Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

By Daniel Alpay, Fabrizio Colombo, Irene Sabadini. 2011

85 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/523b78c208b9ac58063ddef8905d4035efb3cc4f

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A New Resolvent Equation for the $$S$$S-Functional Calculus (2013, unknown): “The reader is referred to [2, 3, 4] for Schur analysis in the slice hyperholomorphic setting and to [1] and [7] for an overview of Schur analysis in the complex setting.”
Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis (2013, background): “theory of linear systems. It is because of these relations that Schur analysis has already been extended to many other settings, for example several complex variables [1], slice-holomorphic functions [4], and hyperholomorphic functions, [11, 12, 13, 5]. In this nal chapter of our work, we show how t”
Entire slice regular functions (2015, background): “In particular, the quaternionic Hardy spaces H2(Ω), where Ω is the quaternionic unit ball B or the half space H+ of quaternions with positive real part, together with the Blaschke products are in [14, 15, 16] and further properties are in [28, 29, 30].”
The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum (2014, unknown): “ operators on a quaternionic Hilbert space. We point out that the S-resolvent operators are also used in Schur analysis in the realization of Schur functions in the slice hyperholomorphic setting see [1, 8, 9, 10] and [3, 11] for the classical case. In the papers [4, 14, 25] the problem of the gener”
Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “As mentioned above, a study of Schur analysis in the setting of slice hyperholomorphic functions has been initiated recently in [2], and it is the purpose of the present paper to continue this study.”