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SUMMARYThis note gives new necessary and sufficient conditions for a linear system to be passive or lossless. It will be shown that, by using the theory of operators on an indefinite J(t)-Hilbert space, both the passivity and isometric conditions for an operator to be lossless can be combined into a single condition-namely an operator is lossless if and only if it is J(t)-contractive.

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Abstract. Let T be a contraction and A the strong limit of {T * n T n } n≥1 . We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class Ꮿ 00 or a nonstrict proper contraction of class Ꮿ 10 for which A is a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator [T * ,T ] is a strict contraction.2000 Mathematics Subject Classification. 47A15, 47B20.

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