Inner Product Spaces and Krein Spaces in the Quaternionic Setting

Alpay, Daniel; Colombo, Fabrizio; Sabadini, Irene

doi:10.1007/978-3-319-10335-8_4

Cited by 12 publications

(4 citation statements)

References 20 publications

(23 reference statements)

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“…Weighted Bergman spaces, Bloch, Besov, and Dirichlet spaces on the unit ball B are considered in [38]. Inner product spaces and Krein spaces in the quaternionic setting, are in [21].…”

Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

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106

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The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.

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“…Weighted Bergman spaces, Bloch, Besov, and Dirichlet spaces on the unit ball B are considered in [38]. Inner product spaces and Krein spaces in the quaternionic setting, are in [21].…”

Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

Self Cite

106

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“…In such a case, one can understand arithmetic functions as Krein-space operators (for fixed primes). (See [2] for more about Krein spaces and Krein-space operators.) These studies are all motivated by results from number theory (e.g., [5], [7], [10], [16], [17] and [22]) under free probability techniques (e.g., [20], [21] and [23]).…”

Section: Ilwoo Cho and Palle Jorgensenmentioning

confidence: 99%

Free probability on <inline-formula><tex-math id="M1">\begin{document}$ C^{*}$\end{document}</tex-math></inline-formula>-algebras induced by hecke algebras over primes

Cho¹,

Jorgense²

2019

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper, we establish free-probabilistic models (H(Gp), ψp) on Hecke algebras H(Gp), and construct Hilbert-space representations of H (Gp), preserving free-probabilistic information from (H(Gp), ψp) , for primes p. From such free-probabilistic structures with representations, we study spectral properties of operators in C *-algebras generated by {H(Gp)} p:primes , via their free distributions.

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“…and hence, by the same argument, the form (6.7) is continuous in f . It follows from the uniform boundedness theorem (which holds in quaternionic Hilbert spaces; see [7,Theorem 3.3…”

Section: The Quaternionic Setting: the Realization Theoremmentioning

confidence: 99%

Realizations of holomorphic and slice hyperholomorphic functions: the Krein space case

Alpay¹,

Colombo²,

Sabadini³

2018

Preprint

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In this paper we treat realization results for operatorvalued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space.

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Inner Product Spaces and Krein Spaces in the Quaternionic Setting

Cited by 12 publications

References 20 publications

Entire Slice Regular Functions

Entire Slice Regular Functions

Free probability on <inline-formula><tex-math id="M1">\begin{document}$ C^{*}$\end{document}</tex-math></inline-formula>-algebras induced by hecke algebras over primes

Realizations of holomorphic and slice hyperholomorphic functions: the Krein space case

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