On compactness of families of second order measures in Danach spaces

Conditions of weak compactness for families of measuies in Banach spaces in terms of their correlation characteristics are considered. For Banach spaces with unconditional basis, not containing $l_{\infty}^n$ uniformly, we give necessary and sufficient conditions for weak compactness and weak convergence of families of Gaussian measures in terms of means and covariance operators. These conditions are necessary and sufficient only for spaces under consideration.