scholarly journals A note on local asymptotic behaviour for Brownian motion in Banach spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 669-676
Author(s):  
Mou-Hsiung Chang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Brownian Motion ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Integral Test ◽  
Real Separable Banach Space ◽  
Real Separable Hilbert Space ◽  
Special Case

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.

scholarly journals On Abstract Wiener Measure

1972 ◽  
Vol 46 ◽  
pp. 155-160 ◽  
Author(s):  
Balram S. Rajput
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Reflexive Banach Space ◽  
Wiener Measure ◽  
Real Separable Hilbert Space ◽  
Cylinder Measure

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

scholarly journals Operator semistable probability measures on a Hilbert space

1980 ◽  
Vol 22 (3) ◽  
pp. 397-406 ◽  
Author(s):  
R.G. Laha ◽  
V.K. Rohatgi
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Real Separable Hilbert Space ◽  
Semistable Probability

A characterization of the class of operator semistable probability measures on a real separable Hilbert space is given.

scholarly journals Domains of elliptic operators on sets in Wiener space

2020 ◽  
Vol 23 (01) ◽  
pp. 2050004
Author(s):  
Davide Addona ◽  
Gianluca Cappa ◽  
Simone Ferrari
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Quadratic Form ◽  
Gaussian Measure ◽  
Adjoint Operator ◽  
The Self ◽  
Affine Function ◽  
Surface Measure ◽  
Regular Convex

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

scholarly journals On a Class of Measure-Dependent Stochastic Evolution Equations Driven by fBm

2007 ◽  
Vol 2007 ◽  
pp. 1-26 ◽  
Author(s):  
Eduardo Hernandez ◽  
David N. Keck ◽  
Mark A. McKibben
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Real Separable Hilbert Space

We investigate a class of abstract stochastic evolution equations driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space. We establish the existence and uniqueness of a mild solution, a continuous dependence estimate, and various convergence and approximation results. Finally, the analysis of three examples is provided to illustrate the applicability of the general theory.

scholarly journals On a Characterization of Convergence in Banach Spaces with a Schauder Basis

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Marat V. Markin ◽  
Olivia B. Soghomonian
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Schauder Basis ◽  
Infinite Dimensional ◽  
Summable Sequence ◽  
Convergent Sequences

We extend the well-known characterizations of convergence in the spaces l p ( 1 ≤ p < ∞ ) of p -summable sequence and c 0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences.“The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications.” Jacob T. Schwartz

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

Biholomorphic Mappings on Banach Spaces

2019 ◽  
Vol 62 (4) ◽  
pp. 913-924
Author(s):  
H. Carrión ◽  
P. Galindo ◽  
M. L. Lourenço
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Open Subset ◽  
Separable Hilbert Space ◽  
Derivative Operator ◽  
Infinite Dimensional ◽  
Dimensional Version ◽  
Dual Banach Space ◽  
Bounded Convex ◽  
Power Bounded

AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals Isometries of Hilbert space valued function spaces

1996 ◽  
Vol 61 (2) ◽  
pp. 150-161 ◽  
Author(s):  
Beata Randrianantoanina
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Invariant Function ◽  
Function Spaces ◽  
Measurable Operator ◽  
Complex Rearrangement ◽  
Rearrangement Invariant Function Spaces ◽  
Surjective Isometry ◽  
Almost All

AbstractLet X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

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