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.

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
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.

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.