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Spectral Characterization of Algebraic Elements
Author(s) -
Ransford Thomas,
White Michael
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/33.1.77
Subject(s) - mathematics , converse , characterization (materials science) , spectrum (functional analysis) , banach algebra , hausdorff space , algebraic number , pure mathematics , element (criminal law) , algebra over a field , banach space , mathematical analysis , geometry , materials science , physics , quantum mechanics , political science , law , nanotechnology
It is known that if a is an algebraic element of a Banach algebra A , then its spectrum σ( a ) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies Δ ( σ ( a + x ) , σ ( a ) ) = 0 ( ‖x ‖ γ) as x → 0.We prove that the converse is also true, provided that A is semisimple.