The description of closed ideals of algebra $\lambda^{(n)}_\omega$

It is proved that all ideals of the space $\lambda^{(n)}_\omega$ are standard in the following two cases: 1) $n\geqslant1$, $\omega$ is a non-decreasing function; $\omega(t)/t$ is a non-increasing function; 2) $n=0$ and there exists $\alpha$, $\alpha>0$, such that $\omega(t)=O(t^\alpha)$.