Unbounded Component Research Articles

The iteration of entire functions of small growth

AbstractWe show that if a transcendental entire function has order zero and sufficiently small growth or has order ρ < ½ and regular growth then its set of normality has no unbounded components.

Pointwise bounded approximation by polynomials

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

Correspondence: Criticality and utility in the next generation

One of the issues raised by the editorial of Stankovic and Ramamritham is that of using unbounded software components within the context of real-time systems. We have attempted to give a different perspective to this issue; specifically how can we increase at run-time, the utility of 100% guaranteed tasks. We believe that many future systems will contain predominantly crucial computations. The need to have full predictability remains. Indeed, there is a need to develop platforms that have realistic worst case performance to probability levels compatible with the overall failure hypothesis.

Separation of two (possibly unbounded) components of the spectrum of a linear operator

LetX be a complex Banach space andA: D(A)→X a densely defined closed linear operator whose resolvent set contains the real line and for which ‖λ(λ−A)−1‖ is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and −A, for the existence of a decompositionX=X+⊕X−, whereX± are closed subspaces, invariant forA, the spectra of the reduced operatorsA± are {λ∈σ(A);Imλ>0} and {λ∈σ(A);Imλ<0} respectively, and ‖λ(λ−A±)−1‖ is bounded forImλ≶0.

Minimal disks and compact hypersurfaces in Euclidean space

Let M n {M^n} be a smooth connected compact hypersurface in ( n + 1 ) (n + 1) -dimensional Euclidean space E n + 1 {E^{n + 1}} , let A n + 1 {A^{n + 1}} be the unbounded component of E n + 1 − M n {E^{n + 1}} - {M^n} , and let κ 1 ⩽ κ 2 ⩽ ⋯ ⩽ κ n {\kappa _1} \leqslant {\kappa _2} \leqslant \cdots \leqslant {\kappa _n} be the principal curvatures of M n {M^n} with respect to the unit normal pointing into A n + 1 {A^{n + 1}} . It is proven that if κ 2 + ⋯ + κ n &gt; 0 {\kappa _2} + \cdots + {\kappa _n} &gt; 0 , then A n + 1 {A^{n + 1}} is simply connected.

Irreducible continua of solutions for Birkhoff-Kellogg type equations

BIRKHoFF-KELLOGG THEOREM. Let E be an infinite dimensional Banach space and let aB(O, 1) be the boundary of its unit ball B(0, 1). Let h: dB(0, l)+ E be a compact map such that llh(x)ll 2 a> 0 for all x E aB(O, 1). Then, there exist z E dZ3(0, 1) and A > 0 such that x = Ah(x), i.e. h has an invariant point, in the terminology adopted in [l]. Sometimes in the literature the Birkhoff-Kellogg Theorem is reported in slightly different terms. Namely, the compact map h is assumed to be defined on the closure B(O, 1) of B(0, 1) and it satisfies the Birkhoff-Kellogg condition on dB(0, 1). The rotation of the unit circle (of the unit disk) in lw2 provides an example of a compact map (over a finite dimensional Banach space) satisfying the Birkhoff-Kellogg condition and not having invariant points. On the other hand, it is not hard to prove that if the compact map h: B(O, 1) --, E satisfies /h(x)ll 2 LY > 0 for all x E B(O, l), then there exists an invariant point for the map h, regardless if E is finite or infinite dimensional. The point is that one can use this strengthened version to prove the Birkhoff-Kellogg Theorem in infinite dimensional Banach spaces by suitably extending the compact map h: dB(0, l)+ E to the whole closed ball B(O, 1). In the first part of this paper, we will see that the situation described above carries over to a much more general context. The first part will also contain two results regarding the existence of unbounded components of eigenvectors (and of solutions) for nonlinear equations with compact “right hand sides” satisfying Birkhoff-Kellogg type assumptions in a neighbourhood of infinity (i.e. on the complement of an open and bounded subset). These two are the main theorems of the paper, which improve and refine some recent results obtained in [7] and in [W

Bounded limits of analytic functions

Let U be a bounded open plane set, and let / be a bounded analytic function on U, which is the pointwise limit of a bounded sequence {fn} of uniformly continuous analytic functions. It is shown that one can find another such sequence {fn}, converging to f, and bounded by the supremum norm off. A similar result is proved for approximation by rational functions. In this paper the following problem is considered: let U be a bounded open subset of the complex plane C, and let A be a set of bounded analytic functions on U. Which bounded analytic functions on U are limits of bounded sequences of functions in A converging pointwise in U? In the case where A consists of the polynomials this question was settled by Rubel and Shields [4], a special case had earlier been treated by Farrell [2]. A function f on U is the pointwise limit of some bounded sequence of polynomials if and only if it is the restriction to U of a bounded analytic function on U*, the Caratheodory hull of U. (U* is the interior of the complement of the unbounded component of the complement of the closure of U.) Rubel and Shields gave an example to show that the set of such pointwise limits need not be closed under uniform convergence. They constructed a set U and a sequence {fn} of bounded analytic functions converging uniformly on U tof, such that eachf, is a pointwise bounded limit of polynomials, but that the bounds on the approximating sequences of polynomials necessarily tended to infinity with n, so that no bounded diagonal subsequence converging to f could be found. The object of this paper is to show that if A is either the algebra of uniformly continuous analytic functions on U or (under mild hypotheses on aU) the rational functions with poles outside U, then this phenomenon cannot occur. The author is very grateful to T. W. Gamelin and B. K. 0ksendal for many valuable conversations. Notation. A(U) denotes the algebra of uniformly continuous analytic functions on the bounded open set U (which we regard as continuous Received by the editors March 26, 1971. AMS 1970 subject classifications. Primary 30A82; Secondary 30A98, 46J15.

A fixed point theorem for manifolds

A Lefschetz type fixed point theorem is proved extending a recent theorem by Robert F. Brown. It deals with compact maps of the form f : ( M − U , X ) → ( M , M − U ) f:(M - U,X) \to (M,M - U) , where M M is an n n -manifold, X X is an ( n − 2 ) (n - 2) -connected ANR which is closed in M M and U U is an unbounded component of M − U M - U . The map f f defines maps u : M − U → M − U u:M - U \to M - U and v : M → M v:M \to M ; the Lefschetz numbers of u u and v v are defined and are shown to be equal; and if this number is nonzero then f f has a fixed point.

Baxter operators and endomorphisms on Banach algebras

A Baxter operator is a linear operator T on a Banach algebra 2I which satisfies the identity Tx.Ty=T(Tx.y+x.Ty-thy) for all x and y in VI, 0 being some fixed element of %. The consequences of this identity were first explored by G. Baxter [l], and subsequently by J. G. Wendel [2], J. F. C. Kingman [3], [4], F. V. Atkinson [S] and the author [6]. The two last-mentioned papers give a partial description of the spectral properties of bounded Baxter operators. While [6] provides a fairly explicit formula for the resolvent R(X, T) = (XI T)-l of T, this formula contains the resolvent of 0, in a way which seems to make the resolvent set Res( T) of T depend upon that of 8; at any rate, the extent of Res( T) in the general case is in doubt, and for the proof of the formula in [6] it was assumed that h belongs to the unbounded component of [Res( 2’) n Res(ffll\(O>. In the present paper we consider only the case 0 = e, where e is the identity element in 91. This case is still of much interest, and has already been studied in some detail. In [6] it was shown that T has a certain summation property when restricted to functions of the element t = Te, which can be written heuristically as

𝑛th roots of operators

Throughout the paper the following notation and terminology will be used. OC is a complex Banach space and ?(O) is the space of continuous linear transformations from C into SC. The identity element of ?(O) will be denoted by I. If A ES2(S), o(A) will denote the spectrum of A and p(A) the resolvent set of A. If C is a simple closed rectifiable curve in the complex plane, C will be assumed to be oriented in a counterclockwise direction. By the interior of C we mean the bounded component of the complement of C; the exterior of C is the unbounded component.

Connectedness in topological linear spaces

The following properties, well known for normed linear spaces of dimension ≧2, are established for an arbitrary topological linear space of dimension ≧2: (a) every neighborhood of 0 contains one whose complement is connected; (b) the complement of a bounded set has exactly one unbounded component.

Bounded approximation by polynomials

Abstract : This paper presents a complete solution to the following problem: if G is an arbitrary bounded open set in the complex plane, characterize those functions in G that can be obtained as the bounded pointwise limits of polynomials in G. Roughly speaking, the answer is that a function is such a limit if and only if it has a bounded analytic continuation throughout a certain bounded open set G* that contains G. This set G* is the inside of the 'outer boundary' of G. More precisely, if G is a bounded open set and if H is the unbounded component of the complement of G- (the closure of G), then G* denotes the complement of H-.

On some extremal functions and their applications in the theory of analytic functions of several complex variables

In the case of one complex variable, the function log FD(z, E, 0) is a generalized Green's function for the unbounded component of CE with pole at so. It is well known that the Green's function plays the primary role in the theory of interpolation and approximation of holomorphic functions of one variable by polynomials (see [27]). It turns out that the function D(z, E, 0), z cn Cn, also plays a quite similar role in the theory of interpolation and approximation of holomorphic functions of several variables by polynomials. For instance, one can obtain the Bernstein-Walsh inequality