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Abstract: Sampling theory is concerned with the problem of reconstructing a signal in a Hilbert space from a given a collection of sampled values of . If a certain decomposition of the Hilbert space is possible (in terms of the sampling and reconstruction subspaces) then a consistent reconstruction can be obtained. In this paper we treat the case in which such a decomposition cannot be found. For this situation, we study the quasi-consistent reconstructions which are an extension of the consistent reconstructions. We relate the previous concepts to generalized inverses. We also present some new results and problems regarding consistent sampling. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a family of examples that can be regarded as spaces of nonpositive curvature, but with the distinct quality that they are not complete as metric spaces. This amounts to the fact that they are modelled on a finite von Neumann algebra, and the metrics introduced arise from the trace of the algebra. In spite of the noncompleteness of these manifolds, their geometry can be studied from the view-point of metric geometry, and several techniques derived from the functional analysis are applied to gain insight on their geodesic structure. [Copyright &y& Elsevier]

Abstract: We study the geometry of the set , with , which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of . These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920–931], we introduced the metric and exposed several results about this metric space. The aim of this work is to prove that the space behaves in many senses like a nonpositive curvature metric space. [Copyright &y& Elsevier]

Abstract: The aim of this work is to apply the complex interpolation method to norms of n-tuples of operators in the p-Schatten classes of a Hilbert space H. The norms considered define Finsler metrics in a certain manifold of positive operators, and can be regarded as weighted p-norms, the weight being a positive invertible operator. As a by-product, we obtain Clarkson''s type inequalities. [Copyright &y& Elsevier]