"To the memory of W.A.J. Luxemburg and his mathematical legacy." Quaestiones Mathematicae, 47(sup1), pp. iii–iv

Abstract Let E E be a complex Banach lattice and T T is an operator in the center Z ( E ) = { T : ∣ T ∣ ≤ λ I for some λ } Z\left(E)=\left\{T:| T| \le \lambda I\hspace{0.33em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\lambda \right\} of E E . Then, the essential norm ‖ T ‖ e \Vert T{\Vert }_{e} of T T equals the essential spectral radius r e ( T ) {r}_{e}\left(T) of T T . We also prove r e ( T ) = max { ‖ T A d ‖ , r e ( T A ) } {r}_{e}\left(T)=\max \left\{\Vert {T}_{}\hspace{-0.35em}{}_{{A}^{d}}\Vert ,{r}_{e}\left({T}_{A})\right\} , where T A {T}_{A} is the atomic part of T T and T A d {T}_{}\hspace{-0.35em}{}_{{A}^{d}} is the nonatomic part of T T . Moreover, r e ( T A ) = limsup ℱ λ a {r}_{e}\left({T}_{A})={\mathrm{limsup}}_{{\mathcal{ {\mathcal F} }}}{\lambda }_{a} , where ℱ {\mathcal{ {\mathcal F} }} is the Fréchet filter on the set A A of all positive atoms in E E of norm one and λ a {\lambda }_{a} is given by T A a = λ a a {T}_{A}a={\lambda }_{a}a for all a ∈ A a\in A .

We prove that polynomial valuations on vector lattices correspond to orthosymmetric multilinear maps. As a consequence we obtain a concise proof of the correspondence between orthosymmetry and orthogonal additivity.

Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^{r}(Y, Z)$ and $B\in L^{r}(W, X)$ let $M_{A,\,B}$ be the two-sided multiplication operator from $L^{r}(X, Y)$ into $L^r(W,\,Z)$ defined by $M_{A,\,B}(T)=ATB$. We show that for every $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A,\, B}|=M_{|A|, \,|B|}$ for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.

Let $E$ and $F$ be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice $E$, which shows that in this case the unbounded disjointness operators from $E\to F$ separate the points of $E$. Then we show that every disjointness preserving operator $T:E\to F$ is norm bounded on an order dense ideal. In case $E$ has order continuous norm, this implies that that every unbounded disjointness preserving map $T:E\to F$ has a unique decomposition $T=R+S$, where $R$ is a bounded disjointness preserving operator and $S$ is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that $E=C(X)$, with $X$ a compact Hausdorff space, we show that every disjointness preserving operator $T:C(X)\to F$ is norm bounded on an norm dense sublattice algebra of $C(X)$, which leads then to a decomposition of $T$ into a bounded disjointness operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.

The Positivity VII conference in Leiden in July 2013 was the Zaanen Centennial Conference, on the occasion of the 100th birth year of Adriaan Cornelis Zaanen (1913-2003). Zaanen held the chair of analysis in Delft for five years and subsequently the chair of mathematical analysis in Leiden for more than twenty-five years, until his retirement in 1982. He played a prominent role in `Positivity' throughout that period. During the conference, Koos Grobler, Rien Kaashoek and Anton Schep, three of his former PhD students, shared their personal recollections of Zaanen with the audience. Pieter Zaanen, his oldest son, told about their family life. The written accounts of their speeches are included here, followed by a chronological list of material on the life and works of Zaanen that has appeared in print and on the internet over the years.

Let E be a Banach lattice and F a Banach space. A bounded linear operator T:E→F is an isomorphism on the positive cone of E if and only if T∗ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E is almost surjective if and only if T∗ is an isomorphism on the positive cone of F∗.

1.1. Normed spaces. Recall that a (real) vector space V is called a normed space if there exists a function ‖ · ‖ : V → R such that (1) ‖f‖ ≥ 0 for all f ∈ V and ‖f‖ = 0 if and only if f = 0. (2) ‖af‖ = |a| ‖f‖ for all f ∈ V and all scalars a. (3) (Triangle inequality) ‖f + g‖ ≤ ‖f ||+ ‖g‖ for all f, g ∈ V . If V is a normed space, then d(f, g) = ‖f−g‖ defines a metric on V . Convergence w.r.t this metric is called norm convergence. If V is a complete metric space w.r.t. this metric, then V is called a Banach space. Let (X,B, μ) be a measure space and 0 < p ≤ ∞. Then we shall first define L(X,μ), where we will usually write L and omit X and μ. For 0 < p < ∞ we define

It is clear that limy↓x f(y) ≤ limy↓x f(y). Analogously we can define limy↑x f(y) and limy↑x f(y) and we also have limy↑x f(y) ≤ limy↑x f(y). One can verify as in the sequential case that e.g. (1) limy↓x f(y) ≤ A if and only if for all > 0 there exists a δ > 0 such that f(y) < A+ for all y such that 0 < y − x < δ. (2) limy↓x f(y) ≤ A if and only if for all > 0 and δ > 0 there exists an y with 0 < y − x < δ such that f(y) < A+ . From these and other similar properties one sees that limy→x f(y) = A if and only if limy↑x f(y) = limy↑x f(y) = limy↓x f(y) = limy↓x f(y) = A. Now let F : (a, b)→ R. Then the Dini derivates of F at x are defined as

Recently, K.M.R. Audenaert (2010), and R.A. Horn and F. Zhang (2010) proved inequalities between the spectral radius of Hadamard products of finite nonnegative matrices and the spectral radius of their ordinary matrix product. We will prove these inequalities in such a way that they extend to infinite nonnegative matrices A and B that define bounded operators on the classical sequence spaces l(p).

At the international conference Positivity VI, held in Escorial, Spain on July 20–24, 2009, the editorial board of Positivity met and decided to dedicate a special issue in honor of Wim Luxemburg’s contributions to the area of positivity on the occasion of his 80th birthday. This issue of Positivity is the outcome of that decision. We will not try to summarize Wim Luxemburg’s contributions here, as Peter Dodds’ article in this issue will do this in much better way than I could do here. We would like to thank Ron Luxemburg for providing us with a current photograph of Wim (Fig. 1). The editors of Positivity wish Professor Luxemburg many further creative years to come.

Given two Banach function spaces we study the pointwise product space E · F , especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M ( E , F ) = F , where M ( E , F ) denotes the space of multiplication operators from E into F .

Let E be a reflexive Banach function space. Let T be a positive order continuous operator with values in E. Then the optimal domain [T, E] is (isomorphic to) a weighted L 1-space if and only if the operator T is an integral operator with kernel T(x, y), the adjoint operator T′ is a Carleman integral operator and there exists 0≤g∈E′ such that ϕ(y)=∥T y ′ (·)∥E≤T′g(y) a.e. on Y. In this case [T, E]=L 1(Y, ϕdυ).

Let Ir(L) denote the collection of all regular integral operators on L (1 < p <∞) with the regular norm ‖ · ‖r. Then Ir(L) is a Banach function space on X ×X with the Fatou property. In this paper we shall give two distinct descriptions of the order continuous dual, or associate space, of Ir(L). Our first description follows from a result on Ir(L). In [3] we showed that Ir(L) is equal to the Calderon space (L∞,1) 1 p′ (L∞,1) 1 p . As a consequence we derive our first description via Lozanovskii’s duality theorem. Then we will present a different description of this space. We will prove that the order ideal generated by the finite rank operators on L provided with an extension of the positive projective tensor norm is a Banach function space with the Fatou property, from which it follows that it is isometric to the order continuous dual of Ir(L).

It is hard not to have Ray Redheffer's title of [2] as a reaction to another article on the Fundamental Theorem of Algebra. In fact at least 28 notes have appeared in this Monthly about this theorem. In this note we present nevertheless two proofs of the Fundamental Theorem of Algebra which do not seem to have been observed before and which we think are worth recording. The first one uses Cauchy's integral theorem and is, in the author's opinion, as simple as the most popular complex analysis proof based on Liouville's theorem (see [3] for this and three other proofs using complex analysis). Problem 5 on p. 126 of [1] gives a proof of the Fundamental Theorem of Algebra based on a complex contour integral that is similar to the one used here, but the details are not quite the same. The second one considers the integral obtained by parameterizing the contour integral from the first proof and uses only results from advanced calculus. This proof is similar to the proof of [4], where the same ideas were used to prove the nonemptiness of the spectrum of an element in a complex Banach algebra. There the companion matrix of a polynomial was then used to derive the Fundamental Theorem of Algebra.