On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Approximating eigenfunctions of Fredholm operators in Banach spaces

A singular Fredholm operator A is perturbed by an operator of finite rank to obtain an invertible operator B. Theory previously developed for A and B in Hilbert spaces is extended here to Banach spaces. The operator B^?1 is used to construct independent elements in the null spaces N(A), N(A²),…, N(A^k), for some positive integer k, and a basis for N(A) and N(A²). The theory is used to compute approximations to eigenfunctions and generalized eigenfunctions of integral operators.     

Characterization of the trace by monotonicity inequalities

Let ?be a positive linear functional on the algebra of n × n complex matrices and p be a number greater than 1. The main result of the paper says that if for any pair A, B of positive semi-definite n × n matrices with A ? B the inequality ?(A^p) ? ?(B^p) holds true, then ?is a nonnegative scalar multiple of the trace.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k?2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let G_m(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from G_m(V) onto itself such that for any A,B in G_m(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.     

Intersection problems and integral equivalence

Let A and B be matrices over a principal ideal domain, Π. Necessary conditions, involving the invariant factors of A and B, are given for B to be a submatrix of A or a principal submatrix of A.If a given nonnegative integral matrix, B, is the intersection matrix of a pair of families of subsets of an n-set, and n is the smallest integer for which this is true, we say that the content of B is n. In that event, B is a submatrix of K(n), the intersection matrix of all subsets of an n-set. More refined results are obtained in certain cases by considering S(n, k, l), the intersection matrix of the k-subsets of an n-set versus its l-subsets. The invariant factors of K(n) and S(n, k, l) are calculated and it is shown how this information may be used to get lower bounds for the content of B. In the more widely studied symmetric version of the content problem, B must be a principal submatrix of K(n) or, possibly, S(n, k) = S(n, k, k). In this case, the invariant factors of K(n) ? xI or S(n, k) ? xI also provide relevant information.     

On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient α_m,k(A,B) of t^k in the polynomial Tr(A+tB)^m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.     

On the stability properties of perturbed linear nonstationary systems

Several results are presented that relate the stability properties of a perturbed linear nonstationary system ?(t) = (A(t) + B(t)) x(t) to those of an unperturbed linear system ?(t) = A(t) x(t). Similarly, the stability properties of the discrete system x_{k + 1} = (A_k + B_k) x_k are related to those of x_{k + 1} = A_kx_k.     

On the growth of solutions of second order linear differential equations with extremal coefficients

In this paper, we consider the differential equation f + A(z)f + B(z)f = 0, where A and B ≡ 0 are entire functions. Assume that A is extremal for Yang's inequality, then we will give some conditions on B which can guarantee that every non-trivial solution f of the equation is of infinite order.     

Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.     

Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

On permanents of positive semidefinite matrices

It is shown in an elementary way that if A and B are positive semidefinite matrices, then per(A + B) ? per A + per B. The conditions under which equality may occur in this inequality are completely described, and some consequences are given.     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.     

On Krull and Valuative Dimension of Tensor Products of Algebras

This article is concerned with the dimension theory of tensor products of algebras over a field k. In fact, we provide formulas for the Krull and valuative dimension of A? _k B when A and B are k-algebras such that the polynomial ring A[n] is an AF-domain for some positive integer n. Also, we compute dim _v (A? _k B) in the case where A ? B.     

Embedding of products Q(k) × B(τ) in absolute A-sets

Theorems about closed embeddings in absolute A-sets of the products Q(k) × B(τ), Q(k) ×N, and Q(k) × C are proved. These are generalizations to the nonseparable case of theorems of Saint-Raymond, van Mill, and van Engelen about closed embeddings in separable absolute Borel sets of the products Q × N and Q × C, where Q is the space of rational numbers, C is the Cantor perfect set, and N is the space of irrational numbers.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Generalizations of Bohr inequality for Hilbert space operators

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A,B∈B(H) and p,q>1 real numbers such that 1/p+1/q=1,

²|A+B|?p²|A|+q²|B|