A fixed point theorem for holomorphic mappings

Earle, Clifford J.; Hamilton, Richard S.

doi:10.1090/pspum/016/0266009

Cited by 101 publications

(84 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…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.

View full text Add to dashboard Cite

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...

show abstract

A fixed point theorem for holomorphic mappings

Cited by 101 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

Contact Info

Product

Resources

About