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Numerical Range

  • Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 19)

Abstract

To prove that the numerical range W of an operator A is convex, it is sufficient to prove that the intersection of W with every straight line in the complex plane is connected. (The ingenious idea of basing the proof on this observation is due to N. P. Dekker [35].) Consider a line with the (real) equation px + qy + r = 0 (where the pair 〈x, y〉 is identified with the complex number x + iy). If A = B + iC, with B and C Hermitian, then the intersection in question consists of all those complex numbers x + iy for which px + qy + r = 0 and for which there exists a unit vector f such that x = (Bf, ƒ), y = (Cf, f). In other words, the intersection is the set of all (Bf, f) + i(Cf, f) with ∥f∥ = 1 and ((pB + qC + r)f, f) = 0. There is still another way of describing the intersection: it is the image under the (continuous) mapping f ↦ (Bf, f) + i(Cf, f) of the set N of all those unit vectors f for which ((pB + qC + r) f, f) = 0. An efficient way to conclude the proof is to show that N itself is connected.

Keywords

Half Plane Positive Real Part Numerical Range Open Unit Disc Diagonal Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Paul R. Halmos

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