Denote by 12(X) the oscillating spectrum of some subspace X of C[0, 1]. In this paper we show M is a 1'sigma set if and only if there exists a subspace X of C[0, 1] such that 12(X) = M; for any 1'sigma set M there exists a complemented subspace Y of C[0, 1] such that 12(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 12(Z) = N. (c) 2023 Elsevier Inc. All rights reserved.
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