By Israel Gohberg, Jürgen Leiterer

This is a booklet on holomorphic operator capabilities of a unmarried variable and functions, that is concerned with the family members among neighborhood and international theories. it's according to tools and technics of complicated research of a number of variables.

The first a part of the speculation starts off with an easy generalization of a few effects from the fundamentals of study of scalar features of 1 complicated variable. within the moment half, that is the most a part of the speculation, effects are got by way of tools and instruments tailored from complicated research of features of numerous variables. we've got in brain the speculation of holomorphic cocycles (fiber bundles) with values in infinite-dimensional non-commutative teams. mostly, those effects don't seem in conventional advanced research of 1 variable, no longer even for matrix valued cocycles. The 3rd half comprises functions to operator idea. right here functions are offered for holomorphic households of subspaces and Plemelj-Muschelishvili factorization. The fourth half provides a generalization of the speculation of cocycles to cocycles with regulations. This half includes additionally purposes 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 (Operator Theory: Advances and Applications)**

**Example text**

7) n=0 is holomorphic on the open disc |z − z0 | < ρ, and ∞ n(ζ − z0 )n−1 an , f (ζ) = |ζ − z0 | < ρ. 6), then an = f (n) (z0 ) , n! n ∈ N. 9) Proof. 3) of the radius of convergence, assertion (i) is equivalent to the equality 1 lim sup n an = 1 n−1 lim sup n→∞ n an . n→∞ But the latter relation follows (for example) from the fact that the scalar power series ∞ ∞ an (z − z0 )n n an (z − z0 )n−1 and n=0 n=1 have the same radius of convergence. 4) it follows that the sequence of partial sums N an (z − z0 )n , N ∈ N, n=0 converges to f , uniformly on each compact subset of the disc |z − z0 | < ρ.

Um such that: • The sets U1 , . . , Um are pairwise disjoint, • D ⊆ U0 ∪ . . ∪ Um ⊆ U , and, for all 1 ≤ j ≤ m, • aj ∈ Uj , • Γj ∩ U0 = Γj \ {aj }, • Uj ∩ U0 consists of precisely two connected components. Proof. Since Γ0 and Γ1 are parts of the piecewise C 1 -boundary of D and D is connected, ﬁrst we can ﬁnd a contour γ1 , diﬀeomorphic to the closed interval [0, 1], which starts at a1 , transversally to Γ1 , which ends at some smooth point b1 ∈ Γ0 , transversally to Γ0 , and which lies, except for these two points, in D.

The integer p then is called the order of the pole z0 . If z0 is not a removable singularity of F and not a pole of f , then z0 is called an essential singularity of f . 3 Theorem (Riemann’s theorem on removable singularities). Let E be a Banach space, and let z0 be an isolated singularity of an E-valued holomorphic function f deﬁned in a deleted neighborhood of z0 . If f is bounded, then z0 is removable as a singularity of f . Proof. Let ∞ fn (z − z0 )n f (z) = n=−∞ be the Laurent series of f at z0 .