GROWTH CONDITIONS FOR OPERATORS WITH SMALLEST SPECTRUM
Keyword(s):
AbstractLet A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(ℤ) (resp. $\mathcal{D}$Aα (ℤ+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that $ \begin{equation*} \Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vert n\right\vert \right) ^{\alpha }, \end{equation*} $ for all n ∈ ℤ (resp. n∈ ℤ+). We present a complete description of the class $\mathcal{D}$Aα (ℤ) in the case when the spectrum of A is real or is a singleton. If T ∈ $\mathcal{D}$A(ℤ) (=$\mathcal{D}$A0(ℤ)), some estimates for the norm of AT-TA are obtained. Some results for the class $\mathcal{D}$Aα (ℤ+) are also given.
1978 ◽
Vol 30
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pp. 1045-1069
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1984 ◽
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pp. 483-493
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1969 ◽
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pp. 592-594
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2013 ◽
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1986 ◽
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2014 ◽
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pp. 709-718
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1967 ◽
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1960 ◽
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pp. 686-693
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