Yu Can ZHU
In this paper, we introduce the concepts of q-Besselian frame and (p, σ)-near Riesz basis in a Banach space, where σ is a finite subset of positive integers and 1/p+1/q=1 with p>1, q>1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and (p, σ)-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X* that if {xn}n=1∞⊂X is a q-Besselian frame for X, then there exists a p-Besselian frame {yn*}n=1∞⊂X for X* such that x= ∑n=1∞ yn*(x)xn for all x∈X, and x*=∑n=1∞ x*(xn)yn* for all x*∈X*. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces.
Y, the injective tensor product of X and Y, has the Radon-Nikodym property (respectively, the analytic Radon-Nikodym property, the near Radon-Nikodym property, non-containment of a copy of c0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.
