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).     

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.     

Saturated packings and reduced coverings obtained by perturbing tilings

Let a normed space X possess a tiling T consisting of unit balls. We show that any packing P of X obtained by a small perturbation of T is completely translatively saturated; that is, one cannot replace finitely many elements of P by a larger number of unit balls such that the resulting arrangement is still a packing.In contrast with that, given a tiling T of Rⁿ with images of a convex body C under Euclidean isometries, there may exist packings P consisting of isometric images of C obtained from T by arbitrarily small perturbations which are no longer completely saturated. This means that there exists some positive integer k such that one can replace k−1 members of P by k isometric copies of C without violating the packing property. However, we quantify a tradeoff between the size of the perturbation and the minimal k such that the above phenomenon occurs.Analogous results are obtained for coverings.     

Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure

We discuss exponential asymptotic property of the solution of a parallel repairable system with warm standby under common-cause failure. This system can be described by a group of partial differential equations with integral boundary. First we show that the positive contraction C₀-semigroup T(t) [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] which is generated by the operator corresponding to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] that 0 is an eigenvalue of the operator with algebraic index one and the C₀-semigroup T(t) is contraction, we conclude that the spectral bound of the operator is zero. By using the above results the exponential asymptotical stability of the time-dependent solution of the system follows easily.     

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.     

The role of directional compactness in the existence and computation of contingent epiderivatives

This paper underlines the role of directional compactness in the scalarization of graphical derivatives of set-valued maps taking values in infinite-dimensional spaces. Two main theorems are given. The first one states the equivalence of contingent epiderivatives and τw-contingent epiderivatives for directionally compact maps. The second main result proves a variational characterization for the contingent epiderivative of stable and directionally compact maps taking values in general image spaces, extending known results in finite-dimensional and reflexive Banach spaces. The hypotheses given are minimal as is shown by means of several examples. Connections of these theorems with other results of the literature are also provided.     

A noncommutative Brooks-Jewett Theorem

In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive-analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C^*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks-Jewett property if, and only if, it is topologically equivalent to an abelian algebra.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

A functional characterization of measures and the Banach-Ulam problem

For a measurable space (Ω,A), let ?_∞(A) be the closure of span{χA:A∈A} in ?_∞(Ω). In this paper we show that a sufficient and necessary condition for a real-valued finitely additive measure μ on (Ω,A) to be countably additive is that the corresponding functional ?μ defined by (for x∈?_∞(A)) is w^*-sequentially continuous. With help of the Yosida-Hewitt decomposition theorem of finitely additive measures, we show consequently that every continuous functional on ?_∞(A) can be uniquely decomposed into the ?₁-sum of a w^*-continuous functional, a purely w^*-sequentially continuous functional and a purely (strongly) continuous functional. Moreover, several applications of the results to measure extension are given.     

Absolutely convergent Fourier series and function classes. II

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes Λ_∗(α) and λ_∗(α) for some 0<α?2. This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168-1177] under the same title, and we keep its notations.     

Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

On convergence of closed convex sets

In this paper we introduce a convergence concept for closed convex subsets of a finite-dimensional normed vector space. This convergence is called C-convergence. It is defined by appropriate notions of upper and lower limits. We compare this convergence with the well-known Painlevé-Kuratowski convergence and with scalar convergence. In fact, we show that a sequence (An)_n∈NC-converges to A if and only if the corresponding support functions converge pointwise, except at relative boundary points of the domain of the support function of A, to the support function of A.     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

On the weak-approximate fixed point property

Let X be a Banach space and C a bounded, closed, convex subset of X. C is said to have the weak-approximate fixed point property if for any norm-continuous mapping , there exists a sequence {xn} in C such that (xn−fn(xn)) converges to 0 weakly. It is known that every infinite-dimensional Banach space with the Schur property does not have the weak-approximate fixed point property. In this article, we show that every Asplund space has the weak-approximate fixed point property. Applications to the asymptotic fixed point theory are given.     

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H_∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.     

Numerical index of absolute sums of Banach spaces

We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of c₀-, ?₁- and ?_∞-sums and giving as a new result the case of E-sums where E has the RNP and n(E)=1 (in particular for finite-dimensional E with n(E)=1). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional Lp(μ)-spaces coincide.     

Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces

The existence of common fixed points is established for three mappings where T is either generalized (f,g)-nonexpansive or asymptotically (f,g)-nonexpansive on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained. Our results unify and substantially improve several recent results existing in the current literature.     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

Growth order and stability of semigroups and cosine operator functions

Equivalent conditions for a trajectory of a C₀-semigroup T(⋅) (resp. cosine function C(⋅)) of operators to have the growth order O(tα) or o(tα) are expressed in terms of Cesàro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T(⋅) and C(⋅). It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T(⋅) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.