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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 141, Pages 100–143
(Mi znsl4181)
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This article is cited in 6 scientific papers (total in 6 papers)
Invariant subspaces of weighted shift operators
D. V. Yakubovich
Abstract:
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.
Citation:
D. V. Yakubovich, “Invariant subspaces of weighted shift operators”, Investigations on linear operators and function theory. Part XIV, Zap. Nauchn. Sem. LOMI, 141, "Nauka", Leningrad. Otdel., Leningrad, 1985, 100–143; J. Soviet Math., 37:5 (1987), 1323–1346
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https://www.mathnet.ru/eng/znsl4181 https://www.mathnet.ru/eng/znsl/v141/p100
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