Selected Problems in Real Analysis

Makarov, B. M.; Lodkin, A. A.; Goluzina, M.; Podkorytov, Anatolii

doi:10.1090/mmono/107

Cited by 29 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As we show in Proposition 3.9 below, if L is a Lipschitz seminorm on a compact metric space (X, d) then Mdim L (C(X)) coincides with the Kolmogorov dimension [17,18], whose definition we recall. Let (X, d) be a compact metric space.…”

Section: Proof the Inequality Mdimmentioning

confidence: 99%

Dimension and Dynamical Entropy for Metrized C * -Algebras

Kerr

2003

Commun. Math. Phys.

View full text Add to dashboard Cite

We introduce notions of dimension and dynamical entropy for unital C *algebras "metrized" by means of cLip-norms, which are complex-scalar versions of the Lip-norms constitutive of Rieffel's compact quantum metric spaces. Our examples involve the UHF algebras Mp∞ and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical cLipnorm coincides with the topological entropy of its commutative analogue.Date: November 4, 2002.Proposition 5.12. If α ∈ Aut L (A), σ is an α-invariant state on A, and k ∈ Z, then Entp L,σ (α k ) = |k| Entp L,σ (α).Proposition 5.13. If α ∈ Aut L (A) and σ is an α-invariant state on A thenProof. It suffices to show that, for a given Ω ∈ P f (L ∩ A 1 ) and δ > 0,and for this inequality we need only observe that if X is a finite-dimensional subspace of A such that Ω ⊂ δ X, then whenever a ∈ Ω and x ∈ X satisfy a − x < δ we haveso that π(X)ξ σ is a subspace of H σ with π(Ω)ξ σ ⊂ δ π(X)ξ σ and dim π(Ω)ξ σ ≤ dim X.Corollary 5.14. If Mdim L (A) is finite and α ∈ Aut L (A) is Lipschitz isometric then Entp L,σ (α) = 0. In particular Entp L,σ (id A ) = 0.

show abstract

Section: Proof the Inequality Mdimmentioning

confidence: 99%

Dimension and Dynamical Entropy for Metrized C * -Algebras

Kerr

2003

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…are equimeasurable, that is, they have equal distribution functions. Hence, if s.s. X belongs to H, then it follows from Theorem 1 and equality (13) that…”

Section: Lemmamentioning

confidence: 96%

Sparse Rademacher chaos in symmetric spaces

Асташкин

Lykov

2016

St. Petersburg Math. J.

View full text Add to dashboard Cite

In this paper we study properties of series with respect to orthogonal systems {r i (t)r j (t)} i =j and {r i (s)r j (t)} ∞ i,j=1 in symmetric spaces on interval and square, respectively. Necessary and sufficient conditions for the equivalence of these systems with the canonical base in l 2 and also for the complementability of the corresponding generated subspaces, usually called Rademacher chaos, are derived. The results obtained allow, in particular, to establish the unimprovability of the exponential integrability of functions from Rademacher chaos. Besides, it is shown that for spaces that are "close" to L ∞ , the systems considered, in contrast to the ordinary Rademacher system, do not possess the property of unconditionality. The degree of this non-unconditionality is explained in the space L ∞ .

show abstract

“…This implies that the measure dM(λ) is equivalent to the Lebesgue measure on R + . This is proven in Problem 2.12 of Chapter X of the book [7]. To be precise, the invariance of measures with respect to translations was considered in [7], but the invariance with respect to dilations reduces to this case by a change of variables.…”

Section: 4mentioning

confidence: 99%

“…This is proven in Problem 2.12 of Chapter X of the book [7]. To be precise, the invariance of measures with respect to translations was considered in [7], but the invariance with respect to dilations reduces to this case by a change of variables. Thus the operator H is absolutely continuous.…”

Section: 4mentioning

confidence: 99%

Quasi-Carleman Operators and Their Spectral Properties

Yafaev

2014

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Abstract. The Carleman operator is defined as integral operator with kernel (t + s) −1 in the space L 2 (R + ). This is the simplest example of a Hankel operator which can be explicitly diagonalized. Here we study a class of self-adjoint Hankel operators (we call them quasi-Carleman operators) generalizing the Carleman operator in various directions. We find explicit formulas for the total number of negative eigenvalues of quasi-Carleman operators and, in particular, necessary and sufficient conditions for their positivity. Our approach relies on the concepts of the sigma-function and of the quasi-diagonalization of Hankel operators introduced in the preceding paper of the author.

show abstract

Selected Problems in Real Analysis

Cited by 29 publications

References 0 publications

Dimension and Dynamical Entropy for Metrized C * -Algebras

Dimension and Dynamical Entropy for Metrized C * -Algebras

Sparse Rademacher chaos in symmetric spaces

Quasi-Carleman Operators and Their Spectral Properties

Contact Info

Product

Resources

About