On slice biregular functions and isomorphisms of Bergman spaces

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 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 . In the following, we discuss two possible definitions of slice hyperholomorphic functions.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…In this section, we consider the special case of weighted Bergman spaces on the unit ball. A first study of these spaces in the slice hyperholomorphic setting has been performed in . We recall the main definitions and results.…”

Section: The Case Of Weighted Bergman Spacesmentioning

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 Bergman space A 2 (U ) has been studied in a series of papers, see [9], [12], [13], [14]. In [7] the authors study the weighted A p (U ) Bergman spaces and, as a special case, we have the following definition: Definition 4.1.…”

Section: Carleson Measures In Bergman Spacesmentioning

confidence: 99%

Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball

Sabadini

Saracco

2017

J. London Math. Soc.

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We study a characterization of slice Carleson measures and of Carleson measures for both the Hardy spaces Hp(B) and the Bergman spaces Ap(B) of the quaternionic unit ball B. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudo-hyperbolic balls is not possible

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“…The slice hyperholomorphic Hardy spaces

{scriptH}^{2} false(double-struckD false)

on the unit ball and

{scriptH}^{2} false(H^{+} false)

on the quaternionic half space

H^{+}

, which consists of those quaternions that have positive real part, were introduced in and Hardy spaces

{scriptH}^{p} false(double-struckD false)

with

0 < p \leq + \infty

on the unit ball were studied in detail in . Slice hyperholomorphic Bergman spaces were treated in and slice hyperholomorphic Fock spaces in . Furthermore contains a first study on slice hyperholomorphic Bloch, Besov and weighted Bergman spaces and on the Dirichlet space in this setting.…”

Section: Introductionmentioning

confidence: 99%

BMO- and VMO-spaces of slice hyperholomorphic functions

2017

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In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO-and VMO-spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space H 1 (D).The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.

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On slice biregular functions and isomorphisms of Bergman spaces

Cited by 33 publications

References 17 publications

Schatten class and Berezin transform of quaternionic linear operators

Schatten class and Berezin transform of quaternionic linear operators

Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball

BMO- and VMO-spaces of slice hyperholomorphic functions

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