The Curvature of a Single Contraction Operator on a Hilbert Space ∗
نویسنده
چکیده
This note studies Arveson’s curvature invariant for d-contractions T = (T1, T2, . . . , Td) for the special case d = 1, referring to a single contraction operator T on a Hilbert space. It establishes a formula which gives an easy-to-understand meaning for the curvature of a single contraction. The formula is applied to give an example of an operator with nonintegral curvature. Under the additional hypothesis that the single contraction T be “pure”, we show that its curvature K(T ) is given by K(T ) = −index (T ) := −(dimker (T ) − dim coker (T )). 1 The curvature of a single operator This note studies Arveson’s curvature invariant for d-contractions T = (T1, T2, . . . , Td) for the special case d = 1, referring to a single contraction operator T on a Hilbert space. It establishes a formula which gives an easy-to-understand meaning for the curvature of a single contraction. The formula is applied to give an example of an operator with nonintegral curvature. Under the additional hypothesis that the single contraction T be “pure”, we show that its curvatureK(T ) (defined below) is given by K(T ) = −index (T ) := −(dim ker (T )− dim coker (T )). Let T be a contraction operator on a Hilbert spaceH , and ∆T := √ 1− TT ∗. Assume that ∆T has finite rank. Then the curvature K(T ) of T (our shorthand AMS Subject Classification: 47A13 (Primary); 47A20 (Secondary).
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