Multiplicative coordinate functionals and ideal-triangularizability
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Multiplicative coordinate functionals and ideal-triangularizability Marko Kandi´c
Received: 11 October 2012 / Accepted: 7 January 2013 © Springer Basel 2013
Abstract In this paper we investigate how strong is the presence of atoms in Banach lattices corresponding to ideal-triangularizability of semigroups of positive operators. In the first part of the paper we prove that a semigroup S of positive operators on an atomic Banach lattice with order continuous norm is ideal-triangularizable if and only if every coordinate functional φa,a associated to an atom a is multiplicative on S for all atoms a in E. We apply this result to the case of positive ideal-triangularizable compact operators on not necessarily atomic lattices. In the second part of the paper we prove that the spectrum of a power compact ideal-triangularizable operator T satisfies σ (T )\{0} = {ϕa (T a) : a is an atom in E}\{0}. We also prove that for a positive operator from some of the trace ideals the equality above between the spectrum σ (T ) and the set of diagonal entries implies that T is ideal-triangularizable. Keywords Banach lattices · Positive operators · Compact operators · Ideal-triangularizability · Spectrum Mathematics Subject Classification (2000)
47A15 · 47B65 · 47A10 · 47B06
1 Introduction and preliminaries A family F of operators on a normed Riesz space is ideal-reducible if there exists a nontrivial closed ideal invariant under every operator from F . If there exists a
M. Kandi´c (B) Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19 , 1000 Ljubljana , Slovenia e-mail: [email protected]
M. Kandi´c
maximal chain C of closed ideals that are invariant under F , then F is said to be ideal-triangularizable, and C is an ideal-triangularizing chain for the family F . R. Drnovšek proved in [5] that every maximal chain of closed ideals is also maximal as a chain of closed subspaces. Let C be a maximal chain of closed ideals in a normed Riesz space and let J be a closed ideal in C . The predecessor J− of J in C is a closed linear span of all closed ideals in C that are properly contained in J . Let F be a family of operators on a Banach lattice E, and let I and J be closed ideals of E satisfying J ⊆ I that are invariant under every member of F . Then F induces a family Fˆ of operators on the quotient Banach lattice I /J as follows. For each T ∈ F the operator Tˆ is defined on I/J by Tˆ (x + J ) = T x + J . Because of the invariance of I and J the operator Tˆ is a well-defined operator on I /J . Any such Fˆ is called a collection of ideal-quotients of the family F . Whenever an operator T is ideal-triangularizable and I is a closed ideal in one of on I /I− will be denoted by its ideal-triangularizing chains, the induced operator T TI . A set P of properties is said to be inherited by ideal-quotients if every family of ideal-quotients of a family of operators satisfying properties in P also satisfies the same properties. We recall the Ideal-triangularization lemma proved in [4]. Lemma 1 Let P b
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