  We construct spherical subgroups in infinite-dimensional classical groups $G$
(usually they are not symmetric and their finite-dimensional analogs are not
spherical). We present a structure of a semigroup on double cosets $L\setminus
G/L$ for various subgroups $L$ in $G$, moreover these semigroups act in spaces
of $L$-fixed vectors in unitary representations of $G$. We also obtain
semigroup envelops of groups $G$ generalizing constructions of operator
colligations.
