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Matriceal Lebesgue spaces and Hölder inequality. (English) Zbl 1093.42002

A matrix \(A=(a_{ij})\) is said to be of \(n-\)band type if \(a_{ij}=0\) for \(| i-j| >n.\) For a sequence of functions \(f=(f_l)_{l\geq1},\) let \(A_f=(a_{ij})\) where \(\widehat f_l(k)=a_{l,l+k}\) if \(k\geq0\), \(l\geq1\), and \(\widehat f_l(k)=a_{l-k,l}\) if \(k<0\), \(l\geq1\), and identify \(A_f\) with its symbol \(f\). Let \(A_f\) and \(A_g\) be two such matrices of finite band type. The authors define a commutative product of \(A_f\) and \(A_g\) as \(A_f\square A_g=A_{fg}\) and for this product prove matriceal versions of the Hölder inequality.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
26D15 Inequalities for sums, series and integrals
15B57 Hermitian, skew-Hermitian, and related matrices
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