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Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 205-218.

The angle of an operator and range-kernel complementarity

Authors:  Dimosthenis Drivaliaris (1) and Nikos Yannakakis (2)
Author institution:(1) Department of Financial and Management Engineering, University of the Aegean, Kountourioti 45, 82100 Chios, Greece
(2) Department of Mathematics, National Technical University of Athens, Iroon Polytexneiou 9, 15780 Zografou, Greece

Summary:  We show that if the angle of a bounded linear operator, with closed range and closed sum of its range and kernel, is less than $\pi$, then its range and kernel are complementary. Applying our result we get simple proofs of two known facts concerning eigenvalues lying in the boundary of the numerical range. For an operator on a Hilbert space we present a sufficient condition for range-kernel complementarity. Finally, we discuss some properties of operators whose spectrum does not intersect all rays emanating from the origin and show that such operators are surjective if and only if they are injective.

Keywords:  angle of an operator, range of an operator, kernel of an operator, ascent, descent, index of an operator, numerical range

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