Holomorphic Extensions of Whitney Jets

Let φ be a Whitney jet on a closed set F ? ?. By Whitney’s extension theorem φ can be extended to an infinitely differentiable function f on ? which is real analytic on ? F. The main purpose of this article is to show that f can be chosen in such a way that f¦_?F has a holomorphic continuation to the open set (? F) × i? ? ?. In the special case that F is a compact interval or a single point we can even achieve that f¦_?F has a holomorphic continuation to all of $\hat {\rm C}\setminus F$ . In particular, this implies an improvement of the well-known theorem of E. Borel. We also investigate the question when such extensions are given by a so-called extension operator.     

Holomorphic Differential Operators and Plane Sets with Dense Images

We prove in this paper that, given a nonempty open set G in the complex plane, a subset A of G which is not relatively compact and a holomorphic infinite order differential or antidiffeärential operator T, then there are holomorphic functions ? on G such that the image of A under T ? is dense in the complex plane. This extends a recent result about a property of boundary behaviour exhibited by the derivative operator.     

Linear extensions,projections, and split faces

The main result is essentially: Let F be a closed split face of a compact convex set K such that A(F) is separable and has the (positive) metric approximation property. Then there is a (positive) linear extension operator from A(F) into A(K) of norm one.This is applied to C¹-algebras thus giving sufficient conditions for the existence of right inverses to surjective ¹-homomorphisms.     

On a characterization of Arakelian sets

Let K be a compact set in the complex plane, such that its complement in the Riemann sphere, (? ∪ {∞}) / K, is connected. Also, let U ? ? be an open set which contains K. Then there exists a simply connected open set V ? ? such that K ? V ? U. We show that if K is replaced by a closed set F ? ?, then the preceding result is equivalent to the fact that F is an Arakelian set in ?. This holds in more general case when ? is replaced by any simply connected open set Ω ? ?. In the case of an arbitrary open set Ω ? ?, the above extends to the one point compactification of Ω. If we do not require (? ∪ {∞}) /K to be connected, we can demand that each component of (? ∪ {∞}) / V intersects a prescribed set A containing one point in each component of (? ∪ {∞}) / K. Using the previous result, we prove that again if we replace K by a closed set F, the latter is equivalent to the fact that F is a set of uniform meromorphic approximation with poles lying entirely in A.     

Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ H_b(U, F), where U ? E is a balanced open subset, and E, F are complex Banach spaces, let A : H_b(F) ∈ H_b(U) be the homomorphism defined by A(f) = fog for all f ∈ H_b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ? U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space H_b(U,F) and prove that E' and F have the Schur property if and only if H_b(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.     

The {\overline{\partial}} operator along the leaves and Guichard’s theorem for a complex simple foliation

In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on ${{\mathbb C}}In (Ann Sc ENS Sér 3 4:361–380, 1887) Guichard proved that, for any holomorphic function g on \mathbb C{{\mathbb C}}, there exists a holomorphic function h (on \mathbb C{{\mathbb C}}) such that h - h °t = g{h - h \circ \tau = g} where τ is the translation by 1 on \mathbb C{{\mathbb C}}. In this note we prove an analogous of this theorem in a more general situation. Precisely, let (M,F){(M,{\mathcal F})} be a complex simple foliation whose leaves are simply connected non compact Riemann surfaces and γ an automorphism of F{{\mathcal F}} which fixes each leaf and acts on it freely and properly. Then, the vector space H_F(M){{\mathcal H}_{\mathcal F}(M)} of leafwise holomorphic functions is not reduced to functions constant on the leaves and for any g ? H_F(M){g \in {\mathcal H}_{\mathcal F}(M)}, there exists h ? H_F(M){h \in {\mathcal H}_{\mathcal F}(M)} such that h - h °g = g{h - h \circ \gamma = g}. From the proof of this theorem we derive a foliated version of Mittag–Leffler Theorem.     

Analytic Continuation of Complex Gauge Fields

The main result of this note treats the problem of unique extension of holomorphic gauge fields across closed subsets of complex Euclidean space, and is based on a corresponding extension theorem for holomorphic vector bundles due to N. P. Buchdahl and the author. Alternatively, let F be a unitary gauge field corresponding to a complex differential form of type (1, 1) (e.g., an anti self-dual Yang–Mills field on a punctured ball in C ²). As a corollary of the main theorem, it is seen that a unique extension of such F , which preserves the curvature type, is obtained if the contraction of F with a holomorphic vector field lies in the image of the ?¯-operator of the associated holomorphic vector bundle.     

Causality structure and the Weierstrass theorem

In a recent paper P. M. Prenter has shown that the Weierstrass theorem can be lifted up to a real separable Hilbert space H. In this paper H is equipped with an identity resolving orthoprojector chain. The Weierstrass type result of Prenter, namely, if ? is any continuous function on H, then there exists a finite order approximating polynomic operator on every compact K ? H, is sharpened by the extension: if ? is strictly causal (strictly anticausal) then the polynomic approximation can also be strictly causal (strictly anticausal). Other extensions in the same spirit are developed and the results are interpreted in the setting of Volterra operators on L₂.     

Entire Functions on Banach Spaces with a Separable Dual

Let E and F be complex Banach spaces. We show that if E has a separable dual, then every holomorphic function from E into F which is bounded on weakly compact sets is bounded on bounded sets.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

Compact operators without extended eigenvalues

A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT=λTX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.     

On the Best Exponent in Markov Inequality

Let E be a compact set preserving the Markov inequality and m(E) be its best exponent i.e., m(E) is the infimum of all possible exponents in this inequality on E. It is known that $\alpha (E) \le \frac1{m(E)}$ where α(E) is the best exponent in Hölder continuity property of the (pluri)complex Green function (with pole at infinity) of E. We show that if E???? ^N (or ? ^N ) with N?≥?2 then the Markov inequality need not be fulfilled with m(E). We also construct a set E????² such that the Markov inequality holds at the tip of exponential cusps composing E but for the whole set E we have m(E)?=?∞. Moreover, we prove that sup m(E)?=?∞ where the supremum is taken over all compact sets E???? preserving the Markov inequality. Finally, we prove that if E is a Markov set in ? then its image F(E) under a holomorphic mapping F is a Markov set too. More precisely, we prove that $m(F(E))\leq m(E)\cdot \Big(1+ \max\limits_{ \partial E\cap\{F'(t)=0\}}\textrm{ord}_t F'\Big)$ .     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

On certain hyperbolic sets

Two invariant sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an F, there is no locally maximal set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the hyperbolicity of F. (This means that if a hyperbolic set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher set, then F′ has a neighborhood in which S′ has no locally maximal set containing F′.)     

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete K?hler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not K?hler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact K?hler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. Received: January 2006, Revision: November 2006, Accepted: March 2007     

SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T^?, the adjoint of T, is quasitriangular. Moreover, if T^? is quasitriangular, then, given an ε>0, there exists a compact operator K on H with ‖K‖<ε such that T+K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.     

Variational measures and the Kurzweil-Henstock integral

For a given continuous function F on a compact interval E in the set ? of reals the problem is how to describe the “total change” of F on a set M ? E. Full variational measures W _F (M) and V _F (M) (see Section 2) in the sense presented by B. S. Thomson are introduced in this work to this aim. They are generated by two slightly different interval functions, namely the oscillation of F over an interval and the value of the additive interval function generated by F, respectively. They coincide with the concept of classical total variation if M is an interval and they are zero if on the set M the function F is of negligible variation. The Kurzweil-Henstock integration is shortly described and some of its properties are studied using the variational measure W _F (M) for the indefinite integral F of an integrable function f.     

Lipschitz image of a measure-null set can have a null complement

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ? ?₂ which contains a translate of any compact set in the unit ball of ?₂ and a Lipschitz isomorphismF of ?₂ onto ?₂ so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA?X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.