Commutators and systems of singular integral equations. I

Abstract. Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own whithin applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.

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“…that is u T is given by an integrable function function g T , called the principal function of the operator T , see [133,134].…”

Section: Semi-normal Operatorsmentioning

confidence: 99%

Random matrices in 2D, Laplacian growth and operator theory

Mineev-Weinstein

Putinar

Teodorescu³

2008

J. Phys. A: Math. Theor.

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“…Such models generalize to higher dimension the Hilbert transform models of hyponormal operators with a self-commutator of rank 1 discovered by Xia [42] and Pincus [33], analyzed in more detail by Pincus and Xia [34] and Picus and Xia and Xia [35], and afterwards set up in full generality for pure hyponormal operators by Kato [17] and Muhly [32]. We should point out that the natural framework for developing Riesz transforms models is provided by n-tuples of decomposable linear operators on a direct integral Hilbert space…”

Section: Riesz Transforms and Joint Hyponormalitymentioning

confidence: 82%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C * -algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C * -algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

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“…Operators where A and B are almost commuting self-adjoint operators, ϕ and ψ are polynomials and g is the Pincus principal function which is uniquely determined by A and B and which was introduced in [59]. The problem considered in [54] was to extend the Helton-Howe trace formula for a reasonably big class of functions.…”

Section: Functions Of Almost Commuting Operators An Extension Of Thementioning

confidence: 99%

Multiple operator integrals in perturbation theory

Peller

2015

Bull. Math. Sci.

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The purpose of this survey article is to give an introduction to double operator integrals and multiple operator integrals and to discuss various applications of such operator integrals in perturbation theory. We start with the Birman-Solomyak approach to define double operator integrals and consider applications in estimating operator differences f (A)− f (B) for self-adjoint operators A and B. Next, we present the Birman-Solomyak approach to the Lifshits-Krein trace formula that is based on double operator integrals. We study the class of operator Lipschitz functions, operator differentiable functions, operator Hölder functions, obtain Schatten-von Neumann estimates for operator differences. Finally, we consider in Chapter 1 estimates of functions of normal operators and functions of d-tuples of commuting self-adjoint operators under perturbations. In Chapter 2 we define multiple operator integrals in the case when the integrands belong to the integral projective tensor product of L ∞ spaces. We consider applications of such multiple operator integrals to the problem of the existence of higher operator derivatives and to the problem of estimating higher operator differences. We also consider connections with trace formulae for functions of operators under perturbations of class S m , m ≥ 2. In the last chapter we define Haagerup-like tensor products of the first kind and of the second kind and we use them to study functions of noncommuting self-adjoint operators under perturbation. We show that for functions f in the Besov class B

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Commutators and systems of singular integral equations. I

Cited by 86 publications

References 11 publications

Random matrices in 2D, Laplacian growth and operator theory

Random matrices in 2D, Laplacian growth and operator theory

Seminormal systems of operators in Clifford environments

Multiple operator integrals in perturbation theory

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