Mappings and Prohorov spaces

We consider completely regular Hausdorff spaces. In this paper we investigate the space of probability Radon measures P(X) on a space X and the property to be a Prohorov space. We prove that the space P(X) is sieve-complete if and only if X is sieve-complete. Every mapping generates the mapping . Some properties of the mapping P(φ) are studied. In particular, we investigate under which conditions the open continuous image of a Prohorov space is Prohorov.     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

Young’s inequality and trace

Let φ be a positive linear functional on M_n(C) and f,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality

     

An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set .Theorem: If dimLX=∞ and , then

     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

Walks and the spectral radius of graphs

Given a graph G, write μ(G) for the largest eigenvalue of its adjacency matrix, ω(G) for its clique number, and w_k(G) for the number of its k-walks. We prove that the inequalities

     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

Holomorphic cohomology groups on compact Kähler complex spaces

Let (X,OX) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups Hq(X,C) carry a mixed Hodge structure. In particular they carry a weight filtration W−lHq(X,C) (l=0,…,q), and the graded quotients have a direct sum decomposition into holomorphic invariants as . Here we investigate the relationships between the above invariants for r=0 and the cohomology groups , where is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c₁(L) in has pure type (1,1).     

Characterization of graphs having extremal Randi? indices

The higher Randi? index R_t(G) of a simple graph G is defined as

     

On the max version of the generalized spectral radius theorem

Let Ψ be a bounded set of n×n non-negative matrices. Recently, the max algebra version μ(Ψ) of the generalized spectral radius of Ψ was introduced. We show that

     

A Dunkl-Williams type inequality for absolute value operators

Dunkl and Williams showed that for any nonzero elements x,y in a normed linear space X

     

Accretive operators and Cassels inequality

Let a,b>0 and let Z∈M_n(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈M_n(R),

     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Stable solutions of linear systems involving long chain of matrix multiplications

This paper is concerned with solving linear system (I_n+B_L?B₂B₁)x=b arising from the Green’s function calculation in the quantum Monte Carlo simulation of interacting electrons. The order of the system and integer L are adjustable. Also adjustable is the conditioning of the coefficient matrix to give rise an extreme ill-conditioned system. Two numerical methods based on the QR decomposition with column pivoting and the singular value decomposition, respectively, are studied in this paper. It is proved that the computed solution by each of the methods is weakly backward stable in the sense that the computed is close to the exact solution of a nearby linear system

     

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip₀(X)→Lip₀(Y) be a surjective map between pointed Lipschitz ^∗-algebras, where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm ^∗-multiplicativity condition: