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Research Papers
The spectrum of some compressions of unilateral shifts
S. Duberneta, J. Esterleb a Professeur de CPES, Epinay sur Seine, France
b Université Bordeaux 1, Talence, France
Abstract:
Let $E$ be a star-shaped Banach space of analytic functions on the open unit disc $\mathbb D$. We assume that the unilateral shift $S\colon z\to zf$ and the backward shift $T\colon f\to\frac{f-f(0)}{z}$ are bounded on $E$ and that their spectrum is the closed unit disc.
Let $M$ be a closed $z$-invariant subspace of $E$ such that $\dim(M/zM)=1$, and let $g\in M$. The main result of the paper shows that if $g$ has an analytic extension to $\mathbb D\cup D(\zeta,r)$ for some $r>0$, with $g(\zeta)\ne 0$, and if $S$ and $T$ satisfy the “nonquasianalytic condition”
$$
\sum_{n\ge 0}\frac{\log\| S^n\|+\log\| T^n\|}{ 1+n^2}<+\infty,
$$
then $\zeta$ does not belong to the spectrum of the compression $S_M\colon f+M\to zf+M$ of the unilateral shift to the quotient space $E/M$. This shows in particular that $\operatorname{Spec}(S_M)=\{1\}$ for some $z$-invariant subspaces $M$ of weighted Hardy spaces constructed by N. K. Nikol'skiĭ in the seventies by using the Keldysh method.
Keywords:
Unilateral shift, nonquasianalyticty condition, spectrum.
Received: 12.08.2006
Citation:
S. Dubernet, J. Esterle, “The spectrum of some compressions of unilateral shifts”, Algebra i Analiz, 20:5 (2008), 83–98; St. Petersburg Math. J., 20:5 (2009), 737–748
Linking options:
https://www.mathnet.ru/eng/aa531 https://www.mathnet.ru/eng/aa/v20/i5/p83
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Abstract page: | 262 | Full-text PDF : | 80 | References: | 40 | First page: | 14 |
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