Small eigenvalues of large Hermitian moment matrices

We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane. We prove that if the polynomials are dense in L2(μ) then the smallest eigenvalue λn of the truncated matrix Mn of M of size (n+1)×(n+1) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results

A Rakhmanov-like theorem for orthogonal polynomials on Jordan arcs in the complex plane

Rakhmanov's theorem establishes a result about the asymptotic behavior of the elements of the Jacobi matrix associated with a measure ¹ which is de¯ned on the interval I = [¡1; 1] with ¹ 0 > 0 almost everywhere on I. In this work we give a weak version of this theorem, for a measure with support on a connected ¯nite union of Jordan arcs on the complex plane, in terms of the Hessenberg matrix, the natural generalization of the tridiagonal Jacobi matrix to the complex plane

Asymptotically Toeplitz Hessenberg Operators and the Rieman mapping

In a recent work the authors have established a relation between the limits of the elements of the diagonals of the Hessenberg matrix D associated with a regular measure, whenever those limits exist, and the coe?cients of the Laurent series expansion of the Riemann mapping ?(z) of the support supp(?), when this is a Jordan arc or a connected nite union of Jordan arcs in the complex plane C. We extend here this result using asymptotic Toeplitz operator properties of the Hessenberg matriz

Computing Hessenberg matrix associated to self-similar measure

Objective: The obtention of the Hessenberg matrix associated to a self-similar measure with compact support in the complex plane in two di_erent ways. Outline of the talk: 1 Preliminaries. Moment and Hessenberg matrices. Self-similar measures. 2 Moment matrices of self-similar measures. Fixed point theorem for moment matrices of self-similar measures (EST2007).Cholesky factorization. 3 Hesssenberg matrix of a sum of measures (generalization of Mantica's spectral techniques). Hessenberg matrix associated to a self-similar measure

A characterization of polynomial density on curves via matrix algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L2(m), with m a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure m. To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, g(M) and l(M), associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index g and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index g that will allow us to give an alternative proof of Thomson's theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients urn:x wiley:mma:media:mma5716:mma5716-math-0001 verifying urn:x-wiley:mma:media:mma5716:mma5716-math-0002, ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D=SR+K2 with K2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that urn:x-wiley:mma:media:mma5716:mma5716-math-0003 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if urn:x-wiley:mma:media:mma5716:mma5716-math-0004, then D=SR+Kp with Kp an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle