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Linearity and weak convergence on the boundary of numerical range

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

K. C. Das  and
B. D. Craven
K. C. Das
Affiliation:
Department of MathematicsUniversity of MelbourneParkville, Victoria 3051, Australia
B. D. Craven
Affiliation:
Department of MathematicsIndian Institute of TechnologyKharagpur721302, W. Bengal, India
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Abstract

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Stampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.

MSC classification

Secondary: 47A12: Numerical range, numerical radius
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 2 , October 1983 , pp. 221 - 226
Copyright
Copyright © Australian Mathematical Society 1983

References

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Stampfli, J. G. (1966), ‘Extreme points of the numerical range of a hyponormal operator’, Michigan Math. J. 13, 87–89, MR 32 # 4551.CrossRefGoogle Scholar