We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p−Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of norm-parallelism problem in p−Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

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The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a puniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of seminegative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. A characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group G. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting.

Abstract. In this paper we present refinements and improvement of the Young inequality in the context of linear bounded operators.