Geodesic
Numerical ranges of an operator on an indefinite inner product space
The electronic journal of linear algebra, Tome 1 (1996).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: For n $\Theta n$ complex matrices A and an n $\Theta n$ Hermitian matrix S, we consider the S-numerical range of A and the positive S-numerical range of A defined by W S (A) = ae hAv; vi S hv; vi S : v 2 IC n ; hv; vi S 6= 0 oe and W + S (A) = fhAv; vi S : v 2 IC n ; hv; vi S = 1g ; respectively, where hu; vi S = v $\Lambda $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 W S (A) and W + S (A) for arbitrary Hermitian matrices S. In particular, it is shown that W + S (A) is always convex and W S (A) is always p-convex for all S. Similar results are obtained for the sets V S (A) = ae hAv; vi hSv; vi : v 2 IC n ; hSv; vi 6= 0 oe ; V + S (A) = fhAv; vi : v 2 IC n ; hSv; vi = 1g ; where hu; vi = v $\Lambda u$. Furthermore, we characterize those linear operators preserving W S (A), W + S (A), V S (A), or V + S (A). Possible generalizations of our results, including their extensions to bounded linear operators on an infinite dimensional Hilbert or Krein space, are discussed.
Classification : 15A60, 15A63, 15A45, 46C20, 52A30
Keywords: field of values, numerical range, generalized numerical range, krein space, convexity, linear preserver, indefinite inner product space 
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     title = {Numerical ranges of an operator on an indefinite inner product space},
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     year = {1996},
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     url = {http://geodesic.mathdoc.fr/item/ELA_1996__1__a5/}
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Li, Chi-Kwong; Tsing, Nam-Kui; Uhlig, Frank. Numerical ranges of an operator on an indefinite inner product space. The electronic journal of linear algebra, Tome 1 (1996). http://geodesic.mathdoc.fr/item/ELA_1996__1__a5/