We observe an infinitely dimensional Gaussian random vector where is a sequence of standard Gaussian variables and is an unknown mean. We consider the hypothesis testing problem alternatives $H_{\epsilon ,\tau }:v\in V_{\epsilon }$ for the sets $V_{\epsilon }=V_{\epsilon }(\tau ,{\rho }_{\epsilon })\subset l_2$. The sets are -ellipsoids of semi-axes with -ellipsoid of semi-axes removed or similar Besov bodies with Besov bodies removed. Here $\tau =(\kappa ,R)$ or $\tau =(\kappa ,h,t,R);\kappa =(p,q,r,s)$ are the parameters which define the sets for given radii , 0 < ; is the asymptotical...