Stetigkeit des Spektrums einer Klasse nichtlinearer Operatoren II
The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces, subspaces of the space of rapidly decreasing smooth complex functions, which are defined by imposing suitable bounds on their elements in terms of a given sequence M. Necessary and sufficient conditions on M are stated for the problem to have a solution, sometimes coming with linear continuous right inverses of the moment map, sending a function to the sequence of its moments. On the way, some results on the...
The classical Banach principle is an essential tool for the investigation of ergodic properties of Cesàro subsequences. The aim of this work is to extend the Banach principle to the case of stochastic convergence in operator algebras. We start by establishing a sufficient condition for stochastic convergence (stochastic Banach principle). Then we prove stochastic convergence for bounded Besicovitch sequences, and as a consequence for uniform subsequences.
This work is concerned with the study of stochastic processes which are continuous in probability, over various parameter spaces, from the point of view of approximation and extension. A stochastic version of the classical theorem of Mergelyan on polynomial approximation is shown to be valid for subsets of the plane whose boundaries are sets of rational approximation. In a similar vein, one can obtain a version in the context of continuity in probability of the theorem of Arakelyan on the uniform...
Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) (t∈ [0,T]), almost surely, where A is the generator of a -semigroup of bounded linear operators on...
Let be a Hilbert space and a Banach space. We set up a theory of stochastic integration of -valued functions with respect to -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter . For we show that a function is stochastically integrable with respect to an -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...
Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is...
Operator version of the Stokeslet method in the theory of creeping flow is suggested. The approach is analogous to the zero-range potential one in quantum mechanics and is based on the theory of self-adjoint operator extensions in the space L2 and in the Pontryagin?s space with an indefinite metric. The problem of Stokes flow in two channels connected through a small opening is considered in the framework of this approach. The case of a periodic system of small openings is studied too. The picture...