On stability of κ-monotonicity of Banach couples

Suppose that E is a separable Banach lattice of two-sided real sequences such that e_n = 1 (n ε ℕ), where {e_n}nε ℤ is the standard basis. One of the main aims of this paper is a characterization of couples E→ = (E, E(2^-κ)) whose κ-monotonicity is stable when multiplying the weight by a constant. It is shown that such a property holds only for a couple E→ constructed upon a shift-invariant lattice. We construct also a non-trivial example of shift-invariant separable Banach lattice E such that the couple E→ is κ-monotone. The last result contrasts with the following well-known theorem due to Kalton: if E is a separable symmetric Banach lattice such that the couple E→ is κ-monotone then either E = l_p (1 ≤ p< ∞) or E = c_0..