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.
The G-bilinear range
Let 1 ⩽ r⩽ n A a linear transformation on an n-dimensional unitary space V and let G = [G1:G2:G3]bean r × 3 r matrix.A collection of 2r vectors x1,…,xr,y1,…,yrin V me said to be G vectors whenever G1= [(xi, Xj)], G2 = [yi,yj], and G3= [(Xi,yj)]. In the present paper we examine the structure of the setW(A;G), of values of the form (Ax1,y1)+ ⋯ +(Axr,yr) where x1,…,xr,y1,…,yr run over all G-vectors. Specifically, conditions for W(A;G) to be convex, the origin, or empty are given along with some upper bounds on the maximum modulus of any element in W(A;G). These results extend some of the classical results of Hausdorff, Toeplitz, von Neumann, Fan, Berger, and Westwick concerning higher numerical ranges.