scholarly journals CONE STRUCTURE OF L2-WASSERSTEIN SPACES

2012 ◽  
Vol 04 (02) ◽  
pp. 237-253 ◽  
Author(s):  
ASUKA TAKATSU ◽  
TAKUMI YOKOTA
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Geometric Structure ◽  
Separable Hilbert Space ◽  
Polish Space ◽  
Cone Structure ◽  
Underlying Space ◽  
Wasserstein Space ◽  
Product Structures

The aim of this paper is to obtain a better understanding of the geometric structure of quadratic Wasserstein spaces over separable Hilbert spaces. For this sake, we focus on their cone and product structures, and prove that the quadratic Wasserstein space over any separable Hilbert space has a cone structure and splits the underlying space isometrically but no more than that. These are shown in more general settings, and one of our main results is that the quadratic Wasserstein space over a Polish space has a cone structure if and only if so does the underlying space.

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

2007 ◽  
Vol 10 (02) ◽  
pp. 261-276 ◽  
Author(s):  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Separable Hilbert Space ◽  
Fock Spaces ◽  
Parameter Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

scholarly journals The Behavior of Basic Fields

2018 ◽  
Vol 1 (3) ◽  
Keyword(s):  
Mathematical Modeling ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Differential Calculus ◽  
Separable Hilbert Space ◽  
Floating Platform ◽  
Basic Field ◽  
Floating Platforms ◽  
The Continuum

A basic field is defined in the realm of a mathematical modeling platform that is based on a collection of floating platforms and an embedding platform. Each floating platform is represented by a quaternionic separable Hilbert space. The embedding platform is a non-separable Hilbert space. A basic field is a continuum eigenspace of an operator that resides in the non-separable embedding Hilbert space. The continuum can be described by a quaternionic function, and its behavior is described by quaternionic differential calculus. The separable Hilbert spaces contain the point-like artifacts that trigger the basic field. The floating platforms possess symmetry, which in combination with the background platform generates the sources of symmetry related fields.

scholarly journals A common unique fixed point theorem for two random operators in Hilbert spaces

2002 ◽  
Vol 32 (3) ◽  
pp. 177-182 ◽  
Author(s):  
Binayak S. Choudhury
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Hilbert Spaces ◽  
Closed Subset ◽  
Separable Hilbert Space ◽  
Unique Fixed Point ◽  
Random Operators ◽  
Nonempty Closed Subset ◽  
Random Fixed Point

We construct a sequence of measurable functions and consider its convergence to the unique common random fixed point of two random operators defined on a nonempty closed subset of a separable Hilbert space. The corresponding result in the nonrandom case is also obtained.

Generalized frames for operators in Hilbert spaces

2014 ◽  
Vol 17 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Mohammad Sadegh Asgari ◽  
Hamidreza Rahimi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Closed Subspace ◽  
Bounded Operator ◽  
Riesz Bases ◽  
Small Perturbations ◽  
Riesz Decomposition ◽  
Analysis And Synthesis ◽  
General Range

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

Frame-cyclic operators and their properties

2021 ◽  
Vol 24 (01) ◽  
pp. 2150009
Author(s):  
Nastaran Alizadeh Moghaddam ◽  
Mohammad Janfada
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Separable Hilbert Space ◽  
Cyclic Vector ◽  
Hypercyclic Operators ◽  
Cyclic Operator ◽  
Infinite Dimensional ◽  
Cyclic Vectors ◽  
Cyclic Operators

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

scholarly journals Rigid subsets in Euclidean and Hilbert spaces

1975 ◽  
Vol 20 (1) ◽  
pp. 66-72 ◽  
Author(s):  
Ludvik Janos
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Metric Space ◽  
Separable Hilbert Space ◽  
Euclidean Spaces ◽  
Point Set

AbstractA subset Y of a metric space (X, p) is called rigid if all the distances p(y1, y2) between points y1, y2 ∈ Y in Y are mutually different. The main purpose of this paper is to prove the existence of dense rigid subsets of cardinality c in Euclidean spaces En and in the separable Hilbert space l2. Some applications to abstract point set geometries are given and the connection with the theory of dimension is discussed.

scholarly journals On Injectivity of an Integral Operator Connected to Riemann Hypothesis

2021 ◽  
Author(s):  
Dumitru Adam
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Fractional Part ◽  
Positive Definite ◽  
Its Sequence ◽  
Bounded Operators ◽  
Linear Bounded Operator

Abstract In 1993, Alcantara-Bode showed ([2]) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel function defined by the fractional part of (y/x), isinjective. Since then, the injectivity of the integral operator used inequivalent formulation of RH has not been addressed nor has beendissociated from RH.We provided in this paper methods for investigating the injectivityof linear bounded operators on separable Hilbert spaces using theirapproximations on dense families of subspaces.On the separable Hilbert space L2(0,1), an linear bounded operator(or its associated Hermitian), strict positive definite on a dense familyof including approximation subspaces in built on simple functions, isinjective if the rate of convergence of its sequence of injectivity pa-rameters on approximation subspaces is inferior bounded by a not nullconstant, that is the case with the Beurling - Alcantara-Bode integraloperator.We applied these methods to the integral operator used in RHequivalence proving its injectivity.

Products of Normal Operators

1988 ◽  
Vol 40 (6) ◽  
pp. 1322-1330 ◽  
Author(s):  
Pei Yuan Wu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Sufficient Conditions ◽  
Bounded Linear ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Necessary And Sufficient

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).

Convergence rates of the LIL for random fields in Hilbert spaces

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Ke-Ang Fu ◽  
Xiao-Rong Yang
Keyword(s):  
Hilbert Space ◽  
Gamma Function ◽  
Hilbert Spaces ◽  
Random Fields ◽  
Convergence Rates ◽  
Separable Hilbert Space ◽  
Partial Ordering ◽  
Covariance Operator ◽  
Dimensional Lattice ◽  
Real Separable Hilbert Space

AbstractConsidering the positive d-dimensional lattice point Z +d (d ≥ 2) with partial ordering ≤, let {X k: k ∈ Z +d} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $$ S_n = \sum\limits_{k \leqslant n} {X_k } $$, n ∈ Z +d. Let σ i2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ 2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $$ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $$. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)d−2(log log ‖X‖)b+4] < ∞ implies $$ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $$, where Γ(·) is the Gamma function and $$ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $$.

scholarly journals Weak Laws of Large Numbers for Negatively Superadditive Dependent Random Vectors in Hilbert Spaces

2021 ◽  
Vol 37 (2) ◽  
Author(s):  
Bui Khanh Hang ◽  
Tran Manh Cuong ◽  
Ta Cong Son
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Weighted Sums ◽  
Random Vectors ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Real Separable Hilbert Space ◽  
Weak Laws ◽  
Random Indices

Let $\{X_{n}, {n}\in \mathbb{N}\}$ be a sequence of negatively superadditive dependent random vectors taking values in a real separable Hilbert space. In this paper, we present the weak laws of large numbers for weighted sums (with or without random indices) of $\{X_{n}, {n}\in \mathbb{N}\}$.

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