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Multiplication Operators with Closed Range in Operator Algebras |
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PP: 1-5 |
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Author(s) |
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P. Sam Johnson,
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Abstract |
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| Let B(H) denote the C ∗ -algebra of all bounded linear operators from a Hilbert space H into itself. Let T ∈ B(H). Define LT : B(H) → B(H) by LT (S) = T S and define RT : B(H) → B(H) by RT (S) = ST. Consider the
following three statements:
(i) T has closed range in H,
(ii) LT has closed range in B(H),
(iii) RT has closed range in B(H).
It is proved that all these three statements are equivalent. Some possibilities of extending this result to Banach spaces have been discussed.
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