Two new open mapping theorems for analytic multivalued functions

Baribeau, Line; Harbottle, Sarah

doi:10.1090/S0002-9939-1992-1086321-6

Two new open mapping theorems for analytic multivalued functions
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by Line Baribeau and Sarah Harbottle PDF

Proc. Amer. Math. Soc. 115 (1992), 1009-1012 Request permission

Abstract:

It is conjectured that the image $U$ of a domain $D$, under an analytic multivalued function $K$, satisfies $U \cap \partial U \subset { \cap _{\lambda \in D}}K(\lambda )$ if the values of $K$ are polynomially convex sets. We prove that this conjecture is true in the two special cases when $K$ is convex-valued and polar-valued.

References

Bernard Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Adv. in Math. 44 (1982), no. 1, 18–60. MR 654547, DOI 10.1016/0001-8708(82)90064-0
B. Berndtsson and T. J. Ransford, Analytic multifunctions, the $\overline \partial$-equation, and a proof of the corona theorem, Pacific J. Math. 124 (1986), no. 1, 57–72. MR 850666
T. J. Ransford, Open mapping, inversion and implicit function theorems for analytic multivalued functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 537–562. MR 759303, DOI 10.1112/plms/s3-49.3.537
T. J. Ransford, On the range of an analytic multivalued function, Pacific J. Math. 123 (1986), no. 2, 421–439. MR 840852

Pseudoconcave sets and algebraic lemniscates

Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703