Gaussian characterizations of certain Banach spaces

The general form of characteristic functionals of Gaussian measures in spaces of type 2 and cotype 2 is found. Under the condition of existence of an unconditional basis this problem is solved for spaces not containing l[infinity]n uniformly. The solution uses the language of absolutely summing operators. For each of mentioned space classes it is shown that the results hold in them only. We consider also the equivalent problems on extension of a weak Gaussian distribution and convergence of Gaussian series. Some limit theorems are formulated.