RIESZ SUMMABILITY OF ORTHOGONAL SERIES IN NONCOMMUTATIVE L2-SPACES

• Published in: Journal of operator theory 2011-12, Vol.65 (1), p.3-15
• Main Authors: LE GAC, BARTHÉLEMY, MÓRICZ, FERENC
• Format: Article
• Language: English
• Subjects: Algebra; Integers; Mathematical theorems; Mathematical vectors; Orthogonal functions; Perceptron convergence procedure; Real numbers; Series convergence; Triangle inequalities; Von Neumann algebra
• ISSN: 0379-4024; 1841-7744

A Riesz summability method is defined by means of a sequence 0 = λ0 < λ1 < ··· < λn → ∞ of real numbers. The following theorem is known in commutative L2-spaces: If a sequence {ξn : n = 0,1,...} of pairwise orthogonal functions in some L2 = L2(X,F,μ) over a positive measure space is such that ${\sum _{\mathrm{n}:{\mathrm{\lambda }}_{\mathrm{n}}\ge 4}(\mathrm{log}\mathrm{log}{\mathrm{\lambda }}_{\mathrm{n}}{)}^{2}\Vert {\mathrm{\xi }}_{\mathrm{n}}\Vert }^{2}