A fixed point theorem for holomorphic mappings

Earle, Clifford J.; Hamilton, Richard S.

doi:10.1090/pspum/016/0266009

Cited by 103 publications

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“…The choice of the correct metric on this space follows naturally from the general theory by Earle and Hamilton [17]. The same line of reasoning has appeared before in a context close to ours in [21,26,27].…”

Section: A Existence and Uniquenessmentioning

confidence: 53%

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Ajanki

Krüger

Erdős

2016

Comm Pure Appl Math

105

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Let S be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation − 1 m = z + Sm with a parameter z in the complex upper half-plane H has a unique solution m with values in H. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on R. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most three.Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles; (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or a cubic root cusps; no other singularities occur.

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Section: A Existence and Uniquenessmentioning

confidence: 53%

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Ajanki

Krüger

Erdős

2016

Comm Pure Appl Math

105

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“…The crucial point is that strict holomorphic mappings on such domains are automatically strict contractions in this metric, and thus Banach's fixed point theorem guarantees a unique fixed point of such mappings. For the reader's convenience we recall here the relevant theorem, due to Earle and Hamilton [1]. , there is some ǫ > 0 such that B ǫ (h(x)) ⊂ D, whenever x ∈ D, where B ǫ (y) is the ball of radius ǫ about y), then h is a strict contraction in the Carathéodory-RiffenFinsler metric ρ, and thus has a unique fixed point in D. Furthermore, one has for all x, y ∈ D that ρ(x, y) ≥ m x − y for some constant m > 0, and thus (h n (x 0 )) n∈N converges in norm, for any x 0 ∈ D, to this fixed point.…”

Section: Contraction Maps and Proofsmentioning

confidence: 99%

Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

Helton

Far

Speicher

2007

International Mathematics Research Notices

141

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Abstract. We show that the quadratic matrix equation V W + η(W )W = I, for given V with positive real part and given analytic mapping η with some positivity preserving properties, has exactly one solution W with positive real part. Also we provide and compare numerical algorithms based on the iteration underlying our proofs.This work bears on operator-valued free probability theory, in particular on the determination of the asymptotic eigenvalue distribution of band or block random matrices.

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“…It now follows by the well-known Earle-Hamilton fixed point theorem [26] that J t/n has a unique fixed point τ in D. This point is also a null point of g. In addition, repeating the proof of the Earle-Hamilton theorem as presented in [30], we obtain the estimate…”

Section: Andmentioning

confidence: 53%

Complex dynamical systems and the geometry of domains in Banach spaces

Elin

Reich

Shoikhet

2004

Dissertationes Math.

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consisting of all the holomorphic self-mappings of D, is a semigroup with respect to composition. Definition 1.1.2. A mapping f ∈ Hol(D, Ω) is said to be univalent on D if for each pair of distinct points x 1 and x 2 in D we have f (x 1 ) = f (x 2 ).In this case one can define the inverse mappingIt is well known (see, for example, [39]) that if X = C n and Y = C m are finite dimensional complex spaces, then f :However, this fact is no longer true in the infinite dimensional case (see counterexamples in [79,4]). Therefore, in the general case a univalent mappingIt is also known (see, for example, [29,49]is a linear isomorphism between X and Y . In this situation we will say that D and Ω are biholomorphically equivalent.As we have already mentioned, in this case X and Y must be linearly isomorphic Banach spaces. Generally speaking, the converse is not true. That is, even if X and Y are isomorphic Banach spaces and f ∈ Hol(D, Y ) has at each point x in D a continuously invertible Fréchet derivative, the mapping f need not be univalent on D.Nevertheless, in this case the mapping f is biholomorphic on a neighborhood of each x ∈ D by the inverse function theorem. In this situation we will say that f ∈ Hol(D, Ω) is locally biholomorphic.The set of all univalent mappings from a domain D ⊂ X into X will be denoted by Univ(D). For the special case when D is the open unit ball of X, the subset of Univ(D) normalized by the conditions f (0) = 0 and f (0) = I will be denoted by S(D). This notation conforms to the one used in the classical onedimensional case, when D = ∆ = {z ∈ C : |z| < 1}. In this case we simply writeThat is, S consists of all the mappings f ∈ Univ(∆) such that f has the following Taylor series at the origin: Convex, star-shaped and spiral-shaped domainsDefinition 1.2.1. A set M in X is called convex if for each pair of points w 1 and w 2 in M , the line segment joining w 1 and w 2 is contained in M , i.e., for each t ∈ [0, 1], the point w = (1 − t)w 1 + tw 2 is also in M .If D is a domain in X, then f ∈ Univ(D) is said to be a convex mapping on D if its image f (D) = Ω is a convex domain in X.Geometry of domains in Banach spaces 9 Definition 1.2.2. A set M in X is called star-shaped (with respect to the origin) if given any w ∈ M , the point w t = tw also belongs to M for every t with 0 ≤ t ≤ 1. That is, if M contains w, then it also contains the entire line segment joining w to the origin. In this definition the origin is in Ω. If, in particular, the origin belongs to Ω, then we will say that f is star-like, to make our definition consistent with the classical one. In this case the mapping f has a null point τ in D. The set of all biholomorphic mappings on D which are star-shaped on D will be denoted by Star(D).If, in addition, there is τ ∈ D such that f (τ ) = 0, (1.2.1) then we will write f ∈ S * τ (D). Of course, in this case such a point τ is unique because. Again, in the one-dimensional case, when X = C, we will simply write S * to denote the family of all biholomorphic (univalent) star-like functions...

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A fixed point theorem for holomorphic mappings

Cited by 103 publications

References 0 publications

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

Complex dynamical systems and the geometry of domains in Banach spaces

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