A Hilbert Space Problem Book pp 314-321 | Cite as

# Numerical Range

## 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## Preview

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