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Operators in the Cowen-Douglas Class and Related Topics
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Thus we have a Hermitian line bundle on the complex projective space ℙ1 given by the frame θ1 ↦ z1 + θ1z2 and θ2 ↦ z2 + θ2z1. The curvature of this line bundle is then an invariant for the Hilbert module H02(D2) as shown in [28]. This curvature is easily calculated and is given by the formula K(θ) = (1 + |θ|2)−2. The decomposition theorem yields similar results in many other examples.
The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Let X be complex torus Tn=Cn/Zn or a complex projective space CPn, being X compact, all holomorphic functions on X are bounded. Since bounded holomorphic entire functions are constant, it easily follows that the only holomorphic functions on such an X are the constant functions, and the algebra O(D) is simply C.
The index theorem
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
Example3.2.2: Let M = CP2k be complex projective space. Let () x∈H2(M;C)=C
Solvable stochastic differential games in rank one compact symmetric spaces1
Published in International Journal of Control, 2018
Tyrone E. Duncan, Bozenna Pasik-Duncan
Initially some important parameters for the complex and the quaternion projective spaces are given that appear in the radial part of the Laplacians and in the analysis of the spaces (e.g. Helgason, 1984, p. 168). The antipodal manifolds, A0, for these spaces are important in the solutions of the stochastic differential games. For each complex projective space , where n = 4, 6, ..., the antipodal manifold is and for each quaternion projective space , the antipodal manifold is , where n = 8, 12, ....