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On the Normality of Numbers to Different Bases
We prove independence of normality to different bases We show that the set of
real numbers that are normal to some base is Sigma^0_4 complete in the Borel
hierarchy of subsets of real numbers. This was an open problem, initiated by
Alexander Kechris, and conjectured by Ditzen 20 years ago
Old and new results on normality
We present a partial survey on normal numbers, including Keane's
contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Riesz Bases for p-Subordinate Perturbations of Normal Operators
For p-subordinate perturbations of unbounded normal operators, the change of
the spectrum is studied and spectral criteria for the existence of a Riesz
basis with parentheses of root vectors are established. A Riesz basis without
parentheses is obtained under an additional a priori assumption on the spectrum
of the perturbed operator. The results are applied to two classes of block
operator matrices.Comment: 30 pages, 4 figures, submitted to J. Funct. Ana
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