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Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

In this survey we give geometric interpretations of some standard results on boundary behaviour of holomorphic self-maps in the unit disc of C and generalize them to holomorphic self-maps of some particular domains of C n .

Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice $X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the family $\{f|_{X_a}: f\in\mathcal F\}$., Comment: 7 pages

Let D C n and G C m be pseudoconvex domains, let A (resp. B) be an open subset of the boundary @D (resp. @G) and let X be the 2-fold cross ((D(A) B)( (A (B(G)): Suppose in addition that the domain D (resp. G) is locallyC 2 smooth on A (resp. B). We shall determine the \envelope of holomorphy" b X of X in the sense that any function continuous on X and separately holomorphic on (A G)( (D B) extends to a function continuous on b X and holomorphic on the interior of b X: A generalization of this result to N-fold crosses is also given.

We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively., 9 pages

We construct a function f holomorphic in a balanced domainD in C such that for every positive-dimensional subspace Π of C , and for every p with 1 ≤ p < ∞, f |Π∩D is not L -integrable on Π ∩D.

Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $\gamma_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $\gamma_n$ vary regularly; i.e. the ratio $\gamma_{2n+2}/\gamma_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\gamma(z),$ the generating function associated with the coefficients $\gamma_n.$

The authors obtain a generalization of Jack–Miller–Mocanu’s lemma and, using the technique of subordinations, deduce some properties of holomorphic mappings from the unit polydisc in C into C.

Let D be a domain in C and F = F(D) be a subclass in the class O = O(D) of functions holomorphic in D. Recall that D is called an F-domain of holomorphy iff there exists a function f ∈ F(D) which cannot be holomorphically extended across the boundary of D. If, for example, we take for F(D) the class Otemp(D) of temperate holomorphic functions in D (i.e. holomorphic functions in D growing less than a power of distance to the boundary of D) then the notion of Otemp-domains of holomorphy will coincide with the usual notion of (O)domains of holomorphy according to Pflug [10]. On the other hand, considering as F(D) the class H∞(D) of bounded holomorphic functions we obtain the notion of H∞-domains of holomorphy which is quite different from the notion of (O)domains of holomorphy (cf. Sibony [13]). In this paper we are interested in the case when D is invariant under the action of a compact Lie group K and F(D) coincides with the class O(D) of K-invariant holomorphic functions in D. From first examples of O-domains of holomorphy it becomes clear that this notion differs much from the usual notion of domains of holomorphy. Consider, e.g., the ring D = {1 < |z| < 2} in C with the action of the circle group S given by rotations. Then the only S-invariant holomorphic functions in D are constants so they extend holomorphically across the boundary of D to all of C (note that D is a domain of holomorphy in this example). Later on we shall give several (less trivial) examples of that sort. This article based on recent results by Peter Heinzner, Xiangyu Zhou and the author (cf. [4], [5], [12], [16]) contains some general assertions about O-domains of holomorphy and their holomorphic hulls

Let E be a complex Hausdorff locally convex space such that the strong dual E′ of E is sequentially complete, let F be a closed linear subspace of E and let U be a uniformly open subset of E. We denote by Π : E → E/F the canonical quotient mapping. In §1 we study the factorization of uniformly holomorphic functions through π. In §2 we study F -quotients of uniform type and introduce the concept of envelope of uF -holomorphy of a connected uniformly open subset U of E. The main result states that the pull-back eu(U) of the envelope of uniform holomorphy of Π(U) constructed by Paques and Zaine [9] is the envelope of uF -holomorphy of U . Introduction. We deal with the concept of uniform holomorphy (cf. [6]–[8]) of a holomorphic function f : U → C in the case when U is a nonvoid uniformly open subset of a complex Hausdorff locally convex space E. Let F be a closed linear subspace of E, letΠ : E → E/F be the canonical quotient mapping and let IU be the set of all continuous seminorms α on E such that U is open in (E,α). Let Hu(U) be the set of all uniformly holomorphic functions from U into C and let HuF (U) be the set of all g ◦Π as g ranges over Hu(Π(U)). It is easy to show that HuF ⊂ Hu(U). In §1 we prove that if U is a balanced uniformly open subset of E and F is a closed linear subspace of (E,α) for each α ∈ IU , then g ◦ Π is uniformly holomorphic if and only if g is uniformly holomorphic. The concepts of Riemann domain of uniform type and F -quotient of a Riemann domain were introduced in [9] and [4] respectively. Given a uniformly open subset U of E it is easy to verify that Π(U) is a uniformly open subset of E/F (cf. Ex. 3, §2). We have been unable to decide if an F -quotient of a Riemann domain of uniform type is always of uniform type. However, we give in §2 some non-trivial examples of F -quotients of a Riemann domain of uniform type which are of uniform type. In particular, we consider 1991 Mathematics Subject Classification: Primary 46G20.