scholarly journals Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 498
Author(s):  
Ilwoo Cho
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Number Fields ◽  
Semicircular Elements ◽  
Banach Space Operators

In this paper, we study certain Banach-space operators acting on the Banach *-probability space ( LS , τ 0 ) generated by semicircular elements Θ p , j induced by p-adic number fields Q p over the set P of all primes p. Our main results characterize the operator-theoretic properties of such operators, and then study how ( LS , τ 0 ).

scholarly journals Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 199 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle Jorgensen
Keyword(s):  
Probability Space ◽  
Free Probability ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Probabilistic Information ◽  
Closed Intervals ◽  
Measurable Functions ◽  
Semicircular Elements

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .

scholarly journals Deformation of semicircular and circular laws via p-adic number fields and sampling of primes

2019 ◽  
Vol 39 (6) ◽  
pp. 773-813
Author(s):  
Ilwoo Cho ◽  
Palle E. T. Jorgensen
Keyword(s):  
Banach Algebra ◽  
Probability Space ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Circular Law ◽  
Semicircular Law ◽  
Different Types ◽  
Semicircular Elements

In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

scholarly journals On the essential spectrum of Banach-space operators

2000 ◽  
Vol 43 (3) ◽  
pp. 511-528 ◽  
Author(s):  
Jörg Eschmeier
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Negative Index ◽  
Resolvent Set ◽  
Image Position ◽  
Nuclear Spaces ◽  
Simply Connected ◽  
Banach Space Operators

AbstractLet T and S be quasisimilar operators on a Banach space X. A well-known result of Herrero shows that each component of the essential spectrum of T meets the essential spectrum of S. Herrero used that, for an n-multicyclic operator, the components of the essential resolvent set with maximal negative index are simply connected. We give new and conceptually simpler proofs for both of Herrero's results based on the observation that on the essential resolvent set of T the section spaces of the sheavesare complete nuclear spaces that are topologically dual to each other. Other concrete applications of this result are given.

Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors

2019 ◽  
Vol 19 (04) ◽  
pp. 1950032
Author(s):  
Yejuan Wang ◽  
Xiangming Zhu ◽  
Peter Kloeden
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Separable Banach Space ◽  
Additional Condition ◽  
Probability Space ◽  
Parabolic Partial Differential Equations ◽  
Mean Square ◽  
Probabilistic Argument ◽  
Random Attractors

Let [Formula: see text] be a probability space and let [Formula: see text] be a separable Banach space. It is shown a subset [Formula: see text] of [Formula: see text], where [Formula: see text], is relatively compact in [Formula: see text] if and only if it is uniformly [Formula: see text]-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument. The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differential equations and stochastic parabolic partial differential equations.

scholarly journals Modified cauchy kernels and functional calculus for operators on Banach space

1997 ◽  
Vol 63 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Edwin Franks
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cauchy Kernel ◽  
Necessary And Sufficient ◽  
Banach Space Operators

AbstractIn Banach space operators with a bounded H∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L1 with respect to the measure ]dz]/]z]. The method is direct and has the advantage that no transforms of the functions are necessary.

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