Some remarks on spectra of nuclear operators

2018/03/27 by Reinov, Oleg I.
#47B06 #47B10 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1803.10021

Abstract

It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric iff the spectral traces of all odd powers of the operator equal zero. The criterium can not be extended to the case of general Banach spaces: It follows from Grothendieck-Enflo results that there exists a nuclear operator U in the space l₁ with the property that trace U=1 and U²=0. B. Mityagin (2016) has generalized Zelikin's criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear. We give sharp generalizations of Zelikin's theorem (to the cases of subspaces of quotients of L_p-spaces) and of Mityagin's result (for the case where the operators are not necessarily compact).

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