  $T$-semi-selfdecomposability and subclasses $L_m(b, Q)$ and $\tilde L_m(b,
Q)$ of measures on complete separable metric vector spaces are introduced and
basic properties are proved. In particular, we show that $\mu$ is
$T$-semi-selfdecomposable if and only if $\mu = T(\mu) \nu$ where $\nu$ is
infinitely divisible and $\mu$ is operator selfdecomposable if and only if $\mu
\in L_0(b, Q)$ for all $0< b < 1$.
