On the compactness of weighted shift operators
This note presents a proof of a rather simple folklore result: Let \(H\) be a Hilbert space and \((A_n)_n\) a sequence of operators \(H \to H\); let \(H^\infty\) denote the ``Hilbertian direct sum'' of a countable quantity of copies of \(H\). Then the operator \(A: H^\infty \to H^\infty\) given by \(A((x_n)_n)=(A_n x_n)_n\) is compact if and only if each \(A_n\) is compact and \(\lim_{n \to \infty} \|A_n \|=0\). Unfortunately, there are even some obscure points in this as presented: at some point (proof of lemma 2.3) it is asserted that \(H^\infty\) coincides with the projective tensor product \(H \otimes_\pi H\), which is wrong if \(H^\infty\) means, as seems to follow from the equality at the introduction, the Hilbert space \(\ell_2(H)\).
scientific article; zbMATH DE number 1310096