On Two Approaches to the Bergman Theory for Slice Regular Functions

Colombo, Fabrizio; González-Cervantes, J. Oscar; Luna-Elizarrarás, M. Elena; Sabadini, Irene; Shapiro, Michael

doi:10.1007/978-88-470-2445-8_3

Cited by 38 publications

(35 citation statements)

References 15 publications

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“…The Hardy spaces

H^{p} (double-struckB)

for p > 2 are considered in . The Bergman spaces are treated in and the Fock spaces in . Weighted Bergman spaces, Bloch, Besov, and Dirichlet spaces on the unit ball

double-struckB

are studied in .…”

Section: Preliminary Resultsmentioning

confidence: 99%

Schatten class and Berezin transform of quaternionic linear operators

Colombo

Gantner

Janssens

2016

Math Methods in App Sciences

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In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum for quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S-functional calculus.AMS Classification: 47A10, 47A60, 30G35. Key words: Schatten class of operators, Spectral theorem and the S-spectrum, Berezin transform of quaternionic operators, Weighted Bergman spaces of slice hyperholomorphic functions.

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“…The Hardy spaces

H^{p} (double-struckB)

for p > 2 are considered in . The Bergman spaces are treated in and the Fock spaces in . Weighted Bergman spaces, Bloch, Besov, and Dirichlet spaces on the unit ball

double-struckB

are studied in .…”

Section: Preliminary Resultsmentioning

confidence: 99%

Schatten class and Berezin transform of quaternionic linear operators

Colombo

Gantner

Janssens

2016

Math Methods in App Sciences

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“…The slice Bergman space of the first and second kind were introduced in [8,9]. In this paper, we focus on the case of the Bergman space of the second kind of the quaternionic unit ball B.…”

Section: Quaternions and Slice Regular Functionsmentioning

confidence: 99%

On Slice Polyanalytic Functions of a Quaternionic Variable

2019

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In this paper, we introduce the quaternionic slice polyanalytic functions and we prove some of their properties. Then, we apply the obtained results to begin the study of the quaternionic Fock and Bergman spaces in this new setting. In particular, we give explicit expressions of their reproducing kernels.Proposition 3.18. Let Ω be an axially symmetric domain and let f : Ω −→ H be a slice polyanalytic function. Assume that there exist J, K ∈ S, with J = K and U J , U K two subdomains of Ω + J and Ω + K respectively where Ω + J := Ω ∩ C + J and Ω + K := Ω ∩ C + K . If f = 0 on U J and U K , then f = 0 everywhere in Ω. Proof. Let f be a slice polyanalytic function on Ω such that f = 0 on U J and U K . Thus, since U J and U K are respectively subdomains of Ω + J and Ω + K . It follows, from the Splitting Lemma for slice polyanalytic functions combined with the classical complex analysis that f = 0 everywhere on Ω + J and Ω + K . Then, we just need to use the Representation Formula which allows to recover a slice function by its values on two semi-slices to complete the proof.

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“…Recently many function theoretic problems on the complex analysis have been studied also on quaternion analysis [1,[4][5][6][7][8]. Let be a domain in H which is the algebra of quaternions.…”

Section: Introductionmentioning

confidence: 99%