For a bounded #-unitary, the existence of a Tomita's triangular matrix representation is equivalent to the existence of an invariant maximal nonnegative subspace due to Pontrjagin, Krein et al. In other words, if a bounded #-unitary u has such an invariant subspace, its spectral analysis can be reduced to the following three cases: (i) u is #-spectral; (ii) u is quasi-#-spectral; and (iii) u is represented in the form of a Tomita's triangular matrix.
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Yoshiomi Nakagami, Spectral Analysis in Krein Spaces. Publ. Res. Inst. Math. Sci. 24 (1988), no. 3, pp. 361–378DOI 10.2977/PRIMS/1195175032