Numerical ranges of an operator on an indefinite inner product space
Chi-Kwong Li, Nam‐Kiu Tsing, Frank Uhlig · Electronic Journal of Linear Algebra · 1996
For n n complex matrices A and an n n Hermitian matrix S, w e consider the S-numerical range of A and the positive S-numerical range of A de ned by WS(A) = hAv;viS hv;viS : v 2 I C n ; hv;viS 6 = 0 and W + S (A) = fhAv;viS : v 2 I C n ; hv;viS = 1 g ; respectively, where hu;viS = v Su.These sets generalize the classical numerical range, and they are closely related to the joint n umerical range of three Hermitian forms and the cone generated by it.Using some theory of the joint n umerical range we can give a detailed description of WS(A) and W + S (A) for arbitrary Hermitian matrices S. In particular, it is shown that W + S (A) is always convex and WS(A) is always p-convex for all S. Similar results are obtained for the sets VS(A) = hAv;vi hSv;vi : v 2 I C n ; hSv;vi 6 = 0 ; V + S (A) = fhAv;vi : v 2 I C n ; hSv;vi = 1 g ; where hu;vi = v u.F urthermore, we c haracterize those linear operators preserving WS(A), W + S (A), VS(A), or V + S (A).Possible generalizations of our results, including their extensions to bounded linear operators on an in nite dimensional Hilbert or Krein space, are discussed.