The Banach-Saks index of intersection

JPRM-Vol. 1 (2012), Issue 1, pp. 22 – 27 Open Access Full-Text PDF
Novikova A. I
Abstract: In this paper we estimate Banach-Saks index of intersection of two spaces with symmetric bases from below by indices of these spaces. We also show on example of Orlicz spaces that we can’t estimate Banach-Saks index of intersection in the same way from above.
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Stability estimate for the multidimensional elliptic obstacle problem with respect to the obstacle function

JPRM-Vol. 1 (2012), Issue 1, pp. 12 – 21 Open Access Full-Text PDF
Naveed Ahmad, Malkhaz Shashiashvili
Abstract: The stability estimate of the energy integral established by Danelia, Dochviri and Shashiashvili [1] for the solution of the multidimensional obstacle problem in case of the Laplace operator is generalized to the case of arbitrary linear second order self-adjoint elliptic operator. This estimate asserts that if two obstacle functions are close in the \(L^{∞}\)-norm, then the gradients of the solutions of the corresponding obstacle problem are close in the weighted \(L^{2}\) -norm.
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Some more remarks on grothendieck-lidskii trace formulas

JPRM-Vol. 1 (2012), Issue 1, pp. 05 – 11 Open Access Full-Text PDF
Oleg Reinov
Abstract: Let \(r ∈ (0, 1]\), \(1 ≤ p ≤ 2\), \(u ∈ X^∗⊗X\) and \(u\) admits a representation \(u=\sum_{i}\lambda_{i}x_{i}^{‘}⊗ x_{i}\) with \((λ_i) ∈ l_r\) bounded and \((x_{i} ∈ l^{w}_{p’} (X)\). If \(1/r + 1/2 − 1/p = 1\) then the system \(\mu_{k}\) of all eigenvalues of the corresponding operator \(\widetilde{u}\) (written according to their algebraic multiplicities) is absolutely summable and trace \(u=\sum_{k}\mu_{k}\). One of the main aim of these notes is not only to give a proof of the theorem but also to show that it could be obtained by A. Grothendieck in 1955.
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