On the closed subideals of \(L(\ell _{p} \oplus \ell _{q})\) (Q2912200)

In this paper, the author first reviews the known results about closed subideals of the space of bounded operators on \(\ell_p \otimes \ell_q\), \(1 <p<p<\infty\). It turns out that, apart from two explicitly stated maximal proper closed subideals (induced by the identity maps \(I_{\ell_p}\) and \(I_{\ell_q}\), respectively), all other closed subideals of \(L(\ell_p \otimes \ell_q)\) can be identified with the closed subideals of \(L(\ell_p,\ell_q)\). The author then exhibits two new closed and incomparable subideals of \(L(\ell_p,\ell_q)\) in the case \(1<p<2<q<\infty\), which increases the count of the closed proper and nontrivial subideals of \(L(\ell_p,\ell_q)\) to \(7\). These two ideals are those which are induced by the formal inclusions \(I(p,2): \ell_p \rightarrow \ell_2\) and \(I(2,q): \ell_2 \rightarrow \ell_q\), respectively.