Numerical ranges of an operator on an indefinite inner product space

Abstract

For n x n complex matrices A and an n x n Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A defined by WS(A) = {〈Av, v〉S/〈v, v〉S : v ∈ ℂn, 〈v, v〉S ≠ 0} and W S + (A) = {〈Av, v〉S : v ∈ ℂn, 〈v, v〉S = 1}, respectively, where 〈u, v〉S = v*Su. These sets generalize the classical numerical range, and they are closely related to the joint numerical range of three Hermitian forms and the cone generated by it. Using some theory of the joint numerical range we can give a detailed description of WS(A) and WS + (A) for arbitrary Hermitian matrices S. In particular, it is shown that WS + (A) is always convex and WS(A) is always p-convex for all S. Similar results are obtained for the sets VS(A) = {〈Av, v〉/〈Sv, v〉: v ∈ ℂn, 〈Sv, v〉 ≠ 0}, VS + (A) = {〈Av, v〉: v ∈ ℂn, 〈Sv, v〉 = 1}, where 〈u, v〉 = v* u. Furthermore, we characterize those linear operators preserving WS(A), WS + (A), V S(A), or VS + (A). Possible generalizations of our results, including their extensions to bounded linear operators on an infinite dimensional Hilbert or Krein space, are discussed.