Characterization of certain classes of Banach spaces by properties of Gaussian measures

The following assertions are proved. 1) The classes of $\gamma$-summing and $\gamma$-radonifying operators with values in a Banach space $X$ coincide iff $X$ does not contain isomorphic copies of $c_0$. 2) An operator $T$ from a Hilbert space into a Banach space of type 2 is $\gamma$-summing iff $T^*$ is absolutely 2-summing. 3) The covariance operator of a strong second order tight measure on a Banach space is nuclear. 4) If $X$ is a Banach space, then every positive symmetric and nuclear linear operator from $X^*$ into $X$ is Gaussian covariance iff $X$ is of type 2.