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THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

Published online by Cambridge University Press:  01 January 2008

THOMAS L. MILLER  and
VLADIMIR MÜLLER
THOMAS L. MILLER
Affiliation:
Dept. of Mathematics and Statistics, Mississippi State University, Drawer MA, Mississippi State, MS 39762 e-mail: miller@math.msstate.edu
VLADIMIR MÜLLER
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67 Prague 1, Czech Republic e-mail: muller@math.cas.cz
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Abstract

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Let T be a bounded operator on a complex Banach space X. Let V be an open subset of the complex plane. We give a condition sufficient for the mapping f(z)↦ (T−z)f(z) to have closed range in the Fréchet space H(V, X) of analytic X-valued functions on V. Moreover, we show that there is a largest open set U for which the map f(z)↦ (T−z)f(z) has closed range in H(V, X) for all V⊆U. Finally, we establish analogous results in the setting of the weak–* topology on H(V, X*).

Keywords

47A11
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 17 - 26
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

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