Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically

This paper presents a theory of noncommutative functions which results in an algorithm for determining where they are “matrix convex”. Of independent interest is a theory of noncommutative quadratic functions and the resulting algorithm which calculates the region where they are “matrix positive”. This is accomplished via a theorem (a type of Positivstellensatz) on writing noncommutative quadratic functions with noncommutative rational coefficients as a weighted sum of squares. Furthermore the paper gives an LDU algorithm for matrices with noncommutative entries and conditions guaranteeing that the decomposition is successful.     

Noncommutative dynamics and generalized master equations

We review the basic concepts of quantum probability and stochastics using the universal Itô B*-algebra approach. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The general Lévy process is defined in terms of the modular B*-Itô algebra, and the corresponding quantum stochastic master equation on the predual space of theW*-algebra is derived as a noncommutative version of the Zakai equation driven by the process. This is done by a noncommutative analog of the Girsanov transformation, which we introduce here in full generality.     

Periodic Cyclic Homology as Sheaf Cohomology Dedicated to Daniel Quillen on his 60th Birthday

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology.     

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.     

Matrice de connexion d'un système aux q-différences confluant vers un système différentiel et matrices de monodromie

We built in [6] a version with elliptic coefficients of the Birkhoff connection matrix and showed its role in the classification of regular singular q-difference systems. We give here a geometrical interpretation: when q tends to 1, P tends to a locally constant matrix P such that the (finitely many) values (a)⁻¹ (b) are the monodromy matrices of the limiting differential system (assumed to be non-resonant at 0 and ∞) at the singularities on C^*.     

Matricial R-transform

We study the addition problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the matricial R-transform related to the associated convolution. It is a linear combination of Voiculescu?s R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the matricial linearization property of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for free, boolean, monotone, orthogonal, s-free and c-free independence.     

Noncommutataive Descent and Non-Abelian Cohomology

We study the descent problem of modules over general extensions of noncommutative rings and give different interpretations of descent data. We consider in particular the case of Hopf–Galois extensions. When descent data exist, we classify them by non-Abelian cohomology sets and deduce a noncommutative version of Hilbert's Theorem 90.     

Fractional-Parabolic Systems

We develop a theory of the Cauchy problem for linear evolution systems of partial differential equations with the Caputo-Dzhrbashyan fractional derivative in the time variable t. The class of systems considered in the paper is a fractional extension of the class of systems of the first order in t satisfying the uniform strong parabolicity condition. We construct and investigate the Green matrix of the Cauchy problem. While similar results for the fractional diffusion equations were based on the H-function representation of the Green matrix for equations with constant coefficients (not available in the general situation), here we use, as a basic tool, the subordination identity for a model homogeneous system. We also prove a uniqueness result based on the reduction to an operator-differential equation.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rosenthal inequalities in noncommutative symmetric spaces

We give a direct proof of the ‘upper’ Khintchine inequality for a noncommutative symmetric (quasi-)Banach function space with nontrivial upper Boyd index. This settles an open question of C. Le Merdy and the fourth named author (Le Merdy and Sukochev, 2008 [24]). We apply this result to derive a version of Rosenthal?s theorem for sums of independent random variables in a noncommutative symmetric space. As a result we obtain a new proof of Rosenthal?s theorem for (Haagerup) Lp-spaces.     

On the Fréchet derivative of matrix functions

In 1956 Rinehart [4] discussed the derivatives of matrix functions by considering differences ƒ(A + E) − ƒ(A) for matrices E commuting with A. In that case the derivative turned out to be ƒ′(A). In this paper the case of noncommutative A and E is treated. This leads to the Fréchet derivative of the matrix function ƒ. An explicit integral representation is obtained. Using an approach that is similar to the one in [5], a finite sum in polynomials of A is obtained. The coefficients may be computed recursively. This is useful in computing the Fréchet derivative which is needed for Newton's method to solve nonlinear matrix equations.     

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦⁿW)⁽ⁿ⁾W^-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.     

On the invertibility of ideals in orders

The problem of invertibility of ideals in orders has been studied by a number of authors. The commutative case has been considered by Dade, Taussky, and Zassenhaus; Frolich; and Singer. Ballew gives a generalization of Frolich's results to a class of noncommutative orders. We examine some of the possible extensions of the results of Dade et al. to noncommutative orders.     

Aspects of Matrix Theory and Noncommutative Geometry

We review some basic foundations of the matrix model of M-theory. We study the important problem of compactifying the matrix theory in relation to noncommutative geometry. We show that there exist solutions of this problem other than the well-known toroidal solutions of Connes, Douglas, and Schwarz. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 181–190, November, 2005.     

The Noncommutative Bispectral Problem for Operators of Order One

We study a noncommutative version of the bispectral problem and consider the corresponding ad-conditions in the case when both operators have order one. These terms are explained in an extended abstract given below.     

Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials

In this paper we prove two consequences of the subnormal character of the Hessenberg matrix D when the hermitian matrix M of an inner product is a moment matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We also prove an extension of the Krein theorem for discrete measures on the complex plane based on properties of subnormal operators.