Small ball probabilities for Gaussian random fields and tensor products of compact operators

We find the logarithmic -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of ``tensor product'. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.