Schatten class and Berezin transform of quaternionic linear operators

Colombo, Fabrizio; Gantner, Jonathan; Janssens, Tim

doi:10.1002/mma.3944

Cited by 12 publications

(18 citation statements)

References 39 publications

(118 reference statements)

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“…so using this formula we can define singular values also for operators which are not compact. Furthermore, with a similar proof as in the complex case we can state that for compact or non-compact bounded operators we can extend Corollary 3.9 in [13]. More precisely we have:…”

Section: Some Results On Schatten Class Of Quaternionic Operatorssupporting

confidence: 52%

“…the eigenvalues of the operator |T | := √ T * T in non-increasing order, the vectors (e n ) n∈N form an eigenbasis of |T | and σ n = W e n with W unitary on ker W ⊥ and such that T = W |T |. See [19] and Remark 3.4 in [13]. where (λ n (T )) n∈N denotes the sequence of singular values of T and ℓ p and ℓ ∞ denote the space of p-summable resp.…”

Section: Some Results On Schatten Class Of Quaternionic Operatorsmentioning

confidence: 99%

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Perturbation of normal quaternionic operators

Cerejeiras

Colombo

Kähler

et al. 2019

Trans. Amer. Math. Soc.

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The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two Sresolvent operators and the S-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

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Section: Some Results On Schatten Class Of Quaternionic Operatorssupporting

confidence: 52%

Section: Some Results On Schatten Class Of Quaternionic Operatorsmentioning

confidence: 99%

Perturbation of normal quaternionic operators

Cerejeiras

Colombo

Kähler

et al. 2019

Trans. Amer. Math. Soc.

Self Cite

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“…We stress that other basis-independence properties of the trace exist. Within the different approach of [CGJ16], for every J = −J * with JJ = −I commuting with A ∈ B 1 (H), tr N (A) is fixed when the Hilbert basis N varies in H Jı . (Such a J does exist at least if A is normal for Theorem 5.9 in [GMP13].)…”

Section: A Basis-independence Property Of Tr(a) For D = Hmentioning

confidence: 99%

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Moretti

Oppio

2018

Ann. Henri Poincaré

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Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Solér's theorem. Quantum states are here pictured in terms of σ-additive probability measures over the non-Boolean lattice of orthogonal projectors of the considered Hilbert space. Gleason's theorem proves that, if the Hilbert space is either real or complex and some technical hypothes are true, then these measures are one-to-one with standard density matrices used by physicists recovering and motivating the familiar notion of state. The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, the formulation of this extension [Va68] is partially mathematically incorrect due to some peculiarities of the notion of trace in quaternionic Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is devoted to present Gleason-Varadarajan's theorem into a technically correct and physically meaningful form valid for the three types of Hilbert spaces. In particular, we prove that only the real part of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

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“…The spectral theorem for normal operators was generalized to the unbounded operators by Alpay et al [17]. Colombo et al [19] presented a singular value decomposition of compact quaternionic operator T as: T x = n∈N σ n λ n e n , x , ∀x ∈ H where λ n > 0 are the singular values (not eigenvalues) of T , the vectors {e n } form an eigenbasis |T |, and σ n = W e n with W unitary on ker W ⊥ such that T = W |T | (for detail, see Remark 3.4 of [19]). In the present paper, we obtain a new and different spectral theorem, that is,…”

Section: Proof Letmentioning

confidence: 99%

“…It is worth pointing out that the authors in [19] also proposed the Schatten classes of quaternionic operators. Moreover, some characterizations of Scatten class operators were drawn.…”

Section: Proof Letmentioning

confidence: 99%

Novel Sampling Formulas Associated with Quaternionic Prolate Spheroidal Wave functions

Dong

Kou

2017

Adv. Appl. Clifford Algebras

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The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated.Mathematics Subject Classification (2010). Primary 47B32; Secondary 94A11, 94A20.

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Schatten class and Berezin transform of quaternionic linear operators

Cited by 12 publications

References 39 publications

Perturbation of normal quaternionic operators

Perturbation of normal quaternionic operators

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Novel Sampling Formulas Associated with Quaternionic Prolate Spheroidal Wave functions

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