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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.
