A GENERALIZATION OF THE 0-NUMERICAL RANGE
Let H be a complex Hilbert space. Given a bounded linear operator A on H, we describe the set R n (A) = fV AW : V;W : C n ! H;V V = W W = In;V W = 0g. It is shown that the closed matricial convex hull of R n (A) is a closed ball of radius minfkA I k : 2 Cg centered at the origin.
Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras
1. Numerical range in unital normed algebras 2. Hermitian elements of a complex unital Banach algebra 3. Operators 4. Some recent developments.
On the Field of Values of a Square Matrix.
Xp Xp has the value unity (i.e., in the language of real geometry the p 1 point x is supposed to lie on the unit sphere). Since the composite form (A x x) has values invariant under unitary transformations of A the region in question is a unitary invariant of A. For normal matrices (i.e., matrices which can be transformed unitarily to the diagonal form) it is well known and immediately evident that the region is the interior and perimeter of the convex polygon whose corners are characteristic numbers of A and which cQntains, either in its interior or on its perimeter all characteristic numbers of A. For not-normal matrices it is known (ToeplitzHausdorff) that the region is convex1 and for a two-rowed matrix the region is, according to a remark of Wintner, an ellipse having the two characteristic numbers as foci as may be shown by direct calculation in Cartesian coordinates. This method is somewhat complicated and not easy to generalize while the method of the present note using the complex variable is quite simple and works as well, in principle, for the general case as for n = 2. The range of values of a Hermitian matrix is the segment of the real axis connecting the smallest and greatest characteristic numbers. Any matrix A can be analyzed unambiguously in the form A = F + iG, where F and G are Hermitian and hence the real part of (A x x) = (F x I x). If r denotes the largest characteristic number of F the line t = r contains one point of the boundary of the sought-for region R (if it contains more than one such point a portion of this line forms part of the boundary of R), and there are no points of R to the right of this line. If now t = ejO is a turn (i.e., a complex number of unit modulus) the field of values R of A = A/t is found by rotating R around the origin through an angle 0. Hence if we find the line t = p "bounding" R on the right (in the sense that the line t = r "bounds" R on the right so that, in Minkowski's terminology, t = p is a line of support of R) the line obtained by 246 PROC. N. A. S.