Some properties of D-weak operator topology

Let 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

Operator self-similar processes on Banach spaces

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.