On a conjecture of Barry Simon an trace ideals (Q1826088)

Let H denote a Hilbert space, T a compact operator on H, \(\{s_ n(T)\}^{\infty}_{n=1}\) the eigenvalues of \(| T|\), and \(S_ p\) \((p>0)\) the set of all such T for which \(\{s_ n(T)\}^{\infty}_{n=1}\) is in \(\ell^ p\). If A and B are bounded linear operators on \(L_ 2\), say that B pointwise dominates A if \(| A(x)(t)| \leq B(| x|)(t)\quad a.e.\) for all x(t) in \(L_ 2\). It is known that if \(p=2n\) for some positive integer n, B is in \(S_ p\), and B pointwise dominates A, then A is also in \(S_ p\). Simon has conjectured that this result fails for \(p<2\), and has given a counterexample for \(0<p\leq 1\). The authors provide a counterexample for the remaining cases where \(1<p<2\).