@MISC{Krasikova_onthe, author = {I. V. Krasikova and V. M. Kadets and M. M. Popov}, title = {ON THE SPECTRUM OF NARROW OPERATORS}, year = {} }

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Abstract

Narrow operators were introduced and considered by A. Plichko and M. Popov in [5] acting from a symmetric function space E to a Banach space (or, more general, to an F-space) X. To some extend, this notion generalizes compact operators, still having some of their properties. Then narrow operators were considered in some other settings by different authors [1-4]. It is well known that the spectrum of a compact operator is a countable set containing zero. As it was communicated to us by M. M. Popov, the following question is due to M. L. Gorbachuk: what subsets of the complex plane can be the spectrum of a narrow operator? We answer this question for the setting of narrow operators defined on a Köthe function space. Let (Ω,Σ, µ) be an atomless σ-finite measure space. A complex Banach space E of equivalence classes of Σ-measurable functions x: Ω → C is called a Köthe function space on (Ω,Σ, µ) if for each x ∈ E, y ∈ L0(µ) the condition |y | ≤ |x| implies that y ∈ E and ‖y ‖ ≤ ‖x ‖ (here L0(µ) denotes the set of all equivalence classes of Σ-measurable functions, and the inequality u ≤ v for u, v ∈ L0(µ) means that u(ω) ≤ v(ω) holds a.e. on Ω). Besides, it is also supposed that 1A ∈ E for each A ∈ Σ with µ(A) <∞. Let E be a Köthe function space on (Ω,Σ, µ) and X be a Banach space. A continuous linear operator T: E → X is called narrow if for every A ∈ Σ and every ε> 0 there is an x ∈ E such that x2 = 1A, Ω xdµ = 0 and ‖Tx ‖ < ε. Theorem 1. Let E be a Köthe function space on an atomless σ-finite measure space (Ω,Σ, µ). Let there exists a purely atomic sub-σ-algebra F ⊆ Σ with Ω ∈ F such that the conditional expectation operator M(·/F) : E → E is correctly defined and bounded. Then a subset K ⊂ C is the spectrum of some narrow operator T: E → E if and only if K is a compact set in C containing zero.