Anton R. Schep

Measures of non-compactness of operators in Banach lattices

Abstract Let E and F be complex Banach lattices. Then a measure of non-semicompactness ϱ(T) is introduced for an order bounded operator T from E into F. If E∗ and F have order continuous norm then for every AM-compact operator ϱ(T) = β(T), where β(T) denotes the ball measure of non-compactness of T. From this result monotonicity properties of β(T) and the essential spectral radius ress(T) are derived for AM-compact operators. Also shown is that ress(T) ϵ σess(T) for positive AM-compact operators. In addition properties of the essential spectrum of norm bounded disjointness preserving operators are proved.

Norms of positive operators on ^{}-spaces

Let 0 < T: LP(Y, v) -+ Lq(X, ) be a positive linear operator and let HITIP ,q denote its operator norm. In this paper a method is given to compute 1Tllp, q exactly or to bound 11Tllp q from above. As an application the exact norm 11VIlp,q of the Volterra operator Vf(x) = fo f(t)dt is computed.

Minkowski’s Integral Inequality for Function Norms

Let ρ and λ be Banach function norms with the Fatou property. Then the generalized Minkowski integral inequality ρ(λ(f x )) ≤ Mλ(ρ(f y )) holds for all measurable functions f(x,y) and some fixed constant M if and only if there exists 1 ≤ ρ ≤ ∞ such that λ is p-concave and ρ is p-convex.

A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra

It is hard not to have Ray Redheffers 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 Cauchys integral theorem and is, in the authors opinion, as simple as the most popular complex analysis proof based on Liouvilles 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.

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.

Up-down theorems in the centre of Lb(E,F)

If E and F are Riesz spaces, several theorems are given which describe the centre of the space of order bounded operators from E to F in terms of the monotone closure of certain sublattices constructed from the centres of E and F. The methods yield an alternative approach to an approximation theorem of Kalton and Saab for positive operators on Banach lattices.

When is the Optimal Domain of a Positive Linear Operator a Weighted L1-space?

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).

Positive Operators on L p -spaces

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

Daugavet type inequalities for operators on Lp–spaces

AbstractLet T be a regular operator from Lp → Lp. Then