Two-state free Brownian motions

Anshelevich, Michael

doi:10.1016/j.jfa.2010.09.004

Cited by 9 publications

(7 citation statements)

References 27 publications

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“…Finally it should be mentioned that particular cases of (1.1) surprisingly arise in the theory of martingale polynomials [9], [30], two-state free Brownian motions [1], and Gaussian processes [27]. Let us also signal references [22] and [16] where similar pencils appear in the investigation of exactly solvable kinetic models and the corresponding combinatorial problems.…”

Section: Introductionmentioning

confidence: 99%

CMV Matrices and Little and Big −1 Jacobi Polynomials

2012

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Abstract. We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter λ which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.

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Section: Introductionmentioning

confidence: 99%

CMV Matrices and Little and Big −1 Jacobi Polynomials

2012

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“…The technique of rank one perturbations was used in some results on orthogonal polynomials and Jacobi matrices, and there are some interesting applications to free probability (see e.g. [10,11]). They also turned out to be useful in the investigation of certain random Hamiltonian systems called Anderson models and the longstanding Anderson localization conjecture [9].…”

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confidence: 99%

Singular Integrals, Rank One Perturbations and Clark Model in General Situation

Liaw

Treil

2017

Association for Women in Mathematics Series

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Abstract. We start with considering rank one self-adjoint perturbations Aα = A+α( · , ϕ)ϕ with cyclic vector ϕ ∈ H on a separable Hilbert space H. The spectral representation of the perturbed operator Aα is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators A and Aα.Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle.This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings.Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role.Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators.These lecture notes give an account of the mini-course delivered by the authors, which was centered around [38,39,37]. Unpublished results are restricted to the last part of this manuscript.

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“…The independence coming from the u.a.u.-product given by the above product on UAlgP 2,1 as in Remark 2, which is also called c-freeness, gives rise to a generalized, non-commutative Brownian motion; see [BS91,Ans11]. The concept for the c-free product can be generalized to a wider class of u.a.u.-products in AlgP 2,1 ; see [Has11] and Remark 3.2.…”

Section: Universal Productsmentioning

confidence: 99%

Non-commutative stochastic independence and cumulants

Manzel

Schürmann

2017

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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In a central lemma we characterize 'generating functions' of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to 'unital, associative universal products' on this category, which again define a notion of non-commutative stochastic independence. Using the central lemma, we can prove the existence of cumulants and of 'cumulant Lie algebras' for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in N. Murakis classification, c-free independence of M. Bożejko and R. Speicher, the indented product of T. Hasebe and the bi-free independence of D. Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.

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Two-state free Brownian motions

Cited by 9 publications

References 27 publications

CMV Matrices and Little and Big −1 Jacobi Polynomials

CMV Matrices and Little and Big −1 Jacobi Polynomials

Singular Integrals, Rank One Perturbations and Clark Model in General Situation

Non-commutative stochastic independence and cumulants

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