Title

The numerical range of operators

Date of Completion

January 1991

Keywords

Mathematics

Degree

Ph.D.

Abstract

We study the relationship between operators and their numerical ranges. The main results are as follows: (i) There does not exist a non-zero tensor product of matrices with minimal length less than the dimension of the space such that its tensor product numerical range is the origin. This proves a conjecture of Marcus and Wang. (ii) The permanental numerical range of a matrix is the origin if and only if the matrix is zero. This proves another conjecture of Marcus and Wang. The permanental numerical range lies on a straight line if and only if the matrix is hermitian, except for a few cases. Such exceptions are also described completely. Unlike the classical numerical range, the permanental numerical range is not necessarily convex. (iii) The generalized numerical range of a matrix lies on a straight line if and only if the matrix is hermitian, except for a few cases. Such exceptions are also described completely. ^