A dichotomy result for a pointwise summable sequence of operators

AbstractLet X be a separable Banach space and Q be a coanalytic subset of XN×X. We prove that the set of sequences (ei)i∈N in X which are weakly convergent to some e∈X and Q((ei)i∈N,e) is a coanalytic subset of XN. The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, (aij)i,j∈N is a regular method of summability and (ei)i∈N is a bounded sequence in X, then there exists a subsequence (ei)i∈L such that either (I) there exists e∈X such that every subsequence (ei)i∈H of (ei)i∈L is weakly summable w.r.t. (aij)i,j∈N to e and Q((ei)i∈H,e); or (II) for every subsequence (ei)i∈H of (ei)i∈L and every e∈X with Q((ei)i∈H,e)the sequence (ei)i∈H is not weakly summable to e w.r.t. (aij)i,j∈N. This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232–234]. Both theorems obtain some considerable generalizations

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.Comment: to appear in the Journal of Functional Analysis 201

Ground states for fractional magnetic operators

We study a class of minimization problems for a nonlocal operator involving an external magnetic potential. The notions are physically justified and consistent with the case of absence of magnetic fields. Existence of solutions is obtained via concentration compactness.Comment: 22 pages, minor corrections and typos fixe