A feature of averaging. (English) Zbl 1281.46005
Let \(1<p<\infty\) and let \(ces(p)\) (named after Cesàro) be the Banach sequence space
\[
ces(p)=\left\{x=(x_n)\::\|x\|_{ces(p)}:=\sum_{n=1}^\infty\left(\frac{1}{n}\sum_{k=1}^n |x_k|\right)^p <\infty\right\}.
\]
For any sequence \(x=(x_n)_n\), let \(Cx=(\frac{1}{n}\sum_{k=1}^n x_k)_n\) and \(|x|=(|x_n|)_n\). Note that (by definition) \(x\in ces(p)\) exactly when \(C|x|\in\ell_p\) and that (thus) \(C|x|\in ces(p)\) exactly when \(CC|x|\in\ell_p\).
So, if \(x\in ces(p)\), \(C|x|\in \ell_p\). But, by Hardy’s classical inequality, then \(C|x|\in ces(p)\). The converse is true as well as is proved in Paragraph 20.31 of G. Bennett [Mem. Am. Math. Soc. 576 (1996; Zbl 0857.26009)].
The main aim of the note under review is a short proof of an apparently new inequality, \[ (CC|x|)_n\geq\frac{1}{6}(C(|x|)_{[n/2]}, \tag{\(*\)} \] where \([\cdot]\) denotes the integer part.
As a consequence, \[ \sum_{n=1}^\infty (C|x|)_n^p\leq 6^p \sum_{n=1}^\infty(CC|x|)_{2n}^p, \] from which Bennett’s result follows directly.
For a linear subspace \(\mathbb{X}\) of \(\mathbb{C}^\mathbb{N}\), denote \(ces(\mathbb{X})=\{x\in\mathbb{C}^\mathbb{N}:C|x|\in \mathbb{X}\}\). Inequality \((*)\) now lets us conclude the “half Bennett result” that \(x\in ces(\mathbb{X})\) if \(C|x|\in\mathbb{X}\) whenever \(\mathbb{X}\) is solid. Furthermore, in the cases when \(x\mapsto Cx\) maps \(\mathbb{X}\) into itself, we obtain the “full Bennett result” that \[ x\in ces(\mathbb{X})\Leftrightarrow C|x|\in ces(\mathbb{X)}. \] The last part of this very cute little note contains a list of a lot of cases where either the half or the full generalized Bennett theorem is obtained.
So, if \(x\in ces(p)\), \(C|x|\in \ell_p\). But, by Hardy’s classical inequality, then \(C|x|\in ces(p)\). The converse is true as well as is proved in Paragraph 20.31 of G. Bennett [Mem. Am. Math. Soc. 576 (1996; Zbl 0857.26009)].
The main aim of the note under review is a short proof of an apparently new inequality, \[ (CC|x|)_n\geq\frac{1}{6}(C(|x|)_{[n/2]}, \tag{\(*\)} \] where \([\cdot]\) denotes the integer part.
As a consequence, \[ \sum_{n=1}^\infty (C|x|)_n^p\leq 6^p \sum_{n=1}^\infty(CC|x|)_{2n}^p, \] from which Bennett’s result follows directly.
For a linear subspace \(\mathbb{X}\) of \(\mathbb{C}^\mathbb{N}\), denote \(ces(\mathbb{X})=\{x\in\mathbb{C}^\mathbb{N}:C|x|\in \mathbb{X}\}\). Inequality \((*)\) now lets us conclude the “half Bennett result” that \(x\in ces(\mathbb{X})\) if \(C|x|\in\mathbb{X}\) whenever \(\mathbb{X}\) is solid. Furthermore, in the cases when \(x\mapsto Cx\) maps \(\mathbb{X}\) into itself, we obtain the “full Bennett result” that \[ x\in ces(\mathbb{X})\Leftrightarrow C|x|\in ces(\mathbb{X)}. \] The last part of this very cute little note contains a list of a lot of cases where either the half or the full generalized Bennett theorem is obtained.
Reviewer: Olav Nygaard (Kristiansand)
MSC:
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
Citations:
Zbl 0857.26009References:
| [1] | Bennett G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120(576), 1-130 (1996) · Zbl 0857.26009 |
| [2] | Hardy G.H.: Divergent series. Oxford University Press, Oxford (1949) · Zbl 0032.05801 |
| [3] | Lindenstrauss J., Tzafriri L.: Classical Banach spaces vol. I. Springer, Berlin (1979) · Zbl 0403.46022 |
| [4] | Meise R., Vogt D.: Introduction to functional analysis. Oxford University Press, Oxford (1997) · Zbl 0924.46002 |
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