On I-convergence of series

Georges Grekos, Ladislav Mišík and Miloš Ziman

Workshop on the Density Concept

May 16 – 18, 2004 Bratislava Definition. A series ∞ ; ∑ ; n=1 a_{n} is said to be I-

convergent if the sequence (s_{n}) of its partial sums, i.e., s_{n} = a_{1} + a_{2} + ⋯ + a_{n}, is I-convergent.

Definition. An admissible ideal I ⊆ 2^{ℕ} has a property (T) if for every I-convergent series ∞ ; ∑ ; n=1 a_{n} there is a set F ∈ F(I), F = {n_{1}, n_{2}, … }, such that the corresponding subseries ∞ ; ∑ ; k=1 a_{n_{k}} is convergent in the usual sense.

Proposition (Červeňanský, Šalát). Ideals I_{c}, I_{d}, I_{δ}, I_{u} do not have property (T).

I_{c} = \{A ⊆ ℕ; _{k∈A}^{∑} \frac{1}{k} < ∞\};
I_{d} = {A ⊆ ℕ; d(A) = 0};
I_{δ} = {A ⊆ ℕ; δ(A) = 0};
I_{u} = {A ⊆ ℕ; u(A) = 0}.

Sufficient conditions

Definition. An I-partition of the set ℕ is a family of pairwise disjoint sets B = {B_{n} ∈

I; n ∈ ℕ} such that ∞ ; ⋃ ; n=1 B_{n} = ℕ.

Theorem. Let an admissible ideal I satisfy the following conditions:

(a) If A ∈ I then A + 1 ∈ I, where A + 1 = {a + 1; a ∈ A}.

(b) For every I-partition B of ℕ there exists a set F ∈ F(I) and M ∈ ℕ such that for all B ∈ B we have |B ∩ F | < M.

Then the ideal I has the property (T).

Proposition. There is no admissible ideal satisfying both the conditions (a) and (b).

Theorem. Let an admissible ideal I satisfy the condition (a) and:

(b’) There exists a sequence of positive reals (m_{n}) such that for every I-partition {B_{n} ∈ I; n ∈ ℕ} of ℕ there exists a set F ∈ F(I) and M ∈ ℕ such that for all n ∈ ℕ we have |B_{n} ∩ F | < Mm_{n}.

Then the ideal I has the property (T).

Proposition. The ideal I_{f} does not satisfy the condition (b’).

Necessary conditions

Theorem. If an admissible ideal I has the property (T) then it satisfies the condition (a).

Lemma. If I is an admissible ideal satisfying the condition (a) then for every set A ∈ I and for every M ∈ ℕ there is an a ∈ ℕ such that A ∩ [a, a + M] = ∅. (A has arbitrary large gaps.)

Corollary. No maximal admissible ideal satisfies the condition (a) – and hence no maximal ideal has the property (T).

Theorem. If an admissible ideal I has the property (T) then it satisfies the condition

(c) For every I-partition B of ℕ there exists a set F ∈ F(I) such that the set B ∩ F is finite for all B ∈ B.

Definition. An admissible ideal I ⊆ 2^{ℕ} is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets {B_{1}, B_{2}, … } belonging to I there exists a countable family of sets {A_{1}, A_{2}, … } such that the set A_{n}△B_{n} is finite for all n ∈ ℕ and A = ∞ ; ⋃ ; n=1 A_{n} ∈ I.

Proposition (Kostyrko, Šalát, Wilczyński). An admissible ideal I satisfies the (AP) condition iff for every I-convergent sequence (x_{n})_{n∈ℕ} there exists a set F ∈ F(I), F = {n_{1} < n_{2} < … } such that the corresponding subsequence (x_{n_{k}})_{k∈ℕ} is convergent in the usual sense and I-lim_{n→∞} x_{n} = lim_{k→∞} x_{n_{k}}.

Proposition. An admissible ideal I satisfies the (AP) condition iff I satisfies the condition (c).

Proposition. Every admissible ideal I containing an infinite set and satisfying the conditions (a) and (c) contains an admissible subideal J ⊆ I which does not satisfy the condition (c).