Square roots of semihyponormal operators have scalar extensions

In this paper, we study some properties of , i.e., square roots of semihyponormal operators. In particular we show that an operator has a scalar extension, i.e., is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoar?-Foia?). As a corollary, we obtain that an operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

Harmonic projection methods for large non-symmetric eigenvalue problems

The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh–Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned. © 1998 John Wiley & Sons, Ltd.     

Restrictions of relatively Tauberian operators

For operators T with the property which in addition satisfy a spectral condition we consider conditions under which the restriction of T to a closed invariant subspace also satisfies the same inclusion on the subspace.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient among functions that vanish in a set contained on the boundary ∂Ω of given boundary measure.We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.     

Estimation of the eigenvalues and the smallest singular value of matrices

The purpose of this paper is to discuss the estimation of eigenvalues and the smallest singular value of matrices. Firstly, we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk around of radius

     

Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in Cn is bounded from above by a constant multiple of for any p>n, where δ is the distance to the boundary. For a class of domains that includes those of D?Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of for any p<−1. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.     

Tensor products of holomorphic representations and bilinear differential operators

Let be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product of the representations for any two unitary weights ν and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of . As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product modulo the subspace of functions vanishing of certain degree on the diagonal.     

A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A^?1. We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A. The crucial observation is that with an appropriate residual s, many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils (A, B) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-white Wishart ensembles

We consider non-white Wishart ensembles , where X is a p×N random matrix with i.i.d. complex standard Gaussian entries and Σ is a covariance matrix, with fixed eigenvalues, close to the identity matrix. We prove that the largest eigenvalue of such random matrix ensembles exhibits a universal behavior in the large-N limit, provided Σ is “close enough” to the identity matrix. If not, we identify the limiting distribution of the largest eigenvalues, focusing on the case where the largest eigenvalues almost surely exit the support of the limiting Marchenko-Pastur's distribution.     

Fixing multiple eigenvalues by a minimal perturbation

Suppose we are given an n×n matrix, M, and a set of values, (m?n), and we wish to find the smallest perturbation in the 2-norm (i.e., spectral norm), ΔM, such that M-ΔM has the given eigenvalues λ_i. Some interesting results have been obtained for variants of this problem for fixing two distinct eigenvalues, fixing one double eigenvalue, and fixing a triple eigenvalue. This paper provides a geometric motivation for these results and also motivates their generalization. We also present numerical examples (both “successes” and “failures”) of fixing more than two eigenvalues by these generalizations.     

Reduced Gutzwiller formula with symmetry: Case of a Lie group

We consider a classical Hamiltonian H on R2d, invariant by a Lie group of symmetry G, whose Weyl quantization is a selfadjoint operator on L²(Rd). If χ is an irreducible character of G, we investigate the spectrum of its restriction to the symmetry subspace of L²(Rd) coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of in an interval of R, and interpret it geometrically in terms of dynamics in the reduced space R2d/G. Besides, oscillations of the spectral density of are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R2d.     

The case of equality in the Dobrushin-Deutsch-Zenger bound

Suppose that A=(a_i,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.     

Properties of cyclic subspace regression

Various properties of the regression vector produced by cyclic subspace regression with regard to the meancentered linear regression equation are put forth. In particular, the subspace associated with the creation of is shown to contain a basis that maximizes certain covariances with respect to , the orthogonal projection of onto a specific subspace of the range of X. This basis is constructed. Moreover, this paper shows how the maximum covariance values effect the . Several alternative representations of are also developed. These representations show that is a modified version of the l-factor principal components regression vector , with the modification occurring by a nonorthogonal projection. Additionally, these representations enable prediction properties associated with to be explicitly identified. Finally, methods for choosing factors are spelled out.     

An isomorphic version of the slicing problem

Here we show that any centrally-symmetric convex body has a perturbation which is convex and centrally-symmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense that the Banach-Mazur distance between T and K is . If K is a body of a non-trivial type then the distance is universally bounded. The distance is also universally bounded if the perturbation T is allowed to be non-convex. Our technique involves the use of mixed volumes and Alexandrov-Fenchel inequalities. Some additional applications of this technique are presented here.     

On the best constants in noncommutative Khintchine-type inequalities

We obtain new proofs with improved constants of the Khintchine-type inequality with matrix coefficients in two cases. The first case is the Pisier and Lust-Piquard noncommutative Khintchine inequality for p=1, where we obtain the sharp lower bound of in the complex Gaussian case and for the sequence of functions . The second case is Junge's recent Khintchine-type inequality for subspaces of the operator space RC, which he used to construct a cb-embedding of the operator Hilbert space OH into the predual of a hyperfinite factor. Also in this case, we obtain a sharp lower bound of . As a consequence, it follows that any subspace of a quotient of (RC)^* is cb-isomorphic to a subspace of the predual of the hyperfinite factor of type III₁, with cb-isomorphism constant. In particular, the operator Hilbert space OH has this property.     

Uniform non-amenability

For any finitely generated group G an invariant ?0 is introduced which measures the “amount of non-amenability” of G. If G is amenable, then . If , we call G uniformly non-amenable. We study the basic properties of this invariant; for example, its behaviour when passing to subgroups and quotients of G. We prove that the following classes of groups are uniformly non-amenable: non-abelian free groups, non-elementary word-hyperbolic groups, large groups, free Burnside groups of large enough odd exponent, and groups acting acylindrically on a tree. Uniform non-amenability implies uniform exponential growth. We also exhibit a family of non-amenable groups (in particular including all non-solvable Baumslag-Solitar groups) which are not uniformly non-amenable, that is, they satisfy . Finally, we derive a relation between our uniform Følner constant and the uniform Kazhdan constant with respect to the left regular representation of G.     

On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices

Let X_n be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and R_n an n×N random matrix independent of X_n. Assume, almost surely, as n→∞, the empirical distribution function (e.d.f.) of the eigenvalues of converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.