By Israel Gohberg, Jürgen Leiterer
ISBN-10: 3034601255
ISBN-13: 9783034601252
ISBN-10: 3034601263
ISBN-13: 9783034601269
This is a e-book on holomorphic operator services of a unmarried variable and purposes, that's excited by the kinfolk among neighborhood and worldwide theories. it really is in line with equipment and technics of advanced research of a number of variables.
The first a part of the idea begins with an easy generalization of a few effects from the fundamentals of research of scalar services of 1 advanced variable. within the moment half, that's the most a part of the idea, effects are got by way of tools and instruments tailored from advanced research of services of a number of variables. we now have in brain the speculation of holomorphic cocycles (fiber bundles) with values in infinite-dimensional non-commutative teams. in general, those effects don't seem in conventional advanced research of 1 variable, no longer even for matrix valued cocycles. The 3rd half includes purposes to operator thought. right here functions are offered for holomorphic households of subspaces and Plemelj-Muschelishvili factorization. The fourth half offers a generalization of the idea of cocycles to cocycles with regulations. This half includes additionally functions to interpolation difficulties, to the matter of holomorphic equivalence and diagonalization.
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Extra info for Holomorphic Operator Functions of One Variable and Applications: Methods from Complex Analysis in Several Variables
Sample text
3) 0 Proof. Let E be the dual of E. Then, by the homotopy version of the Cauchy integral theorem for scalar functions, for each Φ ∈ E , 1 1 f γ1 (z) γ1 (z)dz Φ = Φ f γ1 (z) γ1 (z)dz 0 0 1 = 1 Φ f γ1 (z) γ2 (z) dz = Φ 0 f γ2 (z)γ2 (z)dz . 3). 1 Theorem (Cauchy formula). Let D ⊆ C be a bounded open set with piecewise C 1 -boundary, let E be a Banach space, and let f : D → E be a continuous function which is holomorphic in D. Then f (w) = 1 2πi ∂D f (z) dz, z−w w ∈ D. 1) Proof. Let w ∈ D be given.
Um and D ∪ U 1 ∪ . . ∪ U m are simply connected, and the contours Γ0 , Γ1 , . . , Γm are homeomorphic to the unit circle. 2 Lemma. Let D ⊆ C be a connected bounded open set with piecewise C 1 boundary. Suppose D is not simply connected. Let Γ0 , Γ1 , . . , Γm be the connected components of the boundary of D where Γ0 is the boundary of the unbounded component of C \ D. Suppose, for each 0 ≤ j ≤ m, a smooth1 point aj of Γj is chosen. Let U be a neighborhood of D. Then there exist simply connected open sets U0 , .
2), (∂uε )(z) = ε2 ϕ V ζ −z (∂u)(ζ) dλ(ζ) = (∂u)ε (z). 4 Theorem (Cauchy formula for continuous functions). Let D be a bounded open set with piecewise C 1 -boundary ∂D, and let u : D → E be a continuous function such that also ∂u is continuous on D. Moreover we assume that ∂u admits a continuous extension to D. Then u(z) = 1 2πi ∂D u(ζ) 1 dζ − ζ −z π D ∂u(z) dλ(ζ), ζ −z z ∈ D. 4) 32 Chapter 2. Solution of ∂u = f and applications Proof. First consider the case when u is of class C 1 in a neighborhood of D.