On Stably $\mathcal{K}$-Monotone Banach Couples

The $\mathcal{K}$-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 $\|e_n\|=1$ ($n\in\mathbb{N}$), where $\{e_n\}_{n\in\mathbb{Z}}$ is the canonical basis. It is shown that $\vec{E}=(E,E(2^{-k}))$ is a stably $\mathcal{K}$-monotone couple if and only if $\vec{E}$ is $\mathcal{K}$-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 $\mathcal{K}$-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 $\vec{E}$ is $\mathcal{K}$-monotone, then either $E=l_p$ ($1\le p<\infty$) or $E=c_0$.