Abstract. In this paper we study the set of functions G-valued which can be approximated by G-valued continuous functions in the norm L∞G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain G-valued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem. Also, we characterize the set of functions whic...

Informationally complete measurements on a quantum system allow to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kind of measurements can be achieved through positive-operator valued measures (POVM’s) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help o...

Informationally complete measurements on a quantum system allow to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kind of measurements can be achieved through positive-operator valued measures (POVM’s) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help o...

In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.

‎Given a compact space $X$ and a commutative Banach algebra‎ ‎$A$‎, ‎the character spaces of $A$-valued function algebras on $X$ are‎ ‎investigated‎. ‎The class of natural $A$-valued function algebras‎, ‎those whose characters can be described by means of characters of $A$ and‎ ‎point evaluation homomorphisms‎, ‎is introduced and studied‎. ‎For an‎ ‎admissible Banach $A$-valued function algebra...

where G(u) is a bounded, nondecreasing function with G(— °o)=0 and where 7 is a real-valued constant. Below it is shown that for certain processes of this type the measure of the Hilbert neighborhood of the origin is related to the solution of a certain differential system. In fact, (A) if {x(t), 0fkt< °° } is a separable stochastic process with symmetric, stationary, and independent increments...

We investigate stochastic Volterra equations and their limiting laws. The we consider are driven by a Hilbert space valued \Levy noise integration kernels may have non-linear dependence on the current state of process. Our method is based an embedding into functions which allows to represent solution equation as boundary value partial differential equation. first gather abstract results give mo...

A Hilbert-valued stochastic integration is defined for an integrator that is a cylindrical fractional Brownian motion in a Hilbert space. Since the integrator is not a semimartingale for the fractional Brownian motions considered, a different definition of integration is required. Both deterministic and stochastic operator-valued integrands are used. The approach to integration has an analogue ...

We are concerned with the almost automorphic solutions to the second-order elliptic differential equations of type ü(s) + 2Bu̇(s) + Au(s) = f(s) (∗), where A, B are densely defined closed linear operators acting in a Hilbert space H and f : R 7→ H is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions; every solution to (∗) ...

In this note, we point out that a large family of n × n matrix valued kernel functions defined on the unit disc D ⊆ C, which were constructed recently in [9], behave like the familiar Bergman kernel function on D in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on D can be answered usi...