Invariant subspaces of weighted shift operators

Let $s$ be a weighted shift operator in $l^p$, $p\in[1,+\infty)$:$s(b_0, b_1, ...)=(0,\lambda_0b_0,\lambda_1b_1,\dots)$. One proves its unicellularity under the condition $|\lambda_i|\downarrow 0$ and also under some weaker conditions. One obtains also unicellularity conditions for weighted shift operators in Banach spaces of numerical sequences. One gives a new proof of the following theorem of M. P. Thomas: if $(\prod_{i=0}^{n-1}|\lambda_i|)^{1/n}\downarrow 0$ and $|\lambda_i|=O(i^{-\varepsilon})$, $\varepsilon >0$, then the operator $s$ is unicellular in $l^p$. One considers also a multiple weighted shift, corresponding to the case when $b_i$ are finite-dimensional vectors. Under the condition $\mu_{i+1}\|b\|\le\|\lambda_ib\|\le\mu_i\|b\|$, $\mu_i\downarrow 0$ one obtains the description of the invariant subspaces of this operator, using formal matrix power series.