Abstract
Operator self-similar (OSS) stochastic processes on arbitrary
Banach spaces are considered. If the family of expectations of
such a process is a spanning subset of the space, it is proved
that the scaling family of operators of the process under
consideration is a uniquely determined multiplicative group of
operators. If the expectation-function of the process is
continuous, it is proved that the expectations of the process have
power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a
power-function, provided that the linear space spanned by the
expectations has category 2 (in the sense of Baire) in its
closure. It is shown that OSS processes whose expectation-function
is differentiable on an interval (s0,∞), for some s0≥1, have a unique scaling family of operators of the form
{sH:s>0}, if the expectations of the process span a dense
linear subspace of category 2. The existence of a scaling family
of the form {sH:s>0} is proved for proper Hilbert space
OSS processes with an Abelian scaling family of positive
operators.