[Authors]
 O. I. Reinov

[Title]
 On $Z_d$-symmetry of spectra of linear operators in Banach spaces

[AMS Subj-class]
 47B10 Operators belonging to operator ideals (nuclear, $p$-summing,
       in the Schatten-von Neumann classes, etc.)

[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 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 {\it there exists a nuclear
 operator $U$ in the space $l_1$ with the property that $trace\, U=1$ and
 $U^2=0.$}
  B. Mityagin (2016) generalized Zelikin's criterium to the case of compact
 operators in Banach spaces some of which powers are nuclear (he considered
 even the so-called $Z_d$-symmetry of spectra).
  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). Our results are optimal:
 We present the following (sharp) generalization of Grothendieck-Enflo theorem.
 {\bf Theorem}. Let  $p\in [1,\infty],  p \neq2,$ $1/r= 1+|1/2-1/p|.$
 There exists a nuclear operator $V$ in $l_p$ such that
  1) $V$ is $s$-nuclear for each $s\in (r, 1];$
  2) $V$ is not $r$-nuclear;
  3) $trace  V=1$ and $V^2=0.$
 The material is based on the talks at the 2017 International Conferences in
 Odessa ("Algebraic and geometric methods of analysis"), in Saint-Petersburg
 ("26th St.Petersburg Summer Meeting in Mathematical Analysis") and in Valencia
 ("Conference on Non Linear Functional Analysis-Spain").

[Comments]
 LaTeX, English, 23 pp.

[Contact e-mail]
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