## 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

Zbl 0744.46039
Full Text:

### 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 [4] Smith, B., A strong convergence theorem for $$H^1(\mathbb{T})$$, (), 169-173
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