Anderson's theorem for some class of operators

Document Type : Original Article


Department of Systems Engineering, National Polytechnic School of Oran-Maurice Audin (Ex. ENSET of Oran), BP 1523 Oran-El M'naouar, 31000 Oran, Algeria.



Anderson's theorem states that if the  numerical range of an $n\times n$ matrix is contained in the closed unit disk  and intersects with the unit circle at more than $n$ points, then the numerical range coincides with the closed  unit disk. In an infinite-dimensional setting, an analogue of this result for a  compact operator   was established by Gau and Wu and for operator  being the sum of a normal and compact operator by Birbonshi et al. We consider here three classes of operators:  Operators being the sum of compact and operator with numerical  radius strictly less than 1, operators with essentially numerical range coinciding with the convex hull  of its essential spectrum,  and quasicompact operators.


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