Abstract
It is proved in Benamara-Nikolski [1] that if the spectrum σ(T) of a contractionT with finite defects (rank(I−T * T)=rank (I−TT *)<∞) does not coincide with\(\bar D\), then the contraction is similar to a normal operator if and only if
The examples of Kupin-Treil [9] show that the result is no longer true if we replace the condition rank (I−T * T)<∞ by its weakened versionG, whereG denotes the class of nuclear operators.
We prove in this paper that, however, the following theorem holds
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Kupin, S. Linear resolvent growth test for similarity of a weak contraction to a normal operator. Ark. Mat. 39, 95–119 (2001). https://doi.org/10.1007/BF02388793
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DOI: https://doi.org/10.1007/BF02388793