Continuity of the spectrum on a class of upper triangular operator matrices |
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Authors: | B.P. Duggal |
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Affiliation: | a 8 Redwood Grove, Northfield Avenue, London W5 4SZ, UK b Department of Mathematics of Education, Seoul National University of Education, Seoul 137-742, Republic of Korea c Department of Mathematics, University of Incheon, Incheon 406-840, Republic of Korea |
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Abstract: | Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A○) for some operators X and B such that λ∉σp(B) and σ(A○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP. |
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Keywords: | Continuity of spectrum Upper triangular matrix Berberian extension |
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