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A characterization of upper triangular trace class matrices. (English) Zbl 1192.46035

The starting point for this short paper is Pavlović’s theorem on an equivalent characterisation of an analytic function \(f\) to belong to the Hardy space \(H^1\) in terms of uniform boundedness of weighted partial sums of the Taylor series of \(f\). The main result concerns upper triangular matrices \(A\) for which it is shown to be equivalent that \(A\) is of trace class, \(A\in T_1\), with some special uniform boundedness assertions similar to the ones in Pavlović’s theorem, but now involving weighted partial sums of \(s_j(A) = \sum_{k=0}^jA_k\), where \(A_k\) is the \(k\)th diagonal matrix of \(A\). At the end some interesting consequences of this new theorem are given including also some generalised Shield’s inequality. The essential clue in the argument is the vector-valued Hardy inequality from O.Blasco and A.Pełczyński [“Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces”, Trans.Am.Math.Soc.323, No. 1, 335–367 (1991; Zbl 0744.46039)], \[ \sum_{k\geq 0} (k+1)^{-1} \|\hat{f}(k)\|_1 \leq C\|f\|_1, \quad f\in H^1_X, \]
where \(X\) is a complex Banach space.

MSC:

46E40 Spaces of vector- and operator-valued functions
15B05 Toeplitz, Cauchy, and related matrices
30H10 Hardy spaces

Citations:

Zbl 0744.46039
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References:

[1] Blasco, O.; Pelczynski, A., Theorems of Hardy and Paley for vector valued analytic functions and related classes of Banach spaces, Trans. Amer. Math. Soc., 323, 335-367 (1991) · Zbl 0744.46039
[2] McGehee, O. C.; Pigno, L.; Smith, B., Hardy’s inequality and the \(L^1\) norm of exponential sums, Ann. of Math., 113, 613-618 (1981) · Zbl 0473.42001
[3] M. Pavlović, Introduction to function spaces on the disk, Matematicki Institut SANU, Beograd, 2004; M. Pavlović, Introduction to function spaces on the disk, Matematicki Institut SANU, Beograd, 2004 · Zbl 1107.30001
[4] Smith, B., A strong convergence theorem for \(H^1(T)\), (Banach Spaces, Harmonic Analysis and Probability Theory. Banach Spaces, Harmonic Analysis and Probability Theory, Storrs, CT, 1980/1981. Banach Spaces, Harmonic Analysis and Probability Theory. Banach Spaces, Harmonic Analysis and Probability Theory, Storrs, CT, 1980/1981, Lecture Notes in Math., vol. 995 (1983), Springer-Verlag: Springer-Verlag Berlin), 169-173 · Zbl 0527.42004
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