On holomorphic mappings with relatively p-compact range

Equivalence, linearization, and decomposition of holomorphic operator functions

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

Given a holomorphic mapping of bounded type g ∈ Hb(U, F), where U ⊆ E is a balanced open subset, and E, F are complex Banach spaces, let A : Hb(F) ∈ Hb(U) be the homomorphism defined by A(f) = fog for all f ∈ Hb(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 Hb(U,F) and prove that E' and F have the Schur property if and only if Hb(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.

It is shown that in every complex Banach space E with open unit ball D C E there is a closed C-linear subspace Cc such that V C\ D is the orbit of the origin 0 E E under the group Aut(D) of all biholomorphic automorphisms of D. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.Suppose E, F are complex Banach spaces and U E E is an open subset.Then a mapping/: U -* Fis called holomorphic if the Frchet derivative of/ at x (denoted by f'(x)) exists for every x G U. (For details compare [6].)In this paper we show that the Banach space E is completely determined by the holomorphic structure of its (open) unit ball; more precisely, Theorem.The complex Banach spaces E and are isometrically isomorphic if and only if there is a biholomorphic mapping from the unit ball of E onto the unit ball of .This answers a question L. A. Harris raised in [3, p. 15].For the proof let be a fixed complex Banach space with unit ball DEE.Let 0 denote the set of all holomorphic vector fields on D i.e. of all expressions X = f(z)d/dz where/: D -* E is a holomorphic mapping and z denotes the coordinate of D. Then D is a complex Lie algebra in the product [/W,^)]=(.'W/(z)-/'(z)g(z)).For every integer v the Lie algebra D contains the linear subspace ^v = l h(z)~~ h: E ^> E is a. homogeneous polynomial of degree v + 1 >.Obviously [$r,$J C SJL^ for every v and \i.Since D is a bounded domain in E, the set g of all complete holomorphic vector fields on D has the following properties (compare [7]):(0) Every h(z)d/dz G g is uniquely determined by h(0) and h'(0) (Cartan's uniqueness theorem).

Let C be an open convex cone in ${\bf R}^n $, n-dimensional real space, such that $\bar C$ does not contain any entire straight line. We consider holomorphic functions in tubes $T^C = {\bf R}^n + iC \subset {\bf C}^n $, n-dimensional complex space, which are known to have $\mathfrak{S}'$ distributional boundary values. We prove that if the $\mathfrak{S}'$ boundary value of such a holomorphic function is an $L^p $ function, $1 \leq p \leq \infty $, then the holomorphic function is in the Hardy space $H^p (T^C )$, $1 \leq p \leq \infty $, corresponding to the tube $T^C $ and can be recovered as the Poisson integral of the distributional boundary value. A similar result is proved wih respect to the holomorphic functions in tubes $T^C $ which are known to have boundary values in the distribution spaces $\mathfrak{Z}',K'_r ,(\mathfrak{S}^\alpha )'$, and $(W^\Omega )'$ and which do not necessarily have $S'$ distributional boundary values. In all cases we prove converse theorems. Our basic results are motivated ...

Let E E and F F be complex Banach spaces. We say that a holomorphic mapping f f from E E into F F is compact respectively bounding if f f maps some neighbourhood of every point of E E into a relatively compact respectively bounding subset of F F . Recall that a subset of E E is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on E E . Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.

Entire Functions on Banach Spaces with a Separable Dual

The spectrum of an algebra of weakly continuous holomorphic mappings

Holomorphic Functions on Strong Duals of Fréchet-Montel Spaces

Following Sinha and Karn [9], a relatively compact subset K of a Banach space E is said to be p-compact if for some sequence (x n ) ∈ l p (E), K ⊂ {Σ n a x x n | (a n ) ∈ B l′ p }. In [4], Delgado, Oja, Pineiro, and Serrano investigated the p-approximation property, in which one only requires finite rank approximation of the identity on p-compact subsets. We investigate analogous concepts here for the case of holomorphic mappings between Banach spaces, introducing the space of p-compact holomorphic mappings (cf. [1]). A number of problems related to such holomorphic mappings are discussed.

Abstract. Let $\Omega \subset \C$ be open,X a Banach space and $W\subset X^\prime$ . We show that every $\sigma (X,W)\mbox{-holomorphic function } f: \Omega \to X$ is holomorphic if and only if every $\sigma(X,W)\mbox{-bounded}$ set inX is bounded. Things are different if we assume f to be locally bounded. Then we show that it suffices that $\varphi \circ f$ is holomorphic for all $\varphi \in W$ , where W is a separating subspace of $X^\prime$ to deduce that f is holomorphic. Boundary Tauberian convergence and membership theorems are proved. Namely, if boundary values (in a weak sense) of a sequence of holomorphic functions converge/belong to a closed subspace on a subset of the boundary having positive Lebesgue measure, then the same is true for the interior points of $\Omega$ , uniformly on compact subsets. Some extra global majorants are requested. These results depend on a distance Jensen inequality. Several examples are provided (bounded and compact operators; Toeplitz and Hankel operators; Fourier multipliers and small multipliers).

We discuss the conceptions of analytic dependence, strict analytic dependence, spaces of level sets and complex bases of holomorphic mappings and their relationship. For instance, a theorem of the following type is proved: Let X be a normal complex space, let be holomorphic mappings, which are analytically related (i.e. depends analytically on and depends analytically on ). Let be maximal (i.e. majorizes every holomorphic map which is strictly analytically dependent on ) and let be strongly maximal (i.e. majorizes every holomorphic map which depends analytically on ). Then Z* is a proper modification of , if is proper or nowhere degenerated.

This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

Holomorphic mappings between real analytic submanifolds in complex space

If f: X → Y is a holomorphic map between complex spaces X and Y, the direct images f (q)(S) of a coherent O X -sheaf S in general are not coherent O Y -sheaves. For example if Y is just a single (reduced) point, coherence of the q-th image sheaf f(q)(S) means that f (q) (S)=Hq(X, S) is a finite dimensional complex vector space. There are many examples of non-compact spaces X where this is not the case.KeywordsManifoldCoherenceSteinResoThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.