Entropy numbers of r-nuclear operators between $L_{p}$ spaces

Bernd Carl

Displaying similar documents to “Entropy numbers of r-nuclear operators between $L_{p}$ spaces”

Operator entropy inequalities

M. S. Moslehian, F. Mirzapour, A. Morassaei (2013)

Colloquium Mathematicae

Similarity:

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting $S_{q}^{f} (A | B) : = \sum_{j = 1}^{n} A_{j}^{1 / 2} {(A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2})}^{q} f (A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2}) A_{j}^{1 / 2}$ , and then give upper and lower bounds for $S_{q}^{f} (A | B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004),...

Grothendieck-Lidskiĭ theorem for subspaces of quotients of $L_{p}$ -spaces

Oleg Reinov, Qaisar Latif (2014)

Banach Center Publications

Similarity:

Generalizing A. Grothendieck’s (1955) and V. B. Lidskiĭ’s (1959) trace formulas, we have shown in a recent paper that for p ∈ [1,∞] and s ∈ (0,1] with 1/s = 1 + |1/2-1/p| and for every s-nuclear operator T in every subspace of any $L_{p} (ν)$ -space the trace of T is well defined and equals the sum of all eigenvalues of T. Now, we obtain the analogous results for subspaces of quotients (equivalently: for quotients of subspaces) of $L_{p}$ -spaces.

A local approach to $g$ -entropy

Mehdi Rahimi (2015)

Kybernetika

Similarity:

In this paper, a local approach to the concept of $g$ -entropy is presented. Applying the Choquet‘s representation Theorem, the introduced concept is stated in terms of $g$ -entropy.

Entropy and approximation numbers of embeddings between weighted Besov spaces

Iwona Piotrowska (2008)

Banach Center Publications

Similarity:

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t r_{Γ} : B_{p ₁, q}^{s} (ℝ ⁿ, w_{ϰ}^{Γ}) \to L_{p ₂} (Γ)$ , where Γ is a d-set with 0 < d < n and $w_{ϰ}^{Γ}$ a weight of type $w_{ϰ}^{Γ} (x) d i s t {(x, Γ)}^{ϰ}$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

Similarity:

We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

On the joint entropy of $d$ -wise-independent variables

Dmitry Gavinsky, Pavel Pudlák (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

How low can the joint entropy of $n$ $d$ -wise independent (for $d \geq 2$ ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$ , for $p < 1$ )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version), http://people.cs.uchicago.edu/ laci/papers/13augEntropy.pdf, 2013]. In this paper we improve some...

ε-Entropy and moduli of smoothness in $L^{p}$ -spaces

A. Kamont (1992)

Studia Mathematica

Similarity:

The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^{p} (^{d})$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^{p} (ℝ^{d})$ whose tail function decreases as $O (λ^{- γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^{d}$ and the minimax risk for that class is discussed.

The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

Similarity:

Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $c a r d f^{- 1} (y) \geq 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $c a r d f^{- 1} (y) \geq m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

Bernd Carl, David E. Edmunds (2003)

Studia Mathematica

Similarity:

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1 / 2} c ₙ (c o v (K)) \leq c_{K} (1 + \sum_{k = 1}^{ⁿ} k^{- 1 / 2} e_{k} (K))$ , n ∈ ℕ, and $k^{1 / 2} c_{k + n} (c o v (K)) \leq c [l o g^{1 / 2} (n + 1) ε ₙ (K) + \sum_{j = n + 1}^{\infty} ε_{j} (K) / (j l o g^{1 / 2} (j + 1))]$ , k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and $ε_{k} (K)$ and $e_{k} (K)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K)...

Operators whose adjoints are quasi p-nuclear

J. M. Delgado, C. Piñeiro, E. Serrano (2010)

Studia Mathematica

Similarity:

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with $K \subseteq \sum ₙ α ₙ x ₙ : (α ₙ) \in B_{ℓ_{p^{'}}}$ . We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

Generalized Fokker-Planck equations and convergence to their equilibria

Piotr Biler, Grzegorz Karch (2003)

Banach Center Publications

Similarity:

We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on $ℝ^{d}$ with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.

On a proof of the boundedness and nuclearity of pseudodifferential operators in $R^{n}$

Jan A. Rempała (1990)

Annales Polonici Mathematici

Similarity:

Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and $L^{1}$ -data

Abdelali Sabri, Ahmed Jamea, Hamad Talibi Alaoui (2020)

Communications in Mathematics

Similarity:

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic $p (\cdot)$ -Laplacian problem with Dirichlet-type boundary conditions and $L^{1}$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

Similarity:

We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

Topological disjointness from entropy zero systems

Wen Huang, Kyewon Koh Park, Xiangdong Ye (2007)

Bulletin de la Société Mathématique de France

Similarity:

The properties of topological dynamical systems $(X, T)$ which are disjoint from all minimal systems of zero entropy, $ℳ_{0}$ , are investigated. Unlike the measurable case, it is known that topological $K$ -systems make up a proper subset of the systems which are disjoint from $ℳ_{0}$ . We show that $(X, T)$ has an invariant measure with full support, and if in addition $(X, T)$ is transitive, then $(X, T)$ is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists...

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz Natkaniec, Piotr Szuca (2010)

Colloquium Mathematicae

Similarity:

We prove that if f: → is Darboux and has a point of prime period different from $2^{i}$ , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

Similarity:

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Entropy solutions for nonhomogeneous anisotropic $Δ_{p ⃗ (\cdot)}$ problems

Elhoussine Azroul, Abdelkrim Barbara, Mohamed Badr Benboubker, Hassane Hjiaj (2014)

Applicationes Mathematicae

Similarity:

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

Some logarithmic function spaces, entropy numbers, applications to spectral theory

Haroske Dorothee

Similarity:

AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $i d : H_{q}^{s} (w (x), ℝ ⁿ) \to L ₚ (ℝ ⁿ)$ , s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w (x) = ⟨ x ⟩^{α}$ , α > 0, or $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0, and $⟨ x ⟩ = {(2 + | x | ²)}^{1 / 2}$ . We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w (x) = l o g^{β} ⟨ x ⟩$ , β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $L ₚ {(l o g L)}_{- a} (ℝ ⁿ)$ , a...

A representation theorem for perfect partitions of $Z^{2}$ -actions with finite entropy

B. Kamiński (1988)

Colloquium Mathematicae

Similarity:

On some nonlinear nonhomogeneous elliptic unilateral problems involving noncontrollable lower order terms with measure right hand side

C. Yazough, E. Azroul, H. Redwane (2013)

Applicationes Mathematicae

Similarity:

We prove the existence of entropy solutions to unilateral problems associated to equations of the type $A u - d i v (ϕ (u)) = μ \in L ¹ (Ω) + W^{- 1, p^{'} (\cdot)} (Ω)$ , where A is a Leray-Lions operator acting from $W ₀^{1, p (\cdot)} (Ω)$ into its dual $W^{- 1, p (\cdot)} (Ω)$ and $ϕ \in C ⁰ (ℝ, ℝ^{N})$ .

Displaying similar documents to “Entropy numbers of r-nuclear operators between L p spaces”

Displaying similar documents to “Entropy numbers of r-nuclear operators between $L_{p}$ spaces”