Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. ...

We study integration in a class of Hilbert spaces of analytic functions defined on the Rs. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and log ε−1 and give necessary and sufficie...

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined $\mathbf{R}^d$. We prove that only affine transforms can do so in pretty large class of RKHS. Our result covers not Paley-Wiener real line, studied previous works, but also much more general RKHSs correspond...

In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate t...

Let H be a Hilbert space of analytic functions on the unit disk D. For an analytic function ψ on D, we can define the multiplication operator Mψ : f → ψf, f ∈ H. For an analytic selfmapping φ of D, the composition operator Cφ defined on H as Cφf f ◦ φ, f ∈ H. These operators are two classes of important operators in the study of operator theory in function spaces 1–3 . Furthermore, for ψ and φ,...

Controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract Hilbert spaces. Fusion frames and g-frames generalize frames. Hilbert C*-modules form a wide category between Hilbert spaces and Banach spaces. Hilbert C*-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a C*...

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spac...