Jordan Arcs Research Articles

On Curvilinear cluster sets on open Riemann surfaces

Every boundary point of the Kerékjártó-Stoïlow compactification of an open Riemann surface is the limit of a Jordan arc with this property: for every nonempty continuum in the extended complex plane there is a holomorphic function on the surface having the continuum as its cluster set along the arc.

Sobolev approximation by a sum of subalgebras on the circle

Let ψ be an orientation-reversing homeomorphism of the unit circle onto itself. We consider approximation in certain Sobolev norms by functions of the form f(z) + g(ψ), where / and g are polynomials. The methods involve conf ormal welding and Hardy space theory. We construct a Jordan arc of positive continuous analytic capacity such that the harmonic measures for the two complementary domains are mutually absolutely continuous. Let C(S) denote the space of continuous, complex-valued functions on the unit circle S. For feC(S), let P(f) denote the space of all polynomials in / with complex coefficients. Let z denote the identity function. Browder and Wermer proved the following theorem [3, p. 551]. Let ψ be a direction-reve rsing homeomorphism of S onto S. Then the vector space sum P(z) + P(ψ) is uniformly dense in C(S). In the first part of this paper we prove a result that partially extends this theorem to the C1 norm. We say that a direction-reve rsing homeomorphism ψ:S—*S is an involution if ψoψ = z. Let C^S) denote the space of continuously-differentiable, complex-valued functions on S, with the norm

Closure of Similarity Orbits of Hilbert Space Operators, II: Normal Operators

Let L(H) be the algebra of operators on a complex separable Hilbert space H, let T ε L(H) and let ℓ(T) = {WTW1 : W is invertible in L(H)} be the “ similarity orbit ” of T. The norm closurey ℓ(N)− of ℓ(N) is completely characterized “up to compact perturbations” for N normal. A partial answer is also given for normal operators in non-separable Hilbert spaces. For the separable case, it is shown that if the spectrum of T is the union of a totally disconnected set and finitely many pairwise disjoint subsets of regular analytic Jordan arcs (or Jordan curves) and is perfect, then ℓ(T)− = ℓ(A)− for every A having the same spectrum as T. If T is a spectral operator, then ℓ(T)− contains the “scalar part” of T. The results also include some information about those operators T such that ℓ(T)− is maximal or minimal with respect to inclusion, in the family of the sets {ℓ(A)− : A ɛ L(H)}.

A note on Hall’s lemma

Let H be the half plane { z : Re ⁡ z > 0 } \{ z:\operatorname {Re} z > 0\} . Let y be a Jordan arc joining z = 0 z = 0 and z = e i α ( 0 ≦ α > π / 2 ) z = {e^{i\alpha }}\;(0 \leqq \alpha > \pi /2) in H ∩ { | z | ≦ 1 } H \cap \{ |z| \leqq 1\} . Let γ ∗ {\gamma ^\ast } be the segment z = i y ( 0 ≦ y ≦ 1 ) z = iy\;(0 \leqq y \leqq 1) of the imaginary axis. If ω ( z , γ ) \omega (z,\gamma ) is the harmonic measure of γ \gamma with respect to H ∖ γ H\backslash \gamma and ω ( z , γ ∗ ) \omega (z,{\gamma ^\ast }) the harmonic measure of γ ∗ {\gamma ^\ast } with respect to H, then ω ( x + i y , γ ) > ω ( x − i | y | , γ ∗ ) \omega (x + iy,\gamma ) > \omega (x - i|y|,{\gamma ^\ast }) .

Arcs with increasing chords

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that.

Intersections of Arc-Cluster Sets for Meromorphic Functions

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

On integral functions with distinct asymptotic values

Suppose that p > 1 and let Γ1, Γ2, …, Γp be Jordan arcs going from 0 to ∞ and having no other common point and numbered in anti-clockwise order around the origin. Let Γp+1 = Γ1 and let Dv be the domain bounded by Γv and Γv+1 for v = 1 to p.

Addendum to "Analytic Functions and Jordan Arcs"

Addendum to "Analytic Functions and Jordan Arcs"

Addendum to “Analytic functions and Jordan arcs”

Professor Royden has kindly pointed out that in the theorem of [21 E may be replaced by any compact set K of positive two-dimensional measure and that f may be taken to be quasiconformal on all of S. Indeed, one has only to choose f as the normal solution of the Beltrami equation with ,u = 2XK, where XK is the characteristic function of K [1, p. 97]. Rothberger's example remains of considerable interest, however, since E has measure zero. On the other hand, f (E) has positive measure, so that f is not quasiconformal on S. In conclusion, we note that the arc J may also be taken to have zero measure; this follows from an elementary linearization argument suggested by Professor Rudin.

Shorter Notes: Analytic Functions and Jordan Arcs

Shorter Notes: Analytic Functions and Jordan Arcs

Approximation of functions continuous on Jordan arcs

Approximation of functions continuous on Jordan arcs

Analytic functions and Jordan arcs

Let J be a Jordan arc lying on the complex sphere S, and let F: S>S be a homeomorphism. In several different connections (see e.g., [4]) it has been asked if the conformality of F off J implies that F is a linear fractional transformation. The purpose of this brief note is to point out that a recent example of Rothberger [5] can be used to answer this question in the negative. Rothberger proves the

Bi-arc boundary functions

Introduction. A function f is a boundary function if f is defined on the real line and is the limit in some restricted sense of a function 4) defined on the open upper half-plane or if f is defined on the unit circle and is the limit in some restricted sense of a function 1 defined on the open unit disk. Bagemihl and Piranian [I] and Kaczynski [2] have studied boundary functions for the case where at each point of the boundary the limit of 1? is assumed to exist along some arc with one endpoint on the boundary. The author [5], [6] has studied boundary functions which are obtained by taking boundary limits relative to Stolz angles and approximate limits relative to Stolz angles. In this paper we assume that at each point of the boundary there are two arcs along which the limits of c) exist and are equal. We will let R denote the set of real numbers. DEFINITION. If xER, an arc at (x, 0) is a Jordan arc y with one endpoint at (x, 0) and with all other points lying in the open upper half-plane. DEFINITION. r is a bi-arc at (x, 0) if F is the union of two arcs at (x, 0) which are disjoint except for the point (x, 0). DEFINITION. Let 4b: R X (0, oo ) -R and f: R-*R. The function f is a bi-arc boundary function for 4 if for each xER there is a bi-arc rP, such that lim 4b(u, v) =f(x), as (u, v)->(x, 0), for (u, v)EJ'r. DEFINITION. Let f : R-R. The function f is of honorary Baire class two if there is a function g:R R-R such that g is of Baire class one and f (x) = g (x) except on a countable set.

Hyponormal operators and spectral density

Introduction. We say an operator T on a Hilbert space H is hyponormal if Tx || > || T*x || for xeH, or equivalently T*T-TT* > 0. In this paperwe will first examine some general properties of hyponormal operators. Then we restrict our interest to hyponormal operators with thin spectra. The importance of the topological nature of the spectrum is evident in our main result (Theorem 4) which states that a hyponormal operator whose spectrum lies on a smooth Jordan arc is normal. We continue with a general discussion of a certain growth condition on the resolvent which obtains for hyponormal operators. We conclude with a counterexample to a relation between hyponormal and subnormal operators. The reader is advised that additional facts about hyponormal operators may be found in [l1]. We shall denote the spectrum and the resolvent set of an operator by o(T) and p(T), respectively. The spectral radius R,,(T) sup {j z : z E a(T)}. The numerical range = closure {z: z Tx.x) 11 x = 1} is designated by W(T). Throughout the paper the underlyingvector space is always a separable Hilbert space H.

Almost-Bounded Holomorphic Functions with Prescribed Ambiguous Points

Let f be a function mapping the open unit disk D into the extended complex plane. A point ζ on the unit circle C is called an ambiguous point of f if there exist two Jordan arcs J1 and J2, each having an endpoint at ζ and lying, except for ζ, in D, such that

The shape of the sides of an arc

Associated with certain oriented Jordan arcs is a region which is called the side of the arc. Circles centered on the arc and passing through one of the end points of the arc have an envelope which permits one to find analytically the shape of the side of the arc. A condition, stated geometrically, is imposed on the arcs considered, to insure that the circles have an envelope. An analytic condition is then imposed (Theorem 3) to insure an envelope. As an example a parabola is given.

On Jordan ARCS and Lipschitz classes of functions defined on them

On Jordan ARCS and Lipschitz classes of functions defined on them

Some Remarks on Boundary Behavior of Analytic and Meromorphic Functions

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

A theorem on the accessibility of boundary parts of an open point set

The theorem was established and proved in order to bridge a gap in the proof of a classical theorem on surfaces z(x, y) of nonpositive Gaussian curvature. Theorem 1 might, however, be of interest in other respects. The completed proof of the differential geometric theorem is given in the subsequent paper.-The special case of Theorem 1 needed for the completion is that where the boundaries of the open set Q and of the open and connected sets Q'CQ have only one or two points in common. None of these points need, of course, be accessible from Q'. Theorem 1, however, ascertains that each of these points is accessible from Q. The conclusion of the theorem remains valid if the open set Q'CQ, instead of being connected, is merely supposed to be connected to the vicinity of C. We call an open set Q' connected to the vicinity of a closed set S if there is a neighborhood N* of S (open set containing S) with the following property. For every neighborhood N of S there is a Jordan arc whose interior lies in Q' and whose end points lie in N and on the boundary of N*, respectively. The generalized Theorem 2 is found to be more easily proved than Theorem 1. It is convenient to extend the notion of accessibility of a closed set S from an open set Q to arbitrary closed sets (not necessarily parts of the boundary of Q).

The Existence of a Minimal Surface of Least Area Bounded by Prescribed Jordan Arcs and Prescribed Surfaces

The Existence of a Minimal Surface of Least Area Bounded by Prescribed Jordan Arcs and Prescribed Surfaces