Absolutely \(L_{\exp_q}\)-summing norms of diagonal operators in \(\ell_r\) and limit orders of \(L_{\exp}\)-summing operators (Q2718016)

Let \(L_{\exp_q}\) denote the Orlicz space \(L_{\Phi_q}\), where \({\Phi_q}(\lambda)=e^{\lambda^q}-1\), \(1\leq q<\infty\). For a fixed sequence \(\lambda_1\geq\lambda_2\geq\cdots\geq 0\) and \(1\leq r<\infty\) the corresponding diagonal operator \(D_\lambda\) on the sequence space \(\ell_r\) is defined by \(D_\lambda(\xi_k)_{k=1}^\infty=(\lambda_k\xi_k)_{k=1}^\infty\). The author obtained the following estimations for the \(L_{\exp_q}\)-summing norms \(\pi_{\exp_q}(D_\lambda)\): NEWLINE\[NEWLINE \begin{aligned} \tfrac 1c\pi_{\exp_q}(D_\lambda) &\leq \|\lambda\|_{s,r}\leq c\pi_{\exp_q}(D_\lambda)\quad\text{if}\quad 2<s<\infty; \\ \tfrac 1c\pi_{\exp_q}(D_\lambda) &\leq \|\lambda\|_2\leq c\pi_{\exp_q}(D_\lambda)\quad\text{if}\quad 1\leq s\leq 2, \end{aligned} NEWLINE\]NEWLINE where \(s=\max(q,r)\), \(\|\lambda\|_{s,r}\) denotes the Lorentz quasi-norm, and the constant \(c\) depends on \(q\) and \(r\) only. There are also investigated \(L_{\exp_q}\)-absolutely summing operators on the sequence space \(\ell_2\) and determined the limit orders (in the sense of Pietsch) of the absolutely \(L_{\exp}\)-summing operators.