A class of ℐ-conservative matrices

Let ∞ and c be the Banach spaces of bounded and convergent sequence x = (xk) with the usual supremum norm. Let σ be a one-to-one mapping of N, the set of positive integers, into itself and T : ∞ → ∞ a linear operator defined by Tx = (Txk)= (xσ(k)). An element φ ∈ ′ ∞, the conjugate space of ∞, is called an invariant mean or a σ-mean if and only if (i) φ(x) ≥ 0 when the sequence x = (xk) has xk ≥ 0 for all k, (ii) φ(e) = 1 where e = (1,1,1, . . .), and (iii) φ(Tx) = φ(x) for all x ∈ ∞. Let M be the set of all σ-means on ∞. A sublinear functional P on ∞ is said to generate σ-means if φ ∈ ′ ∞ and φ ≤ P ⇒ φ is a σ-mean, and to dominate σ-means if φ ≤ P for all φ ∈ M, where φ ≤ P means that φ(x)≤ P(x) for all x ∈ ∞. It is shown [8] that the sublinear functional