Characterizations of series in Banach spaces.

*(English)*Zbl 0952.46009The authors of this paper consider weakly unconditionally Cauchy series in Banach spaces. It is well known that:

1. A weakly unconditionally Cauchy series in a Banach space can be characterized as a series \(\sum_i x_i\) such that, for every null sequence \((t_i)_i\), \(\sum_i t_ix_i\) is convergent.

2. In a normed space \(X\), \(\sum^\infty_{i=1} x_i\) is a weakly unconditionally Cauchy series if and only if the set \[ E= \Biggl\{ \sum^n_{i=1} \alpha_i x_i:n\in\mathbb{N},\;|\alpha_i|\leq 1, i\in\{1,\dots, n\}\Biggr\} \] is bounded.

3. If \(X\) is a Banach space then the following conditions are equivalent:

(a) There exists a weakly unconditionally Cauchy series which is convergent, but is it not unconditionally convergent, in \(X\).

(b) There exists a weakly unconditionally Cauchy series which is weakly convergent, but is not convergent.

(c) There exists a weakly unconditionally Cauchy series which is not weakly convergent.

(d) The space \(X\) has a copy of \(c_0\).

For any given series \(\zeta= \sum_i x_i\) in \(X\) consider the sets.

1. \({\mathcal S}={\mathcal S}(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) converges.

2. \({\mathcal S}_w={\mathcal S}_w(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) is weakly convergent.

3. \({\mathcal S}_0={\mathcal S}_0(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) is \(*\)-weakly convergent (i.e. \(\left(\sum^n_{i= 1}a_ix_i\right)_n\) converges with respect to the topology \(\sigma(X^{**},X^*))\).

These sets, endowed with the sup norm, will be called the spaces of convergence, of weakly convergence and of weak-\(*\) convergence of the series \(\zeta\), respectively.

It is clear that if the Banach space \(X\) does not have a copy of \(c_0\), then every weakly unconditionally Cauchy series is unconditionally convergent and we have \({\mathcal S}={\mathcal S}_w={\mathcal S}_0= \ell_\infty\). The authors obtain the following

Theorem. Let \(\sum_i x_i\) be a series in a Banach space \(X\). The following statements are equivalent:

1. \({\mathcal S}_0= \ell_\infty\).

2. The space \({\mathcal S}_0\) is complete.

3. The series \(\sum_i x_i\) is a weakly unconditionally Cauchy series.

Using sequence spaces of reals such as \({\mathcal S}\), \({\mathcal S}_w\), \({\mathcal S}_0\), they considered some characterization of unconditionally Cauchy series in a Banach space.

1. A weakly unconditionally Cauchy series in a Banach space can be characterized as a series \(\sum_i x_i\) such that, for every null sequence \((t_i)_i\), \(\sum_i t_ix_i\) is convergent.

2. In a normed space \(X\), \(\sum^\infty_{i=1} x_i\) is a weakly unconditionally Cauchy series if and only if the set \[ E= \Biggl\{ \sum^n_{i=1} \alpha_i x_i:n\in\mathbb{N},\;|\alpha_i|\leq 1, i\in\{1,\dots, n\}\Biggr\} \] is bounded.

3. If \(X\) is a Banach space then the following conditions are equivalent:

(a) There exists a weakly unconditionally Cauchy series which is convergent, but is it not unconditionally convergent, in \(X\).

(b) There exists a weakly unconditionally Cauchy series which is weakly convergent, but is not convergent.

(c) There exists a weakly unconditionally Cauchy series which is not weakly convergent.

(d) The space \(X\) has a copy of \(c_0\).

For any given series \(\zeta= \sum_i x_i\) in \(X\) consider the sets.

1. \({\mathcal S}={\mathcal S}(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) converges.

2. \({\mathcal S}_w={\mathcal S}_w(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) is weakly convergent.

3. \({\mathcal S}_0={\mathcal S}_0(\zeta)\), of sequences \((a_i)_i\in\ell_\infty\) such that \(\sum_i a_ix_i\) is \(*\)-weakly convergent (i.e. \(\left(\sum^n_{i= 1}a_ix_i\right)_n\) converges with respect to the topology \(\sigma(X^{**},X^*))\).

These sets, endowed with the sup norm, will be called the spaces of convergence, of weakly convergence and of weak-\(*\) convergence of the series \(\zeta\), respectively.

It is clear that if the Banach space \(X\) does not have a copy of \(c_0\), then every weakly unconditionally Cauchy series is unconditionally convergent and we have \({\mathcal S}={\mathcal S}_w={\mathcal S}_0= \ell_\infty\). The authors obtain the following

Theorem. Let \(\sum_i x_i\) be a series in a Banach space \(X\). The following statements are equivalent:

1. \({\mathcal S}_0= \ell_\infty\).

2. The space \({\mathcal S}_0\) is complete.

3. The series \(\sum_i x_i\) is a weakly unconditionally Cauchy series.

Using sequence spaces of reals such as \({\mathcal S}\), \({\mathcal S}_w\), \({\mathcal S}_0\), they considered some characterization of unconditionally Cauchy series in a Banach space.

Reviewer: Shozo Koshi (Sapporo)

##### MSC:

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

46B45 | Banach sequence spaces |

40A05 | Convergence and divergence of series and sequences |