Size of the peripheral point spectrum under power or resolvent growth conditions
Let \(X\) be a complex infinite-dimensional Banach space, and let \({\mathcal L}(X)\) be the algebra of all bounded linear operators on \(X\). An operator \(T\in{\mathcal L}(X)\) is said to be power-bounded if \(\sup_{n\geq0}\| T^n\| <+\infty\), and is said be to partially power-bounded with respect to an increasing sequence \((n_k)_k\) of positive integers if \(\sup_{k\geq0}\| T^{n_k}\| <+\infty\). In [Proc.\ Am.\ Math.\ Soc.\ 16, 375--377 (1965; Zbl 0133.07202)], \textit{B.\,Jamison} proved that if \(T\in{\mathcal L}(X)\) is a power-bounded operator, then the peripheral point spectrum \(\sigma_p(T)\cap\mathbb{T}\) is at most countable, where \(\sigma_{p}(T)\) denotes the point spectrum of \(T\) and \(\mathbb{T}\) stands as usual for the unit circle. This result does not always hold when replacing the power-boundedness by partial power-boundedness, as shown in [\textit{T.\,Ransford} and \textit{M.\,Roginskaya}, J.~Funct.\ Anal.\ 230, No.\,2, 432--445 (2006; Zbl 1087.47004); \textit{T.\,Ransford}, Isr.\ J.\ Math.\ 146, 93--110 (2005; Zbl 1103.47002)]. Thus the choice of the following definition, introduced by the authors in [Adv.\ Math.\ 211, No.\,2, 766--793 (2007; Zbl 1123.47006)], is justified: A sequence \((n_k)_k\) of positive integers is said to be a \textit{Jamison sequence} if every bounded linear operator on a separable Banach space which is partially power-bounded with respect to \((n_k)_k\) has a countable peripheral point spectrum. The authors first characterize Jamison sequences and show that an increasing sequence \((n_k)_k\) is a Jamison sequence if and only if \[ \inf\big\{\{\sup| \lambda^{n_k}-1| :k\geq0\}:\lambda\in\mathbb{T}\backslash\{1\}\big\}>0. \] They next provide and discuss several examples of Jamison sequences including the ones which appeared in the above cited papers. For any sequence \((n_k)_k\) of positive integers for which \(\lim_{k\to+\infty}\frac{n_{k+1}}{n_k}=+\infty\), they construct a separable complex Banach space \(X\) and a partially power-bounded operator \(T\in{\mathcal L}(X)\) with respect to \((n_k)_k\) whose peripheral point spectrum is of Hausdorff dimension \(1\). They further show that this construction can even be carried out on a Hilbert space provided that the series \(\sum_{k}\left(\sup_{j\geq k}(\frac{n_j}{n_{j+1}})^\epsilon\right)\) is convergent for any \(\epsilon>0\). This completes some sharp related results obtained in [\textit{T.\,Ransford} and \textit{M.\,Roginskaya}, op.\,cit.]\ where it was shown, in particular, that if \(T\) is partially power-bounded with respect a sequence \((n_k)_k\) for which \(P:=\liminf\frac{n_{k+1}}{n_k}>1\) and \(Q:=\limsup{n_k}^{1/k}<+\infty\), then the peripheral point spectrum of \(T\) has Hausdorff dimension at most \(1-\frac{\log P}{\log Q}\). Examples of partially power-bounded operators with peripheral point spectrum of Hausdorff dimension \(1\) were also constructed in the last cited paper. Furthermore, the authors prove that if \(\Omega\) is a Lavrentiev domain and if an operator \(T\in{\mathcal L}(X)\) whose spectrum is contained in the closure of \(\Omega\) satisfies \[ \| (T-z)^{-1}\| \leq\frac{\text{Const}}{\text{dist}(z,\partial\Omega)},\;z\not\in\overline{\Omega}, \] the Kreiss condition with respect to \(\Omega\), then the family of eigenvectors corresponding to the eigenvalues of the peripheral point spectrum, \(\sigma_p(T)\cap\partial\Omega\), is uniformly minimal and thus is uniformly separated. This implies, in particular, that the peripheral point spectrum of \(T\) is countable provided that \(X\) is separable. This result opens the way for the authors to introduce and study the notion of \(\Omega\)-Jamison sequence \((n_k)_k\), which is defined by replacing the partial power-boundedness condition \(\sup_{k\geq0}\| T^{n_k}\| <+\infty\) by \(\sup_{k\geq0}\| F_{n_k}^\Omega(T)\| <+\infty\), where \(F_{j}^\Omega(T)\) is the \(j\)th Faber polynomial of \(\Omega\). Lastly, the authors show that if \(\Omega\) is a bounded domain whose boundary is a curve of class \(C^{\alpha+1}\) for some \(\alpha>0\), then an increasing sequence of positive integers is an \(\Omega\)-Jamison sequence if and only if it is a Jamison sequence.
scientific article; zbMATH DE number 5167475
scientific article