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Opening gaps in the spectrum of strictly ergodic Schrödinger operators

Artur Avila, Jairo Bochi, David Damanik (2012)

Journal of the European Mathematical Society

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...

Operations between sets in geometry

Richard J. Gardner, Daniel Hug, Wolfgang Weil (2013)

Journal of the European Mathematical Society

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space ℝ n . It is proved that if n ≥ 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, G L ( n ) covariant, and associative if and only if it is L p addition for some 1 ≤ p ≤ ∞ . It is also demonstrated that if n ≥ 2 ,...

Operators commuting with translations, and systems of difference equations

Miklós Laczkovich (1999)

Colloquium Mathematicae

Let ℬ = f : ℝ → ℝ : f i s b o u n d e d , and ℳ = f : ℝ → ℝ : f i s L e b e s g u e m e a s u r a b l e . We show that there is a linear operator Φ : ℬ → ℳ such that Φ(f)=f a.e. for every f ∈ ℬ ∩ ℳ , and Φ commutes with all translations. On the other hand, if Φ : ℬ → ℳ is a linear operator such that Φ(f)=f for every f ∈ ℬ ∩ ℳ , then the group G Φ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from ℂ ℝ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every f ( x ) = e c x , then G Φ must...

Opial inequalities on time scales

Martin Bohner, Bıllûr Kaymakçalan (2001)

Annales Polonici Mathematici

We present a version of Opial's inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential equations and difference equations as special cases. Various extensions of our inequality are offered as well.

Opial's type inequalities on time scales and some applications

S. H. Saker (2012)

Annales Polonici Mathematici

We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equations on time scales. We also apply these inequalities to obtain a lower bound for the smallest eigenvalue of a Sturm-Liouville eigenvalue problem on time scales. The results contain as special cases some results obtained for second order differential...

Orthogonally additive mappings on Hilbert modules

Dijana Ilišević, Aleksej Turnšek, Dilian Yang (2014)

Studia Mathematica

We study the representation of orthogonally additive mappings acting on Hilbert C*-modules and Hilbert H*-modules. One of our main results shows that every continuous orthogonally additive mapping f from a Hilbert module W over 𝓚(𝓗) or 𝓗𝓢(𝓗) to a complex normed space is of the form f(x) = T(x) + Φ(⟨x,x⟩) for all x ∈ W, where T is a continuous additive mapping, and Φ is a continuous linear mapping.

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