Weyl and Browder S-spectra in a right quaternionic Hilbert space

Muraleetharan, B.; Thirulogasanthar, K.

doi:10.1016/j.geomphys.2018.09.006

Cited by 16 publications

(8 citation statements)

References 17 publications

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“…The H ∞ -functional calculus was further extended following the book [52] in [14]. For more recent developments associated with quaternionic quantum mechanics see [59,60,61,62]. Remark 6.6.…”

Section: Noncommuting Matrix Variables and Some Final Remarksmentioning

confidence: 99%

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

Colombo,

Gantner,

Kimsey

et al. 2021

Preprint

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The spectral theory on the S-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n-tuples of noncommuting operators. With this paper we show that the spectral theory on the S-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. More precisely, we show that the S-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S-functional calculus goes beyond quaternionic and Clifford analysis. Indeed we show that the S-functional calculus has a certain universality property. This fact makes the spectral theory on the S-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

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Section: Noncommuting Matrix Variables and Some Final Remarksmentioning

confidence: 99%

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

Colombo,

Gantner,

Kimsey

et al. 2021

Preprint

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“…Only in 2015 the authors with D. Alpay proved the spectral theorem for quaternionic normal operators based on the S-spectrum σ S (T ) in [4] which was published in 2016. Fairly recently, there has been a renewed quaternionic quantum mechanics, which utilises the notion of S-spectrum (see, e.g., [57,58,59,68]).…”

Section: Proofmentioning

confidence: 99%

The spectral theorem for normal operators on a Clifford module

Colombo¹,

Kimsey²

2021

Preprint

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In this paper, using the recently discovered notion of the Sspectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be −1). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

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“…The set of right eigenvalues coincides with the point S−spectrum, see [19,Proposition 4.5]. Now, we recall the definition of essential S−spectrum, we refer to [29,30] for more details. Let V R H be a separable right quaternionic Hilbert space equipped with a left scalar multiplication.…”

Section: This Implies Thatmentioning

confidence: 99%

“…We recall that in [29,30], the study of the essential S−spectrum is established using the theory of Fredholm operators. The aim of this section is to show that the essential S−spectrum does not contain discrete element of the S−spectrum.…”

Section: Riesz Projection and Essential Spectrummentioning

confidence: 99%

“…Motivated by the new concept of S−spectrum in the quaternionic setting, Muraleetharan and Thiralogasanthar, in [29,30], introduced the Weyl and essential S-spectra and gave a characterization using Fredholm operators. We refer to [7] for the study of the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Riesz projection and essential $S$-spectrum in quaternionic setting

Baloudi¹,

Belgacem²,

Jeribi³

2021

Preprint

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This paper is devoted to the investigation of the Weyl and the essential S−spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S−eigenvalue of finite type is both introduced and studied.In particular, we have shown that the Weyl and the essential S−spectra do not contain eigenvalues of finite type. We have also described the boundary of the Weyl S−spectrum and the particular case of the spectral theorem of the essential S−spectrum. Contents 1. Introduction 1 2. Mathematical preliminaries 3 2.1. Quaternions 3 2.2. Right quaternionic Hilbert space and operator 4 2.3. The quaternionic functional calculus 7 3. Riesz projection and essential spectrum 9 4. Some results on the Weyl S-spectrum 18 References 22

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Weyl and Browder S-spectra in a right quaternionic Hilbert space

Cited by 16 publications

References 17 publications

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

The spectral theorem for normal operators on a Clifford module

Riesz projection and essential $S$-spectrum in quaternionic setting

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