Abstract
The Open image in new window-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 ‖en‖ = 1 (n ∈ ℕ), where enn∈ℤ is the canonical basis. It is shown that \( \vec E \) = (E, E(2-k)) is a stably Open image in new window-monotone couple if and only if \( \vec E \) is Open image in new window-monotone and E is shift-invariant. A non-trivial example of a shift-invariant separable Banach lattice E such that the couple \( \vec E \) is Open image in new window-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 \( \vec E \) is Open image in new window-monotone, then either E = lp (1 ≤ p < ∞) or E = c0.
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