arXiv subjects

Hari Bercovici

Publications and source records attributed to Hari Bercovici.

29 records · Page 2Linked to original sources

arXiv2012

Outliers in the spectrum of large deformed unitarily invariant models

We investigate the asymptotic behavior of the eigenvalues of the sum A+U*BU, where A and B are deterministic N by N Hermitian matrices having respective limiting compactly supported distributions μ, ν, and U is a random N by N unitary matrix distributed according to Haar measure. We assume that A has a fixed number of fixed eigenvalues (spikes) outside the support of μ, whereas the distances between the other eigenvalues of A and the support of μ, and between the eigenvalues of B and the support of ν, uniformly go to zero as N goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of A+U*BU outside the support of the limiting spectral measure, called outliers. This phenomenon is fully described in terms of free probability involving the subordination function related to the free additive convolution of μ and ν. Only finite rank perturbations had been considered up to now.

math.PR↗

arXiv2012

The numerical range of a contraction with finite defect numbers

An n-dilation of a contraction T acting on a Hilbert space H is a unitary dilation acting on H \oplus C^n. We show that if both defect numbers of T are equal to n, then the closure of the numerical range of T is the intersection of the closures of the numerical ranges of its n-dilations. We also obtain detailed information about the geometrical properties of the numerical range of T in case n=1.

math.FA↗

arXiv2012

A note on composition operators in a half-plane

Conditions for a composition operator on the Hardy space of the disk to have closed range or be similar to an isometry are well known. We provide such conditions for composition operators on the Hardy space of the upper half-plane. We also show that the operator of composition with an analytic self-map Φ of the upper half-plane can be similar to an isometry even when Φ is far from being an inner function.

math.FA↗

arXiv2012

Factorizations of analytic self-maps of the upper half-plane

We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has fewer zeros and singularities than the original one. The result is used to construct functions with given zeros and poles on the real line.

math.CV↗

arXiv2010

Canonical Models For Bi-Isometries

A canonical model, analogous to the one for contraction operators, is introduced for bi-isometries, two commuting isometries on a Hilbert space. This model involves a contractive analytic operator-valued function on the unit disk. Various pureness conditions are considered as well as bi-isometries for which both isometries are shifts. Several families of examples are introduced and classified.

math.FA↗

arXiv2009

The unbounded commutant of an operator of class C_0

We describe the closed, densely defined linear transformations commuting with a given operator T of class C_0 in terms of bounded operators in {T}'. Our results extend those of Sarason for operators with defect index 1, and Martin in the case of arbitrary finite index.

math.FA↗

arXiv2007

On freely indecomposable measures

We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use analytic subordination.

math.OA↗

arXiv2006

The asymptotic behavios of free convolution

We give a streamlined proof of the limit theorems for the free additive convolution of infinitesimal triangular arrays of probability measures on the real line. The result was first proved by Chistyakov and Götze using analytic subordination.

math.OA↗

arXiv2006

Limit theorems for free multiplicative convolutions

We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically distributed rows. The case of unitary variables is considered as well.

math.OA↗

arXiv2004

Multiplicative Monotone Convolution

We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. We discuss infinite divisibility in the multiplicative monotonic context as well.

math.OA↗

arXiv1999

Stable laws and domains of attraction in free probability theory

In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

math.OA↗