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2023•Journal of Functional AnalysisOpen access

On the existing set of the oscillating spectrum

Rui Xie

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Abstract

Denote by Ω ( X ) the oscillating spectrum of some subspace X of C [ 0 , 1 ] . In this paper we show M is a F σ set if and only if there exists a subspace X of C [ 0 , 1 ] such that Ω ( X ) = M ; for any F σ set M there exists a complemented subspace Y of C [ 0 , 1 ] such that Ω ( Y ) = M , which answers two questions posed in [7] . Besides, we introduce the concept of large (small) oscillating elements. In particular, we show N is a nonempty closed set if and only if there exists a large oscillating element Z of C [ 0 , 1 ] such that Ω ( Z ) = N .

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What this paper is about

Denote by Ω ( X ) the oscillating spectrum of some subspace X of C [ 0 , 1 ] . In this paper we show M is a F σ set if and only if there exists a subspace X of C [ 0 , 1 ] such that Ω ( X ) = M ; for any F σ set M there exists a complemented subspace Y of C [ 0 , 1 ] such that Ω ( Y ) = M , which answers two questions posed in [7] . Besides, we introduce the concept of large (small) oscillating elements. In particular, we show N is a nonempty closed set if and only if there exists a large oscillating element Z of C [ 0 , 1 ] such that Ω ( Z ) = N .

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Available abstract

Denote by Ω ( X ) the oscillating spectrum of some subspace X of C [ 0 , 1 ] . In this paper we show M is a F σ set if and only if there exists a subspace X of C [ 0 , 1 ] such that Ω ( X ) = M ; for any F σ set M there exists a complemented subspace Y of C [ 0 , 1 ] such that Ω ( Y ) = M , which answers two questions posed in [7] . Besides, we introduce the concept of large (small) oscillating elements. In particular, we show N is a nonempty closed set if and only if there exists a large oscillating element Z of C [ 0 , 1 ] such that Ω ( Z ) = N .

Key concepts: Subspace topology, Mathematics, Spectrum (functional analysis), Existential quantification, Set (abstract data type), Element (criminal law), Combinatorics, Pure mathematics

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