Reproducing Kernel Spaces of Series of Fueter Polynomials

Integr. Equ. Oper. Theory

Colombo

Sabadini

2011

Self Cite

In this paper we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allow to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels and positive definite functions in this setting and we show how they can be obtained using the extension operator and the slice hyperholomorphic product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space.Mathematics Subject Classification (2010). Primary 47B32, 30G35; Secondary 93A25.

Section: Discussionmentioning

confidence: 99%

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

Integr. Equ. Oper. Theory

Colombo

Sabadini

2011

Self Cite

“…The argument follows the one given in the hyper-holomorphic setting in [15]. For an earlier result in the setting of power series in finite number of variables, see [8].…”

Section: The Gleason Problem In Certain Spaces Of Power Seriesmentioning

confidence: 95%

“…Proof We follow the proof given in [15,Section 4.2]. Let H(k i ) be the reproducing kernel Hilbert space on the real numbers of functions with reproducing kernel k i and let…”

Section: Theorem 72 a Schur-agler Multiplier Is The Hermite Transformentioning

confidence: 99%

Rational Functions Associated with the White Noise Space and Related Topics

Levanony

2008

Potential Anal

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Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida's white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the CauchyKovalevskaya product is replaced by the Wick product.Keywords Rational functions · Hida's white noise space · Gleason's problem · Wick product · Hyperholomorphic functions Mathematical Subject Classifications (2000) Primary 30G35 · 26C15 · 60H40 · Secondary 47A99 · 32A05

“…In [22][23][24] a different approach was used and a similar construction yields the Drury-Arveson kernel…”

Section: The Hardy Space and Schur Multipliersmentioning

confidence: 99%

On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

Integr. Equ. Oper. Theory

Colombo

Diki

et al. 2021

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In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers.