Trace, determinant and eigenvalues of nuclear operators

It is shown how the results in the theory of determinants and traces as well as in the theory of quasi-normed tensor products can be applied for getting new theorems on distribution of eigenvalues of nuclear operators in Banach spaces and on coincidence of spectral and nuclear traces such operators. As examples, it is considered the new classes of operators – generalized nuclear operators of Lorentz–Lapresté $N_{(r,s),p}\mathrm{.}$