A note on the range of compact multipliers of mixed-norm sequence space (Q2724024)

Let \(\ell^{p,q}\), \(0 < p,q \leq \infty\), be the usual mixed-norm sequence spaces and \(T_{\lambda}\) the multipliers of the spaces \(\ell^{p,q}\). In this paper the authors proved the following theorem:NEWLINENEWLINENEWLINE\textbf{Theorem 3.} Let \(T_\lambda : \ell^{r,s} \rightarrow \ell^{u,v}\), \(0 < r,s,u,v, \leq \infty\), be a compact multiplier. If \(R(T_\lambda)\) is the range of \(T_\lambda\) and \(T_{\lambda} : \ell^{r,s} \rightarrow R(T_\lambda)\) is the reduced operator corresponding to \(T_\lambda\), then \(T_{0\lambda}\) is also compact.