Weak unconditional Cauchy series

Weakly unconditional Cauchy series , also called weak unconditionally convergent series or, for shorter, WUC series , are examined in the mathematical sub-area of functional analysis. They are not necessarily convergent series in Banach spaces with a certain additional property.

definition

Let it be a Banach space, its dual space and a series in , which means, as always, the sequence of the partial sums . The series is called weak unconditionally Cauchy or weak unconditionally convergent if for every continuous , linear functional out holds. ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle \textstyle \sum _{n}x_{n}}$ ${\displaystyle X}$ ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }|x^{*}(x_{n})|<\infty }$ ${\displaystyle X^{*}}$

This property is abbreviated to WUC after the English term weakly unconditionally Cauchy or weakly unconditionally convergent .

Remarks

The term weak in the above definition means that it is a property that must apply to everyone . ${\displaystyle x^{*}\in X^{*}}$

The name component necessarily stems from the fact that the condition can also be replaced by the unconditional convergence of the series , because unconditional convergence and absolute convergence coincide in the basic body . A direct consequence of this observation is that every rearrangement of a WUC series is again WUC. ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }|x^{*}(x_{n})|<\infty }$ ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }x^{*}(x_{n})}$

Since the sequence of the partial sums of a WUC series is obviously a weak Cauchy sequence , the name component Cauchy is also explained . Using convergent can be misleading because there is generally no weak convergence of the series.

characterization

For a series in a Banach space , the following statements are equivalent: ${\displaystyle \textstyle \sum _{n}x_{n}}$ ${\displaystyle X}$

• ${\displaystyle \textstyle \sum _{n}x_{n}}$ is WUC
• There's a constant , so that ${\displaystyle C>0}$

${\displaystyle \sup _{n\in \mathbb {N} }\left\|\sum _{k=1}^{n}t_{k}x_{k}\right\|\leq C\sup _{n\in \mathbb {N} }|t_{n}|}$

applies to all episodes from the episode space . ${\displaystyle (t_{n})_{n}}$ ${\displaystyle \ell ^{\infty }}$

• There's a constant , so that ${\displaystyle C>0}$

${\displaystyle \sup _{n\in F}\left\|\sum _{k=1}^{n}t_{k}x_{k}\right\|\leq C}$

holds for every finite subset and every choice of sign . ${\displaystyle F\subset \mathbb {N} }$ ${\displaystyle t_{n}\in \{-1,+1\}}$

• For every zero sequence in converges in ${\displaystyle (t_{n})_{n}\in c_{0}}$ ${\displaystyle \textstyle \sum _{n}t_{n}x_{n}}$ ${\displaystyle X}$
• There is a continuous, linear operator with for all , where the n th unit sequence is in, that is , the sequence that has a 1 in the n th position and a 0 in all other positions. ${\displaystyle T:c_{0}\rightarrow X}$ ${\displaystyle T(e_{n})=x_{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle e_{n}}$ ${\displaystyle c_{0}}$ ${\displaystyle e_{n}}$

Comparison with unconditional convergence

It is clear that necessarily convergent series are WUC. The reverse is generally not true. Consider the series of unit sequences in . As is well known, each is given by an absolutely convergent series . thats why ${\displaystyle \textstyle \sum _{n}e_{n}}$ ${\displaystyle c_{0}}$ ${\displaystyle x^{*}\in c_{0}^{*}}$ ${\displaystyle (\xi _{n})_{n}\in \ell ^{1}}$

${\displaystyle \sum _{n\in \mathbb {N} }|x^{*}(e_{n})|=\sum _{n\in \mathbb {N} }|\xi _{n}\cdot 1|<\infty }$,

that is, is WUC. But this series does not converge in , so in particular it is not necessarily convergent. The following sentence specifies conditions under which a WUC series necessarily converges. ${\displaystyle \textstyle \sum _{n}e_{n}}$ ${\displaystyle c_{0}}$

• Let it be a WUC series in a Banach space and let the existing operator with according to the above characterization . Then the following statements are equivalent: ${\displaystyle \textstyle \sum _{n}x_{n}}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle T:c_{0}\rightarrow X}$ ${\displaystyle T(e_{n})=x_{n}}$
  • ${\displaystyle \textstyle \sum _{n}x_{n}}$ is necessarily convergent.
  • T is compact .
  • T is weakly compact
  • T is strictly singular .

According to the following sentence, which goes back to Czesław Bessaga and Aleksander Pełczyński , one can characterize the spaces in which each WUC series necessarily converges. This sentence also shows that the counterexample given above is essentially the only one.

• A Banach space has the property that every WUC series converges unconditionally if and only if it does not contain a sub- Banach space that is too isomorphic. ${\displaystyle c_{0}}$

Individual evidence

1. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 2.4.3
2. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , page 39 below
3. ^ J. Diestel: Sequences and series in Banach spaces , Springer-Verlag (1984), ISBN 0-387-90859-5 , Chapter V, Theorem 6
4. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Lemma 2.4.6 and Theorem 2.4.7
5. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 2.4.8 and Theorem 2.4.10
6. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 2.4.11