On the nuclearity of a dual space with the convergence in probability topology

1982 ◽  
Vol 59 (2) ◽  
pp. 197-201 ◽  
Author(s):  
S. Kwapien ◽  
W. Smolenski
Keyword(s):  
Dual Space ◽  
Convergence In Probability

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals A harmonically weighted filter for cyclical long memory processes

2021 ◽  
Author(s):  
Federico Maddanu
Keyword(s):  
Spectral Density ◽  
Long Memory ◽  
Convergence In Probability ◽  
Simulation Methods ◽  
Integrated Process ◽  
Paleoclimatic Data ◽  
Memory Time ◽  
Long Memory Processes ◽  
Short Memory ◽  
Memory Parameter

AbstractThe estimation of the long memory parameter d is a widely discussed issue in the literature. The harmonically weighted (HW) process was recently introduced for long memory time series with an unbounded spectral density at the origin. In contrast to the most famous fractionally integrated process, the HW approach does not require the estimation of the d parameter, but it may be just as able to capture long memory as the fractionally integrated model, if the sample size is not too large. Our contribution is a generalization of the HW model, denominated the Generalized harmonically weighted (GHW) process, which allows for an unbounded spectral density at $$k \ge 1$$ k ≥ 1 frequencies away from the origin. The convergence in probability of the Whittle estimator is provided for the GHW process, along with a discussion on simulation methods. Fit and forecast performances are evaluated via an empirical application on paleoclimatic data. Our main conclusion is that the above generalization is able to model long memory, as well as its classical competitor, the fractionally differenced Gegenbauer process, does. In addition, the GHW process does not require the estimation of the memory parameter, simplifying the issue of how to disentangle long memory from a (moderately persistent) short memory component. This leads to a clear advantage of our formulation over the fractional long memory approach.

Dual space directional occlusion

2013 ◽  
Vol 29 (9) ◽  
pp. 917-926 ◽  
Author(s):  
Sebastian Herholz ◽  
Jens-Uwe Hahn ◽  
Andreas Schilling
Keyword(s):  
Dual Space

scholarly journals A note on supervised classification and Nash-equilibrium problems

2017 ◽  
Vol 51 (2) ◽  
pp. 329-341
Author(s):  
Nicolas Couellan
Keyword(s):  
Nash Equilibrium ◽  
Support Vector Machines ◽  
Supervised Classification ◽  
Dual Space ◽  
Equilibrium Problems ◽  
Support Vector ◽  
Generalized Nash Equilibrium ◽  
Class Separation ◽  
Vector Machines ◽  
Generalized Nash Equilibrium Problems

In this note, we investigate connections between supervised classification and (Generalized) Nash equilibrium problems (NEP & GNEP). For the specific case of support vector machines (SVM), we exploit the geometric properties of class separation in the dual space to formulate a non-cooperative game. NEP and Generalized NEP formulations are proposed for both binary and multi-class SVM problems.

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