A variant of Yano’s extrapolation theorem on Hardy spaces

In this note, a variant of Yano’s classical extrapolation theorem for sublinear operators acting on analytic Hardy spaces over the torus is obtained.     

Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces

In this paper, we investigate new properties of the generalized projection operators on convex closed cones in uniformly convex and uniformly smooth Banach spaces; establish decompositions theorems for arbitrary elements both in primary and dual spaces; and prove the Banach space analogue of the Pythagorean-type theorem. Earlier, all these results were known only in Hilbert spaces.     

Application of Michael's theorem and its converse to sublinear operators

A theorem of Michael on continuous selectors and its converse are used in this article to study subdifferentials of continuous sublinear operators with values in a cone of lower semicontinuous functions. It is proved that such operators are subdifferentiable (i.e., have nonempty subdifferentials) if their domains are separable Banach spaces. Sublinear operators that are not subdifferentiable are found.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 67–75, July, 1992.     

A pseudospectral mapping theorem

The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.

     

Factorization theorem for 1-summing operators

We study some classes of summing operators between spaces of integrable functions with respect to a vector measure in order to prove a factorization theorem for 1-summing operators between Banach spaces.     

Besov spaces via wavelets on metric spaces endowed with doubling measure,singular integral,and the T1 type theorem

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

Spectral mapping theorem for ascent,essential ascent,descent and essential descent spectrum of linear relations

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.     

A generalized version of the Birkhoff-Kellogg theorem

The present note deals with generalizations of the classical Birkhoff-Kellogg theorem. By means of the Leray-Schauder degree and the introduced notion of a flat neighborhood, some results on the existence of eigenvectors for compact operators defined on the boundary of an unbounded set in a separated linear topological space being admissible in the sense of Klee are obtained. Positive operators on ordered spaces are considered, as well.     

Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-H¨older continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.     

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.     

An inverse function theorem in Fréchet-Spaces

A technical inverse function theorem of Nash-Moser type is proved for maps between Fréchet spaces allowing smoothing operators. A counterexample shows that the growth requirements on the rightinverse of the linearized map needed are minimal.     

A proof of Parrott's theorem on quotient norms

We prove, as an application of our positive extension argument, a theorem of Parrott concerning the quotient norm with respect to spaces of Hilbert space operators.     

The new forms of Voronovskaya’s theorem in weighted spaces

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.     

An eigenvector existence theorem in idempotent analysis

In this paper, we prove an eigenvector existence theorem for linear operators on abstract idempotent spaces in the framework of the algebraic approach. Earlier, an algebraic version of a similar statement was known only for operators in free finite-dimensional semimodules. The corresponding result for compact operators in semimodules of real continuous functions is known in the case of topological semimodules.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

Spectral mapping theorem for generalized Kato spectrum

In this paper, we give an affirmative answer to Mbekhta's conjecture (Mbekhta, 1990 [13]) about the pseudo Fredholm operators in Hilbert space. As a consequence, we characterize pseudo Fredholm operators and we prove that the generalized Kato spectrum satisfies the spectral mapping theorem in the Hilbert spaces setting.