0 Note on the Curvature and Index of Almost Unitary Contraction Operator ∗
نویسنده
چکیده
In the recent preprint [1] S. Parrott proves the equality between the Arveson's curvature and the Fredholm index of a " pure " contraction with finite defect numbers. In the present note one derives a similar formula in the " non-pure " case. The notions of d-contraction T = (T 1 , T 2 ,. .. , T d) and its curvature was introduced by W. Arveson in a series of papers (see [2], [3], and [4]). In the case of a single contraction (d = 1) the curvature is thoroughly investigated in the paper of Parrott [1]. Namely, let T be a contraction operator on a Hilbert space H, and suppose that ∆ T := √ I − T T * has finite rank. Parrott shows that the curvature K(T) of T can be defined on three equivalent ways:
منابع مشابه
Note on the Curvature and Index of Almost Unitary Contraction Operator *
In the recent preprint [1] S. Parrott proves the equality between the Arveson's curvature and the Fredholm index of a " pure " contraction with finite defect numbers. In the present note one derives a similar formula in the " non-pure " case. The notions of d-contraction T = (T 1 , T 2 ,. .. , T d) and its curvature was introduced by W. Arveson in a series of papers (see [2], [3], and [4]). In ...
متن کاملThe Curvature of a Single Contraction Operator on a Hilbert Space ∗
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 additio...
متن کاملThe Curvature Invariant for a Class of Homogeneous Operators
For an operator T in the class Bn(Ω), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle ET determine the unitary equivalence class of the operator T . In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant d...
متن کاملRelating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold
Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M,g) is nearly Kähler, th...
متن کاملOn weak orbits of operators
Let T be a completely nonunitary contraction on a Hilbert space H with r(T ) = 1. Let an > 0, an → 0. Then there exists x ∈ H with |〈Tnx, x〉| ≥ an for all n. We construct a unitary operator without this property. This gives a negative answer to a problem of van Neerven. Let X be a complex Banach space. Then each operator T ∈ B(X) has orbits that are ”large” in the following sense [M1], [B]: Let...
متن کامل