On stably -monotone Banach couples

The -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 ‖ e n ‖ = 1 ( n ∈ ℕ), where e n n ∈ℤ is the canonical basis. It is shown that = ( E, E (2 - k )) is a stably -monotone couple if and only if is -monotone and E is shift-invariant. A non-trivial example of a shift-invariant separable Banach lattice E such that the couple is -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 -monotone, then either E = l p (1 ≤ p < ∞) or E = c 0 .