Trace, determinant and eigenvalues of kernel operators

In this paper, we show how new results in the theory of determinants and traces and in the theory of quasi-normed tensor products can be applied for obtaining new theorems on the distribution of eigenvalues of nuclear operators in Banach spaces and on the coincidence of the spectral and nuclear traces of such operators. As examples, we consider new classes of operators — generalized nuclear Lorentz–LaPreste operators $N_{(r,s),p}$.