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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in ${ℂ}^n$ of codimension 1, $vo{l}_{2n-2}\left(D^{n-1}\right)\le vo{l}_{2n-2}\left(H\cap D^n\right)\le 2vo{l}_{2n-2}\left(D^{n-1}\right)$. The lower bound is attained if and only if H is orthogonal to the versor $e_j$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_j+\sigma e_k$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify ${ℂ}^n$ with ${ℝ}^{2n}$; by $vo{l}_k\left(·\right)$ we denote the usual k-dimensional volume in ${ℝ}^{2n}$. The result is a complex counterpart of Ball’s [B1] result for...

Consider the set of all Toeplitz-Schur multipliers sending every upper triangular matrix from the trace class into a matrix with absolutely summable entries. We show that this set admits a description completely analogous to that of the set of all Fourier multipliers from H₁ into ℓ₁. We characterize the set of all Schur multipliers sending matrices representing bounded operators on ℓ₂ into matrices with absolutely summable entries. Next, we present a result (due to G. Pisier) that the upper triangular...

comité de rédaction: Czesław Bessaga, Stanisław mazur, Władysław Orlicz, Aleksander Pełczyński, Stefan Rolewicz, Wiesław Żelazko Front page of Volume II, p.1-1 Tables des Matières, p.1-4 Préface, p.5-5 Publications de Stefan Banach, p.7-11 S. Banach: Front page and preface to "THÉORIE DES OPÉRATIONS LINÉAIRES" (MONOGRAFIE MATEMATYCZNE V.1), p.13-18 S. Banach: THÉORIE DES OPÉRATIONS LINÉAIRES (MONOGRAFIE MATEMATYCZNE T.1), p.19-222 C. Bessaga, A. Pełczyński: SOME ASPECTS OF THE PRESENT THEORY OF...