OPERATOR SELF-SIMILAR PROCESSES ON BANACH SPACES

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are consid-ered. 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 determinedmultiplicative 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 ex-pectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS pro-cesses 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. Copyright © 2006 M. T. Matache and V. Matache. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1