Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

Variations of Y-dominating functions on graphs

Let Y be a subset of real numbers. A Y-dominating function of a graph G=(V,E) is a function f:V→Y such that for all vertices v∈V, where N_G[v]={v}∪{u|(u,v)∈E}. Let for any subset S of V and let f(V) be the weight of f. The Y-domination problem is to find a Y-dominating function of minimum weight for a graph G=(V,E). In this paper, we study the variations of Y-domination such as {k}-domination, k-tuple domination, signed domination, and minus domination for some classes of graphs. We give formulas to compute the {k}-domination, k-tuple domination, signed domination, and minus domination numbers of paths, cycles, n-fans, n-wheels, n-pans, and n-suns. Besides, we present a unified approach to these four problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. This paper also gives complexity results for the problems on doubly chordal graphs, dually chordal graphs, bipartite planar graphs, chordal bipartite graphs, and planar graphs.     

Some applications of free Fisher information on frame theory

We introduce the notion of operator-valued free Fisher information with respect to a positive map of a random variable in an operator-valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations. Then we can apply this notion on the study of frame theory, especially on the disjointness problem of modular frames arising from conditional expectations.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

Property (w) and perturbations II

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) 683-692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T. A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T, also under the assumption that T is a-isoloid.     

Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H²(S)) basis of polynomials.     

Global bifurcation for a class of degenerate elliptic equations with variable exponents

We are concerned with the following nonlinear problem

     

Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators

We adapt recent results of Albrecht and Ricker to obtain conditions under which growth constraints on the left resolvent of a Banach space operator are preserved under suitable perturbations. As an application, we establish Bishop's property (β) for certain generalized Cesàro operators on the classical Hardy spaces Hp, 1<p<∞. Our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j(A) (resp., k(A)) as the smallest nonnegative integer j (resp., k) such that ker(I−Aj∗Aj) (resp., ker(I−Ak*Ak)∩ker(I−AkAk∗)) equals the subspace of H on which the unitary part of A acts. We show that if , then j(A)?n (resp., k(A)?⌈n/2⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with ‖An−2‖=1 and ‖An−1‖<1, and (3) n is even and A=U⊕A^′, where U is unitary on a one-dimensional space and A^′ is of class Sn−1).     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

Improved bounds on the supremum of autoconvolutions

We give an improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions.     

Fixed point theorems for better admissible multimaps on almost convex sets

We obtain new fixed point theorems on multimaps in the class Bp defined on almost convex subsets of topological vector spaces. Our main results are applied to deduce various fixed point theorems, coincidence theorems, almost fixed point theorems, intersection theorems, and minimax theorems. Consequently, our new results generalize well-known works of Kakutani, Fan, Browder, Himmelberg, Lassonde, and others.     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

Addendum to “Variations of selective separability” [Topology Appl. 156 (7) (2009) 1241-1252]

Two proofs omitted in the paper mentioned in the title are presented.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

Estimates of the dilatation function of Beurling-Ahlfors extension

In this paper we estimate the dilatation function of the Beurling-Ahlfors extension in the most general case. By introducing ?h,m-function, we obtain an inequality which is sharp up to a constant.     

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u^′(t)+B(t,u(t))∋f(t) is studied. In this problem, the operators are of type (M) or type (S₊), which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.     

Multiplicity of characteristics with Lagrangian boundary values on symmetric star-shaped hypersurfaces

In this paper, the multiplicity of Lagrangian orbits on C² smooth compact symmetric star-shaped hypersurfaces with respect to the origin in R2n is studied. These Lagrangian orbits begin from one Lagrangian subspace and end on another. An infinitely many existence result is proved via Z₂-index theory. This is a multiplicity result about the Arnold Chord Conjecture in some sense, and is a generalization of the problem about the multiplicity of Lagrangian orbits beginning from and ending on the same Lagrangian subspace which was considered in the authors' previous paper [F. Guo, C. Liu, Multiplicity of Lagrangian orbits on symmetric star-shaped hypersurfaces, Nonlinear Anal. 69 (4) (2008) 1425-1436].     

Riesz means associated with certain product type convex domain

In this paper we study the maximal operators and the convolution operators Tδ associated with multipliers of the form