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##### FURTHER DEVELOPMENTS OF FURUTA INEQUALITY OF INDEFINITE TYPE

**Authors**: Bebiano N, Lemos R, da Providencia J, et al.

**Ref.**: MATHEMATICAL INEQUALITIES & APPLICATIONS **13**, 523 (2010)

**Abstract**: A selfadjoint involutive matrix J endows C-n with an indefinite inner product [., .] given by [x, y] := < Jx, y >, x, y is an element of C-n. We study matrix inequalities for J-selfadjoint matrices with nonnegative eigenvalues. Namely, Furuta inequality of indefinite type is revisited. Characterizations of the J-chaotic order and of the J-relative entropy are obtained via Furuta inequality. The parallelism between the definite versions of the inequalities on Hilbert spaces and the corresponding indefinite versions on Krein spaces is pointed out.