On stably K-monotone Banach couples

The K-monotonicity of Banach couples which is stable with respect to multiplication of weight by a constant is studied. Suppose that E is a separable Banach lattice of two-sided sequences of reals such that {double pipe}e_n{double pipe} = 1 (n ∈ ℕ), where {e_n}_n∈ℤ is the canonical basis. It is shown that, is a stably K-monotone couple if and only if, is K-monotone and E is shift-invariant. A non-trivial example of a shift-invariant separable Banach lattice E such that the couple, is K-monotone is constructed. This result contrasts with the following well-known theorem of Kalton: If E is a separable symmetric sequence space such that the couple, is K-monotone, then either E = l_p (1 ≤ p < ∞) or E = c₀.