Theorem Of Bishop Research Articles

On a theorem of Bishop and commutants of Toeplitz operators in $$\mathbb {C}^n$$

We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-\v{C}u\v{c}kovi\'c-Rao Theorem in higher dimensions.

Several constructions in the Eremenko–Lyubich class

We use a theorem of Bishop in [2] to construct several functions in the Eremenko–Lyubich class B. First it is verified, that in Bishop's initial construction [2] of a wandering domain in B, all wandering Fatou components must be bounded. Next we modify this construction to produce a function in B with wandering domain and uncountable singular set. Finally we construct a function in B with unbounded wandering Fatou components. It is shown that these constructions answer two questions posed in [9].

On Šilov boundary for function spaces

In this paper, we develop the notion of Šilov boundary for closed, point separating subspaces of spaces of continuous functions on a compact Hausdorff space. We give an example to show that the set of peak points need not be a boundary. We give a proof of Bishop's theorem that describes the Choquet boundary in the case of closed, point separating, subalgebras of continuous functions that do not contain the constant function. For closed point separating subspaces, using results of R.R. Phelps on integral representation theory for subspaces, we show that the closure of the Choquet boundary as a bundle is the Šilov boundary bundle similar in the sense to the one considered by D. Blecher [5].

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in $$ \mathbb{C}$$ are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ $$ \mathbb{R}^3$$ is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ $$ \mathbb{C}$$ (when f is assumed to be in C 1 with $$\bar \partial {\kern 1pt}f$$ small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.

A density theorem for locally convex lattices

Let E be a real, locally convex, locally solid vector lattice of (AM)‐type. First, we prove an approximation theorem of Bishop′s type for a vector subspace of such a lattice. Second, using this theorem, we obtain a generalization of Nachbin′s density theorem for weightedspaces.

Continuation of analytic structure in the maximal ideal space of a uniform algebra

Let A be a uniform algebra with maximal ideal space M and Shilov boundary Σ; see (8), (18) or (21) for the basic definitions. If Σ is different from M, there is often analytic structure in M/Σ. However this is not always the case, as is shown by the classical example of Stolzenberg in (16). Hence much of the considerable amount of research on this topic has been devoted to finding conditions which ensure the presence of analytic structure in M/Σ One particularly fruitful line of development has been concerned with one-dimensional analytic structure; in particular we have in mind the classical theorem of Bishop (see (2), chapter 11) and the more recent result of Aupetit and Wermer(2).

The Dual of a Theorem of Bishop and Phelps

The Dual of a Theorem of Bishop and Phelps

Non-existence of continuous convex functions on certain Riemannian manifolds

The present note is written in response to a question of Wu. In [51 Wu proved that every complete non-compact convex hypersurface in euclidean space has infinite volume. He then asked whether this is true for all complete non-compact non-negatively curved manifold or not. The structure theorem of Cheeger and Gromoll, I-2], says that such a manifold has a continuous convex exhaustion. On the other hand, a theorem of Bishop and O'Neill, [1], says that there is no non-trivial smooth convex function on a complete manifold with finite volume. However, as was remarked by Greene and Wu [3], the passage from continuity to smoothness is highly non-trivial. The main theorem here is that there is no non-trivial continuous convex function on a complete manifold with finite volume. I would like to thank Professor Wu for his constant interest in this problem. Recently he informed us that he and Greene also proved that a complete non-compact manifold with non-negative curvature has infinite volume.

An Extension to the Theorem of Bishop and Phelps

An Extension to the Theorem of Bishop and Phelps

A remark on the theorem of bishop

A remark on the theorem of bishop

Bishop’s generalized Stone-Weierstrass theorem for the strict topology.

1. Let X be a locally compact Hausdorff space, C(X), the locally convex topological vector space obtained from all bounded complex continuous functions on X by employing the strict topology [2 ]. The present note is devoted to a version of Bishop's generalized StoneWeierstrass theorem [1] applicable to certain subspaces of C(X),; essentially it is a footnote to an earlier paper [4], in which a modification of de Branges' proof of the Stone-Weierstrass theorem [3] was used to obtain Bishop's theorem. Insofar as possible the notation will be that of [4]. The version of Bishop's theorem we shall write down was motivated by, and has application to, the spectral theory of bounded continuous functions on locally compact abelian groups, where the strict topology enjoys a useful role [5]. However, our applications to spectral theory amount at best to new proofs of known results. Recall that the strict topology has a base of neighborhoods of OC(X)g of the form (1) V = {f E C(X)m: Ilfgil ? 1 ,

Measures orthogonal to algebras and sets of antisymmetry

1. We shall be concerned with two items concerning function algebras, connected in large part by their relation to the measures orthogonal to a given algebra: the expression of a general function algebra in terms of antisymmetric algebras recently obtained by Bishop [4], and the question of interpolation in such algebras. Let X be a compact Hausdorff space, C(X) the usual algebra of continuous complex functions on X, and A a closed subalgebra of C(X) containing the constants. Silov [16] called A antisymmetric if every real valued element of A is constant, and inquired whether (as a result of his [13, p. 127] partially suggested) every algebra of continuous functions could be expressed in terms of antisymmetric algebras. Recently Bishop [4] obtained the desired expression, following an approach suggested by Silov's work. In the first part of this paper we shall give a simpler approach to Bishop's theorem and some consequences. Call subset K of X a set of antisymmetry (or an antisymmetric set) of A if,