scholarly journals PIP-Space Valued Reproducing Pairs of Measurable Functions

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 52 ◽  
Author(s):  
Jean-Pierre Antoine ◽  
Camillo Trapani
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Measure Space ◽  
Abstract Space ◽  
Initial Space ◽  
Measurable Functions ◽  
Continuous Frames

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ) , where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Stochastic Fubini Theorem for Semimartingales in Hilbert Space

1990 ◽  
Vol 42 (5) ◽  
pp. 890-901 ◽  
Author(s):  
Jorge A. León
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Fubini Theorem ◽  
Stochastic Integrals ◽  
Bounded Linear ◽  
Image Position ◽  
Stochastic Fubini Theorem

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

APPLICATIONS OF KATO'S INEQUALITY TO OPERATOR-VALUED INTEGRALS ON HILBERT SPACES

2013 ◽  
Vol 06 (04) ◽  
pp. 1350059
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Bochner Integral ◽  
Measurable Functions ◽  
Kato’S Inequality ◽  
Operator Exponential

By the use of the celebrated Kato's inequality we obtain in this paper some inequalities for operator-valued integrals on a complex Hilbert space H. Among others, we show that [Formula: see text] for any x, y ∈ H, provided [Formula: see text] and p : E → [0, ∞) are μ-measurable functions on E and such that [Formula: see text] and [Formula: see text] are Bochner integrable on E for some α ∈ [0, 1]. Natural applications for various norms and numerical radii associated with the Bochner integral of operator-valued functions and some examples for the operator exponential are presented as well.

scholarly journals Measurable Hilbert sheaves

1996 ◽  
Vol 61 (2) ◽  
pp. 189-215 ◽  
Author(s):  
Michael A. Wendt
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Measure Space ◽  
Direct Integral ◽  
Space Objects ◽  
Set Up

AbstractWe describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

scholarly journals The space of operator valued functions seen as Hilbert H*-module

2020 ◽  
pp. 1-1
Author(s):  
Zlatko Lazovic
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Orthonormal Basis ◽  
Nuclear Operator ◽  
Measurable Functions

Let M be a space of weakly*-measurable functions F : ? ? B(H) on measure space (?,?,?), for which the function F*F is Gel'fand integrable and Gel'fand integral ? ? F*F d? is a nuclear operator on Hilbert space H. We show that M is Hilbert H*-module which contains an orthonormal basis.

scholarly journals Isometries of measurable functions

1981 ◽  
Vol 24 (1) ◽  
pp. 13-26 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Finite Measure ◽  
Measurable Operator ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Linear Isometries

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

scholarly journals The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 389
Author(s):  
Jeong-Gyoo Kim
Keyword(s):  
Hilbert Space ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Double Sequence ◽  
Intermediate Stage ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Abstract Space ◽  
Double Sequences ◽  
Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

scholarly journals Translation and modulation invariant Hilbert spaces

2021 ◽  
Author(s):  
Joachim Toft ◽  
Anupam Gumber ◽  
Ramesh Manna ◽  
P. K. Ratnakumar
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Zero Element ◽  
Multiplicative Constant ◽  
Space Of Distributions ◽  
Feichtinger Algebra

AbstractLet $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

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