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**Fixed points of holomorphic mappings for domains in Banach spaces.**
*(English)*
Zbl 1034.47021

The author presents a fine exposition of fixed point theory for holomorphic mappings in Banach spaces. This is an important topic of infinite-dimensional holomorphy.

A basic result of this theory is the so-called Earle-Hamilton theorem, which asserts that any holomorphic function of a bounded domain in a Banach space mapping it strictly inside itself has a unique fixed point. The author gives a self-contained proof of this result under slightly more general assumptions. Namely, he requires the given mapping to be bounded rather than its domain of definition.

In fact, this result is a nice application of the Banach fixed point principle since a bounded holomorphic mapping which maps a domain strictly inside itself is a strict contraction with respect to the Carathéodory-Riffen-Finsler hyperbolic metric on a bounded domain containing the image of the mapping.

For a bounded convex domain \(D\), the Earle-Hamilton theorem can be partially extended as follows. If \(h:D\to D\) has a uniformly continuous extension to \(Cl(D)\), the closure of \(D\), and the norm of \(x-h(x)\) is strictly separated from zero on the boundary of \(D\), then \(h\) has a unique fixed point in \(D\). We note, in addition, that this point is regular in the sense that the Fréchet derivative at it is an invertible linear operator. However, in contrast with the Earle-Hamilton theorem, we cannot claim the convergence of the iterates of \(h\) to this point. This theorem still applies in cases where the holomorphic mapping does not necessarily maps its domain strictly inside itself.

The author deduces some recent results in this direction in terms of the numerical range of holomorphic mappings introduced earlier by him in [L. Harris, Am. J. Math. 93, 1005–1019 (1971; Zbl 0237.58010)] as an extension of Lumer’s definition to the nonlinear case. By applying a nonlinear analog of the Lumer-Phillips theorem and the so-called resolvent method, he states a result from [L. Harris, S. Reich and D. Shoikhet, J. Anal. Math. 82, 221–232 (2001; Zbl 0972.46029)]: A holomorphic mapping of the open unit ball \(D\) in a Banach space with the numerical range lying to the left of the vertical line \(x=1\) (i.e., the real part of it is strictly less than 1) has a unique fixed point in \(D\).

Note finally that this result can be applied to the study of the stationary points of a one-parameter semigroup of holomorphic self-mappings.

A basic result of this theory is the so-called Earle-Hamilton theorem, which asserts that any holomorphic function of a bounded domain in a Banach space mapping it strictly inside itself has a unique fixed point. The author gives a self-contained proof of this result under slightly more general assumptions. Namely, he requires the given mapping to be bounded rather than its domain of definition.

In fact, this result is a nice application of the Banach fixed point principle since a bounded holomorphic mapping which maps a domain strictly inside itself is a strict contraction with respect to the Carathéodory-Riffen-Finsler hyperbolic metric on a bounded domain containing the image of the mapping.

For a bounded convex domain \(D\), the Earle-Hamilton theorem can be partially extended as follows. If \(h:D\to D\) has a uniformly continuous extension to \(Cl(D)\), the closure of \(D\), and the norm of \(x-h(x)\) is strictly separated from zero on the boundary of \(D\), then \(h\) has a unique fixed point in \(D\). We note, in addition, that this point is regular in the sense that the Fréchet derivative at it is an invertible linear operator. However, in contrast with the Earle-Hamilton theorem, we cannot claim the convergence of the iterates of \(h\) to this point. This theorem still applies in cases where the holomorphic mapping does not necessarily maps its domain strictly inside itself.

The author deduces some recent results in this direction in terms of the numerical range of holomorphic mappings introduced earlier by him in [L. Harris, Am. J. Math. 93, 1005–1019 (1971; Zbl 0237.58010)] as an extension of Lumer’s definition to the nonlinear case. By applying a nonlinear analog of the Lumer-Phillips theorem and the so-called resolvent method, he states a result from [L. Harris, S. Reich and D. Shoikhet, J. Anal. Math. 82, 221–232 (2001; Zbl 0972.46029)]: A holomorphic mapping of the open unit ball \(D\) in a Banach space with the numerical range lying to the left of the vertical line \(x=1\) (i.e., the real part of it is strictly less than 1) has a unique fixed point in \(D\).

Note finally that this result can be applied to the study of the stationary points of a one-parameter semigroup of holomorphic self-mappings.

Reviewer: David M. Shoikhet (Karmiel)