H ¨ Older Type Inequalities for Orthosymmetric Bilinear Operators 1

The homogeneous functional calculus on vector lattices is a useful tool in a variety of areas. One of the interesting application is the study of powers of Banach lattices initiated by G. Ya. Lozanovskĭı [18]. Recently G. Buskes and A. van Rooij [8] introduced the concept of squares of Archimedean vector lattices which allows to represent orthoregular bilinear operators as linear regular operators. In particular, it is proved in [9] that the square of a relatively uniformly complete vector lattice can be constructed by well known p-convexification procedure (with p = 1/2) which is also based on the homogeneous functional calculus, see [17, 22]. The aim of this paper is to consider some interplay between squares of vector lattices and homogeneous functional calculus and obtain Hölder type inequalities for orthosymmetric bilinear operators. We also collect several useful facts concerning homogeneous functional calculus on relatively uniformly complete vector lattices some of which despite of their simplicity does not seem appeared in the literature. There are different ways to introduce the homogeneous functional calculus on vector lattices, see [6, 13, 17, 19, 21, 22]. We follow the approach [6, 9] going back also to G. Ya. Lozanovskĭı [19]. For the theory of vector lattices and positive operators we refer to the books [2] and [14]. All vector lattices in this paper are real and Archimedean. 1.1. We start by recalling some definitions and results from [7]. Let E and G be vector lattices. A bilinear operator b : E×E → G is said to be orthosymmetric if |x|∧ |y| = 0 implies b(x, y) = 0 for arbitrary x, y ∈ E, see [8]. If b(x, y) > 0 for all 0 6 x ∈ E and 0 6 y ∈ E, then b is named positive. The difference of two positive orthosymmetric bilinear operators is called orthoregular. Denote by BLor(E;G) the space of all orthoregular bilinear operators from E×E