A nuclearity criterion for linear operators (Q1288076)

Let \( (X,\mu)\) be a space with \(\sigma\)-finite positive not purely atomic measure and let \(L_2=L_2 (X,\mu)\) be a separable Hilbert space. Denote by \(C_1\) the set of all nuclear operators and by \(\widetilde\Delta_1\) the set of all strong Akhiezer operators. The main result of the article is the following theorem. Let \(T\) be a continuous linear operator in \(L_2\). Then \(T\) is nuclear if and only if the operator \(KLTM\) belongs to \(\widetilde\Delta_1\) for every compact Akhiezer integral operator \(K\) in \(L_2\), every compact Akhiezer and Carleman integral operator \(L\) in \(L_2\), and every compact Carleman integral operator \(M\) in \(L_2\). As a corollary, the author proves that every two-sided ideal of the set of all compact Akhiezer integral operators in \(L_2\) contained in \(\widetilde\Delta_1\) is included in the two-sided ideal \(C_1\) of this set.