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Register a new userGrowth estimates for pseudo-dissipative holomorphic maps in Banach spaces
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In this paper we introduce a class of pseudo-dissipative holomorphic maps which contains, in particular, the class of infinitesimal generators of semigroups of holomorphic maps on the unit ball of a complex Banach space. We give a growth estimate for maps of this class. In particular, it follows that pseudo-dissipative maps on the unit ball of (infinite-dimensional) Banach spaces are bounded on each domain strictly contained inside the ball. We also present some applications.
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A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call {sl pseudo-contractivity}, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer-Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.
The numerical range of holomorphic mappings arises in many aspects of nonlinear 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 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, nonlinear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.
Let $mathcal{G}$ resp. $M$ be a positive dimensional Lie group resp. connected complex manifold without boundary and $V$ a finite dimensional $C^{infty}$ compact connected manifold, possibly with boundary. Fix a smoothness class $mathcal{F}=C^{infty}$, Holder $C^{k, alpha}$ or Sobolev $W^{k, p}$. The space $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ of all $mathcal{F}$ maps $V to mathcal{G}$ resp. $V to M$ is a Banach/Frechet Lie group resp. complex manifold. Let $mathcal{F}^0(V, mathcal{G})$ resp. $mathcal{F}^{0}(V, M)$ be the component of $mathcal{F}(V, mathcal{G})$ resp. $mathcal{F}(V, M)$ containing the identity resp. constants. A map $f$ from a domain $Omega subset mathcal{F}_1(V, M)$ to $mathcal{F}_2(W, M)$ is called range decreasing if $f(x)(W) subset x(V)$, $x in Omega$. We prove that if $dim_{mathbb{R}} mathcal{G} ge 2$, then any range decreasing group homomorphism $f: mathcal{F}_1^0(V, mathcal{G}) to mathcal{F}_2(W, mathcal{G})$ is the pullback by a map $phi: W to V$. We also provide several sufficient conditions for a range decreasing holomorphic map $Omega$ $to$ $mathcal{F}_2(W, M)$ to be a pullback operator. Then we apply these results to study certain decomposition of holomorphic maps $mathcal{F}_1(V, N) supset Omega to mathcal{F}_2(W, M)$. In particular, we identify some classes of holomorphic maps $mathcal{F}_1^{0}(V, mathbb{P}^n) to mathcal{F}_2(W, mathbb{P}^m)$, including all automorphisms of $mathcal{F}^{0}(V, mathbb{P}^n)$.
We consider holomorphic semicocycles on the open unit ball in a Banach space taking values in a Banach algebra. We establish criteria for a semicocycle to be linearizable, that is, cohomologically equivalent to one independent of the spatial variable.
We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.
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