characterization of the minimality of the maximal subspace is possible in these settings, too.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

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Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation

Chalmoukis¹,

Vassilis²

2021

Preprint

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It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BM OA), 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 (ϕ t ) that can replace the semigroup of rotations in Sarason's Theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (ϕ t ). In addition we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch space and surprisingly enough we find that the semigroups for which the Bloch version of Sarason's Theorem holds are exactly the same as in the BM OA case.

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“…For some K > 0 large enough ω K satisfies the hypothesis of Lemma 3.1. Then f ∈ B log,0 is equivalent to (14), hence it satisfies (15) which is equivalent to f ∈ V M OA log .…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

Section: Proof Of Main Resultsmentioning

confidence: 99%

Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation

Chalmoukis¹,

Vassilis²

2021

Preprint

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“…In the proof of Theorem 4.1, we will make use the following lemma, which can be extracted from [1], see also [13].…”

Section: Absence Of Non-trivial C 0 -Semigroups Of Weighted Compositi...mentioning

confidence: 99%

On generators of C0-semigroups of composition operators

Gallardo-Gutiérrez

Yakubovich

2019

Isr. J. Math.

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Avicou, Chalendar and Partington proved in [3] that an (unbounded) operator (Af ) = G · f ′ on the classical Hardy space generates a C 0 semigroup of composition operators if and only if it generates a quasicontractive semigroup. Here we prove that if such an operator A generates a C 0 semigroup, then it is automatically a semigroup of composition operators, so that the condition of quasicontractivity of the semigroup in the cited result is not necessary. Our result applies to a rather general class of Banach spaces of analytic functions in the unit disc.2010 Mathematics Subject Classification. 47B35 (primary).

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