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 .) 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.
KeywordsHalf Plane Positive Real Part Numerical Range Open Unit Disc Diagonal Operator
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