Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) _n=1 ^∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= _n=1 ^∞ inL ¹(M,τ), adapted to (M _n ) _n=1 ^∞ , there is a decomposition into two sequences (x _n ) _n=1 ^∞ and (z _n ) _n=1 ^∞ withx _n=y _n+z _nfor everyn≥1 and such that . This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces. Supported in part by NSF grant DMS-0096696.     

Fixed point properties in Hardy spaces

We prove that the Hardy space H¹ on the unit disc of C has the weak^∗ fixed point property for left reversible semigroups.     

Pelczynski's property (V*) for symmetric operator spaces

We show that if a rearrangement invariant Banach function space on the positive semi-axis contains no subspace isomorphic to then the corresponding space of -measurable operators, affiliated with an arbitrary semifinite von-Neumann algebra equipped with a distinguished faithful, normal and semifinite trace , has Pe{\l}czy\'{n}ski's property (V*).

     

Asymptotically isometric copies of czynski

We introduce the notion of a Banach space containing an asymptotically isometric copy of . A well known result of Bessaga and Peczynski states a Banach space contains a complemented isomorphic copy of if and only if contains an isomorphic copy of if and only if contains an isomorphic copy of . We prove an asymptotically isometric analogue of this result.

     

Selections, extensions and collectionwise normality

We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well.     

Solitary wave dynamics in shallow water over periodic topography

The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.     

A remark on maximal functions for noncommutative martingales

Let ${\mathcal{M}}$ be a finite von Neumann algebra equipped with a normal tracial state τ. It is shown that if ${\{x_n\}_{n\geq1}}$ is a sequence of positive marginales that is bounded in ${L^1(\mathcal{M},\mathcal{T})}$ , then for every 0 < p < 1, there exists ${y \in L^p(\mathcal{M},\mathcal{T})}$ satisfying the property that ${x_n \leq y}$ for all ${n\geq 1}$ . Thus we obtain a noncommutative analogue of a maximal function theorem from classical martingale theory.