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| On the J-Normal Extensions of Operators |
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Citation： |
Wu Jingbo.On the J-Normal Extensions of Operators[J].Chinese Annals of Mathematics B,1982,3(5):609~616 |
| Page view： 885
Net amount： 714 |
Authors： |
Wu Jingbo; |
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| Abstract： |
A bounded linear operator T acting on a Hilbert space H is said to be J- subnormal with order n if on some \Pi _n-Pontrjagin space \Pi containing H, there exists a bounded J-normal operator \tilde T such that \tilde Tf=Tf for every f in H and that \Pi is spanned by the elements of the form $\tilde T^{*k}f$, where f \in H and k = 0, 1, 2,\cdots.
Let H be a Hilbert space and let I7 be in B(H). The main purpose of this paper is to prove that the following statements are equivalent:
(1) Tis J-subnormal with order n;
(2) For each non-negative integer r and for each set {x_ik: i, k = 0, 1,\cdots, r} of
elements of H, the Hermitian form $\sum\limits_{i,j,k,l=0}^r(T^jx_ik,T^ix_jl)\alpha_ik\bar \alpha_jl$ has at most n negative squares, and for at least one choice of r and {x_ik}, it has exactly n negative squares;
(3) The operator function is quasi-positive befinite with order n in the
complex plane. This result is an extension of the theorems of Halmos and Bram, |
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