Ideals of compact operators with Nakano type norms in a Hilbert space (Q2914868)

Let \(H\) be a separable Hilbert space. For a compact linear operator \(A\) acting in \(H\), let \(A^{\ast}\) denote the adjoint of \(A\), \(\lambda_{k}(A)\) represent the eigenvalues of \(A\) and \(s_{k}(A)=\sqrt{\lambda_{k}(A^{\ast}A)}\), where \(k=1,2,\dots\), are the singular values taken with their multiplicities and ordered decreasingly. Let \(\pi=\left\{ p_{k}\right\} _{k=1}^{\infty}\) be a nondecreasing sequence of numbers \(p_{k}\geq1.\) Let NEWLINE\[NEWLINE \gamma_{\pi}(A)= \sum_{j=1}^{\infty}\frac{s_{j}^{p_{j}}(A)}{p_{j}}, NEWLINE\]NEWLINE and assume that the series converges. Let \(X_{\pi}\) be the set of compact operators in \(H\) such that \(\gamma_{\pi}(tA)<\infty\) for all \(t>0\).NEWLINENEWLINEIf \(SN_{p}\) represents the Schatten-von Neumann ideal of operators \(A\) with the finite norm NEWLINE\[NEWLINEN_{p}(A):=\left[\text{Trace}\left( A^{\ast}A\right) ^{\frac {p}{2}}\right] ^{\frac{1}{p}},NEWLINE\]NEWLINE it is well-known that NEWLINE\[NEWLINE\sum_{k=1}^{\infty}\left| \lambda_{k}\left( A\right) \right| ^{p}\leq N_{p}^{p}\left( A\right) NEWLINE\]NEWLINE for any \(A\in SN_{p}.\) The main motivation of this paper is to generalize the above inequality to the operators from \(X_{\pi}.\)