A Characterization of Compact-friendly Multiplication Operators

Answering in the affirmative a question posed in [Y.A.Abramovich, C.D.Aliprantis and O.Burkinshaw, Multiplication and compact-friendly operators, Positivity 1 (1997), 171--180], we prove that a positive multiplication operator on any $L_p$-space (resp. on a $C(\Omega)$-space) is compact-friendly if and only if the multiplier is constant on a set of positive measure (resp. on a non-empty open set). In the process of establishing this result, we also prove that any multiplication operator has a family of hyperinvariant bands -- a fact that does not seem to have appeared in the literature before. This provides useful information about the commutant of a multiplication operator.Comment: To appear in Indag. Math., 12 page

Invariant Subspaces of Operators on lp-Spaces

AbstractWhile the algebra of infinite matrices is more or less reasonable, the analysis is not. Questions about norms and spectra are likely to be recalcitrant. Each of the few answers that is known is considered a respectable mathematical accomplishment.P.R. Halmos [3, p. 24]A continuous operator T: X → X on a Banach space is quasinilpotent at a pointx0 whenever limn→∞||Tnx0||1/n = 0. Several results on the existence of invariant subspaces of operators which act on lp-spaces and are quasinilpotent at a non-zero point are obtained. For instance, it is shown that if an infinite positive matrix A = [aij] defines a continuous operator on an lp-space (1 ≤ p < ∞) and A is quasinilpotent at a positive vector, then for any bounded double sequence of complex numbers {wij: i,j = 1, 2, ... } the operator defined by the weighted infinite matrix [wijaij] has a non-trivial complemented invariant closed subspace

An Elementary Proof of Douglas′ Theorem on Contractive Projections on L1-Spaces

AbstractDouglas (Pacific J. Math.15 (1965), 443-462) has shown that the conditional expectation operators are the only contractive projections on L1(σ) that leave the constant functions invariant. This remarkable result has applications to several areas and our objective is to present here an elementary and self-contained proof. Douglas* theorem has been proven and generalized in various contexts by many authors; for details see Ando (Pacific J. Math.17 (1966), 391-405), Bernau and Lacey (Pacific J. Math.53 (1974), 21-41), and Dodds et al. (Pacific J. Math.141 (1990), 55-77) and the references therein

The Invariant Subspace Problem: Some Recent Advances

This paper is devoted to recent developments regarding the invariant subspace problem for positive operators on Banach lattices. Some of this material was presented by Y. A. Abramovich at "Workshop di Teoria della Misura e Analisi Reale" Grado (Italia) 18-30 September 1995