On the tensor stability of \(s\)-number ideals
Let \(S_{p,q}\) denote the class of operators having approximation numbers belonging to the Lorentz sequence space \(\ell_{p,q}\), \(0<p<\infty\), \(0<q\leq\infty \). We show that these ideals are unstable under suitable tensor norms \(\tau\) for p,q with \(1/q<(1/p)+1/2\), in particular for \(p=q\); i.e. there are \(S_{p,q}\)-operators S,T between Banach spaces such that the tensor product operator \(S{\hat\otimes }_{\tau}T\) does not belong to \(S_{p,q}\), answering a question of Pietsch. These operator ideals \(S_{p,q}\) are, however, ''almost'' stable under arbitrary tensor norms, namely stable up to logarithmic factors. For Hilbert space operators, \(S,T\in S_{p,q}(H,H)\) implies \(S{\hat\otimes }_{\tau}T\in S_{p,q}(H{\hat\otimes }_{\tau}H,H{\hat\otimes }_{\tau}H),\) by contrast, if \(q\leq p\). Using this, factorization theorems, and bilinear interpolation we show that \(S_{p,q}\)-operators are stable under tensor products, if \(1/q\geq (1/p)+1\) and the operators act between dual cotype 2 and cotype 2 Banach spaces. If the cotype conditions are not satisfied, one can still conclude that the Weyl-numbers of the tensor product operator belong to \(\ell_{p,q}\), at least for certain tensor norms.
scientific article; zbMATH DE number 3872116
scientific article