Entropy numbers of diagonal operators of logarithmic type.
On the r-Nuclearity of Gaussian Covariances and the Composition of Nuclear Operators.
An Operator Ideal in Connection with Gaussian Measures and Applications to Nuclearity of Integral Operators.
Entropy numbers of r-nuclear operators in Banach spaces of type q
Entropy numbers of general diagonal operators.
We determine the asymptotic behavior of the entropy numbers of diagonal operators D: l → l, (x) → (sx), 0 < p,q ≤ ∞, under mild regularity and decay conditions on the generating sequence (σ). Our results extend the known estimates for polynomial and logarithmic diagonals (σ). Moreover, we also consider some exotic intermediate examples like (σ)=exp(-√log k).
Entropy and Eigenvalues of Certain Integral Operators.
Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces
We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also...
Local entropy moduli and eigenvalues of operators in Banach spaces.
In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators...
Extreme points of the complex binary trilinear ball
We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space . This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space . As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.
Compact embeddings of Brézis-Wainger type.
Let Ω be a bounded domain in R and denote by id the restriction operator from the Besov space B (R) into the generalized Lipschitz space Lip(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like e(id) ~ k if α > max (1 + 2/p + 1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.
Remarks on symmetries of trilinear forms.
On the dependence of parameters in the equivalence theorem for the real method
We determine the exact dependence on of the constants in the equivalence theorem for the real interpolation method with pairs of -normed spaces.
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