Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications
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Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications Filippo Bracci1 · Marina Levenshtein2 · Simeon Reich3 · David Shoikhet2
Received: 11 June 2015 / Revised: 9 November 2015 / Accepted: 19 November 2015 © Springer-Verlag Berlin Heidelberg 2016
Abstract The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper
Communicated by Stephan Ruscheweyh. Filippo Bracci was partially supported by the ERC grant “HEVO—Holomorphic Evolution Equations” No. 277691. Simeon Reich was partially supported by the Israel Science Foundation (Grant No. 389/12), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. Marina Levenstein and David Shoikhet were partially supported by the European Commission under the project STREVCOMS PIRSES-2013-612669.
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Filippo Bracci [email protected] Marina Levenshtein [email protected] Simeon Reich [email protected] David Shoikhet [email protected]
1
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy
2
Department of Mathematics, ORT Braude College, 21982 Karmiel, Israel
3
Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel
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F. Bracci et al.
bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness. Keywords Numerical range · Growth estimates · Bloch radii · Holomorphic maps · Banach spaces Mathematics Subject Classification
47A12 · 46G20 · 46T25 · 58B12
1 Introduction and Preliminaries Let X ∗ denote the dual of a complex Banach space X and let x, x ∗ denote the duality pairing of x ∈ X and x ∗ ∈ X ∗ . For each x ∈ X , the set J (x), defined by J (x) := {x ∗ ∈ X ∗ : x, x ∗ = x2 = x ∗ 2 }, is not empty by the Hahn–Banach theorem, and is a closed and convex subset of X ∗ . Let D be a domain in X and let f :D → X be a mapping. We use the notation supx∈D Re f (x), x ∗ for the supremum of Re f (x), x ∗ over all pairs x ∈ D and x ∗ ∈ J (x). We denote by B R := {x ∈ X :x < R} the open ball centered at the origin of radius R in the complex Banach space X . Definition 1 (cf. [11,12]) Let h:B R → X be continuous on the closure B R of B R . We define the set
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