Traces and symmetric linear forms
It is known due to Calkin theorem that there is a one-to-one correspondence between operator ideals \(\mathfrak U(H)\) on separable Hilbert spaces and symmetric ideals of bounded sequences \(\mathfrak a(\mathbb N)\). It is also known that continuous traces \(\tau\) on operator ideals are in one-to-one correspondence with continuous symmetric linear forms on symmetric ideals of sequences. In the paper under review, the author gives an approach that shows that the result holds without any topological assumption. For a given bounded sequence \(s=(\sigma_n)\) and a given orthonormal basis \((w_n)\) in a separable Hilbert space \(H\), the diagonal operator \(W_s\) is defined by \(\sum_{n=1}^\infty \sigma_n w_n^*\otimes w_n\). Now, if \(\tau\) is a trace on an operator ideal \(\mathfrak U(H)\) of operators, the author writes \(\omega_\tau(s)= \tau(W_s)\) for any \(s\in \mathfrak a(\mathbb N)\). On the other hand, if \(\psi\) is a symmetric linear form on \(\mathfrak a(\mathbb N)\), the author writes \(\lambda_\psi(S)=\psi (s)\) where \(s=(\sigma_n)\) and \(S=\sum_{n=1}^\infty \sigma_n u_n^*\otimes v_n\) is any Schmidt representation of \(S\in \mathfrak U(H)\). The main theorem shows that \(\tau\longrightarrow \omega_\tau\) and \(\psi\longrightarrow \lambda_\psi\) are one-to-one correspondences between traces on \(\mathfrak U(H)\) and symmetric linear forms on \(\mathfrak a(\mathbb N)\).
scientific article; zbMATH DE number 7005662