Espaces de Branges Rovnyak et fonctions de Schur : le cas hyper-analytique

Alpay, Daniel; Shapiro, Michael; Volok, Dan

doi:10.1016/j.crma.2004.01.013

Cited by 22 publications

(17 citation statements)

References 4 publications

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“…(3) We have mentioned Schur-Agler classes associated to the unit ball and the poly-disk. Their counterpart in the setting of Fueter series were presented in [16,14]. The study of Schur multipliers slice hyperholomorphic in several quaternionic variables should be a topic of interest.…”

Section: Discussionmentioning

confidence: 99%

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

Alpay

Colombo

Sabadini

2011

Integr. Equ. Oper. Theory

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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.

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Section: Discussionmentioning

confidence: 99%

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

Alpay

Colombo

Sabadini

2011

Integr. Equ. Oper. Theory

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“…Therefore the elements in N + are slice hyperholomorphic and so are the elements in P. This concludes the proof. 9. Generalized Schur functions Definition 9.1.…”

Section: Negative Squaresmentioning

confidence: 99%

Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

Alpay

Colombo

Sabadini

2013

JAMA

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108

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We study reproducing kernel Hilbert and Pontryagin spaces of slice\ud hyperholomorphic functions which are analogs of the Hilbert\ud spaces of analytic functions introduced by de Branges and Rovnyak.\ud In the first part of the paper we focus on the case of Hilbert\ud spaces, and introduce in particular a version of the Hardy space. Then we\ud define Blaschke factors and Blaschke products and we consider an\ud interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized\ud Schur functions and of generalized Carath\'eodory functions

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“…The counterparts of the space A and of the operators R ' have been recently introduced in the quaternionic setting [5,6], and the purpose of this article is to prove a von Neumann inequality in that setting. The lack of commutativity of the quaternions forces the choice of polynomials to be with real coefficients (Theorem 2.1).…”

Section: M N þK Amentioning

confidence: 99%