Volume 8 (2012) Issue 1

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.

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.

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.

Abstract: We present a new short proof of Stirling’s formula for the Gamma function. Our approach is based on the Gauss product formula and on a remark concerning the existence of horizontal asymptotes.