W. A. J. Luxemburg

Multiplicativity factors for seminorms. II

Let S be a seminorm on an algebra A. In this paper we study multiplicativity and quadrativity factors for S, i.e., constants μ > 0 and λ > 0 for which S(xy) ⩽ μS(x)S(y) and S(x2) ⩽ λS(x)2 for all x, y ∈ A. We begin by investigating quadrativity factors in terms of the kernel of S. We then turn to the question, under what conditions does S have multiplicativity factors if it has quadrativity factors? We show that if A is commutative then quadrativity factors imply multiplicativity factors. We further show that in the noncommutative case there exist both proper seminorms and norms that have quadrativity factors but no multiplicativity factors.

Stable Subnorms II

In this paper we continue our study of stability properties of subnorms on subsets of finite-dimensional, power-associative algebras over the real or the complex numbers.

Operator Theory in Function Spaces and Banach Lattices

This volume is dedicated to A.C. Zaanen, one of the pioneers of functional analysis, and eminent expert in modern integration theory and the theory of vector lattices, on the occasion of his 80th birthday. The book opens with biographical notes, including Zaanens curriculum vitae and list of publications. It contains a selection of original research papers which cover a broad spectrum of topics about operators and semigroups of operators on Banach lattices, analysis in function spaces and integration theory. Special attention is paid to the spectral theory of operators on Banach lattices; in particular, to the one of positive operators. Classes of integral operators arising in systems theory, optimization and best approximation problems, and evolution equations are also discussed. The book will appeal to a wide range of readers engaged in pure and applied mathematics.

Maharam extensions of positive operators and f-modules

The principal result of this paper is the construction of simultaneous extensions of collections of positive linear operators between vector lattices to interval preserving operators (i.e., Maharam operators). This construction is based on some properties of so-called f-modules. The properties and structure of these extension spaces is discussed in some detail.

The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any L_∞/C_0 without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.

Diagonals of the Powers of an Operator on a Banach Lattice

This paper is devoted to a detailed study of the properties of the band projection D of the complete lattice ordered algebra − r(E) of the regular (or order bounded) operators of a Dedekind complete Banach lattice E onto the center Z(E) of E. We recall that the center Z(E) is the commutative subalgebra of − r(E) of all T satisfying |T| ≤ λI, where I is the identity operator. In the finite dimensional case, with respect to the standard numerical basis, Z(E) is the algebra of all diagonal matrices. For this reason the band projection D is called the diagonal map of E.

Not all quadrative norms are strongly stable

A norm N on an algebra A is called quadrative if N(x2) ≤ N(x)2 for all x ∈ A, and strongly stable if N(xk) ≤ N(x)k for all x ∈ A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable.

Curriculum Vitae of A.C. Zaanen

Adriaan Cornelis Zaanen was born June 14, 1913, Rotterdam, The Netherlands. He is married with Ada Jacoba van der Woude. They have four sons, born in 1946, 1948, 1957 and 1959.