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Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces

100%

Kühn T. , Mastyło M.

nr 3

275-296

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 able to show the optimality of our eigenvalue estimates in the Lorentz spaces $L_{2,q}$ with 1 ≤ q < 2 and in Zygmund spaces $L_{p}(log L)_a$ with 2 ≤ p < ∞ and a > 0.

Extreme points of the complex binary trilinear ball

80%

Cobos F. , Kühn T. , Peetre J.

Studia Mathematica

2000

tom 138

nr 1

81-92

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space $ℂ^2$. This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space $ℝ^2$. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

On the dependence of parameters in the equivalence theorem for the real method

80%

Cobos F. , Fernández-Cabrera L. M. , Karadzhov G. E. , Kühn T.

Commentationes Mathematicae

2015

tom 55

nr 2

We determine the exact dependence on $\theta,q,p$ of the constants in the equivalence theorem for the real interpolation method $(A_0,A_1)_{\theta,q}$ with pairs of $p$-normed spaces.