On lower and upper bounds of matrices

Using an approach of Bergh, we give an alternate proof of Bennett's result on lower bounds for non-negative matrices acting on non-increasing non-negative sequences in lp when p?1 and its dual version, the upper bounds when 0<p?1. We also determine such bounds explicitly for some families of matrices.     

Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices

New upper and lower bounds are given for the arithmetic,geometric and harmonic means of a set of positive definite matrices     

Upper and lower bounds for the Perron root of a nonnegative matrix

We derive upper and lower bounds for the Perron root of a nonnegative matrix by using generalized Gershgorin inclusion regions. Our bounds seem particularly effective for certain sparse matrices.     

The polynomial numerical hull of a matrix and algorithms for computing the numerical range

Let A be an n × n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A.     

Factoring matrices into the product of two matrices

Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix. The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of these ideas are outlined. Several examples on factoring are presented. AMS subject classification (2000)  15A23, 65F30     

Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

In this paper we prove mixed-means inequalities for integral power means of an arbitrary real order, where one of the means is taken over the ball , centered at and of radius , δ>0. Therefrom we deduce the corresponding Hardy-type inequality, that is, the operator norm of the operator S_δ which averages over , introduced by Christ and Grafakos in Proc. Amer. Math. Soc. 123 (1995) 1687-1693. We also obtain the operator norm of the related limiting geometric mean operator, that is, Carleman or Levin-Cochran-Lee-type inequality. Moreover, we indicate analogous results for annuli and discuss estimations related to the Hardy-Littlewood and spherical maximal functions.     

A Note on Pluripolar Hulls of Graphs of Blaschke Products

In the paper we study the problem of characterization of pluripolar hulls of a class of functions containing Blaschke products. Mathematics Subject Classifications (2000) 32U30, 31A15, 30D50.Research partially supported by the KBN grant No. 5 P03A 033 21 and the Rectors Scholarship Fund at the Jagiellonian University.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

Equality conditions for lower bounds on the smallest singular value of a bidiagonal matrix

Several lower bounds have been proposed for the smallest singular value of a square matrix, such as Johnson’s bound, Brauer-type bound, Li’s bound and Ostrowski-type bound. In this paper, we focus on a bidiagonal matrix and investigate the equality conditions for these bounds. We show that the former three bounds give strict lower bounds if all the bidiagonal elements are non-zero. For the Ostrowski-type bound, we present an easily verifiable necessary and sufficient condition for the equality to hold.     

Relaxation results for functions depending on polynomials changing sign on rank-one matrices

In this paper, we are interested in computing the different convex envelopes of functions depending on polynomials, especially those having it is main part change sign on rank-one matrices. Our main result applies to functions of the type W(F)=φ(P(F)), W(F)=φ(P(F))+f(detF) or W(F)=φ(P(F))+g(adjnF) defined on the space of matrices, where φ, f:R→R and g:R³→R are three continuous functions, and P=P₀+P₁+?+Pd is a polynomial such that Pd has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.     

Additive multiplicative increasing functions on nonnegative square matrices and multidigraphs

It is known that if f is a multiplicative increasing function on , then either f(n)=0 for all or f(n)=n for some 0. It is very natural to ask if there are similar results in other algebraic systems. In this paper, we first study the multiplicative increasing functions over nonnegative square matrices with respect to tensor product and then restrict our result to multidigraphs and loopless multidigraphs.     

On a method for interpolating functions on chaotic nets

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x| ^m form odd,K(x)=|x| ^m In |x| form even (x∈ℝ ⁿ ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝ ⁿ . We construct a function of the form

Research problem: on the norms of infinite Cauchy-Toeplitz-plus-Cauchy-Hankel matrices

We give a conjecture and research problem related to ℓ_p operator and spectral norms of the matrix T_n + H_n.     

Maps on spaces of symmetric matrices preserving idempotence

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let M_n(F) be the space of all n×n matrices over F, and let S_n(F) be the subset of M_n(F) consisting of all symmetric matrices. Let V{S_n(F),M_n(F)}, a map Φ:V→V is said to preserve idempotence if A-λB is idempotent if and only if Φ(A)-λΦ(B) is idempotent for any A,BV and λF. It is shown that: when the characteristic of F is not 2, |F|>3 and n4, Φ:S_n(F)→S_n(F) is a map preserving idempotence if and only if there exists an invertible matrix PM_n(F) such that Φ(A)=PAP^-1 for every AS_n(F) and P^tP=aI_n for some nonzero scalar a in F.