Characterization of certain classes

W. Linde ; V. I. Tarieladze ; S. A. Čobanyan

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 162-167

Résumé

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.