Paweł Domański

COMPOSITION OPERATORS BETWEEN WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

We characterize those analytic self-maps ’ of the unit disc which generate bounded or compact composition operators C’ between given weighted Banach spaces H 1 v or H 0 v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

Real Analytic Curves in Fréchet Spaces and Their Duals.

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on ℝ xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifTμ is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationTμu=f which depends holomorphically on the parameterz∈ℂ wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let Ω ⊆ IRn be an open set and let A(Ω) denote the class of real analytic functions on Ω. It is proved that for every surjective linear partial differential operator P (D,x) : A(Ω) → A(Ω) and every family (fλ) ⊆ A(Ω) depending holomorphically on λ ∈ C there is a solution family (uλ) ⊆ A(Ω) depending on λ in the same way such that P (D,x)uλ = fλ, for λ ∈ C. The result is a consequence of a characterization of Frechet spaces E such that the class of “weakly” real analytic E-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if C is replaced by another set.

Sampling sets and sufficient sets for A

Abstract We give new characterizations of the subsets S of the unit disc D of the complex plane such that the topology of the space A−∞ of holomorphic functions of polynomial growth on D coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A−∞. Our approach is based on a study of the so-called (p,q)-sampling sets which generalize the A−p-sampling sets of Seip. A characterization of (p,q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khoi and Thomas.

Interpolation of Vector-Valued Real Analytic Functions

Let ω ⊆ R be an open domain. The sequentially complete DF-spaces E are characterized such that for each (some) discrete sequence (zn) ⊆ ω, a sequence of natural numbers (kn) and any family (xn,α)n∈N, |α|6kn ⊆ E the infinite system of equations ( ∂|α|f ∂zα ) (zn) = xn,α for n ∈ N, α ∈ N, |α| 6 kn, has an E-valued real analytic solution f. Introduction Let us consider an open connected set (domain) ω ⊆ R and an arbitrary discrete sequence (zn)n∈N in ω. It is a classical problem to find an analytic function f :ω −→ C which takes prescribed values at (zn). We are interested in the analogous problem when the given values belong to a fixed locally convex space E. More precisely, let E be a sequentially complete locally convex space and let (kn)n∈N be an arbitrary sequence. We ask when for any family ((xn,α)α∈Nd, |α|6kn )n∈N ⊆ E there is a real analytic E-valued function f:ω −→ E with ∂|α| ∂xα f(zn) = xn,α for every n ∈ N, α ∈ N, |α| 6 kn. (1) The problem is of special interest if E is a function space. Then we can interpret our problem as if in the infinite system of equations (1) xn,α are scalars depending ‘nicely’ on some parameter (for instance, holomorphically, smoothly, etc.) and we look for a family of scalar solutions of the system depending as ‘nicely’ as xn,α on the parameter. There are two different natural definitions of vector-valued real analytic functions (see [1, 21], cf. [4, Definition 7, 8] or [5]) and certainly the solution of our problem should depend on the choice of the definition. First, we call a function f :ω −→ E real analytic, f ∈ A(ω,E), if for every linear continuous functional y ∈ E ′, y ◦ f ∈ A(ω). Secondly, f is called topologically real analytic, f ∈ At(ω,E), if for every point x ∈ ω, f develops into a Taylor series convergent around x to f in the topology of E. The relation between these two classes is completely clarified in [4] and [5], see also Section 1 below. Let us mention only that (contrary to the analogous classes for holomorphic functions) they are different for some Fréchet spaces but for sequentially complete DF-spaces they coincide (see [4, Proposition 9]). It is known, and follows from the theory of π-tensor products, that every Fréchet Received 26 April 2001; revised 2 November 2001. 2000 Mathematics Subject Classification 46E40 (primary), 46E10, 32E30, 46A63, 46A04, 46A13, 26E05 (secondary). 408 josé bonet, pawe l domański and dietmar vogt space has an analogous interpolation property for holomorphic functions. Consequently, for Fréchet spaces E each interpolation problem (1) has a solution both in A(ω,E) and At(ω,E). The corresponding problem for complete DF-spaces turns out to be more complicated although in that case A(ω,E) = At(ω,E). The main result of our paper (Theorem 4.1) solves the problem completely, showing that such a space E has the interpolation property if and only if E has the property (A): whenever (Bn) is a fundamental sequence of Banach discs in E, then there is n such that for every m there are k, p > 0 and C with Bm ⊆ rBn + C rp Bk for all r > 0, (2) which is equivalent to the existence of a fundamental sequence B0 ⊆ B1 ⊆ . . . of bounded Banach discs in E and positive numbers εk such that Bk ⊆ rB0 + 1 rεk Bk+1 for all k ∈ N and r > 0. This condition is closely related to the well-known (DN) condition (see [29, § 29]); in fact, for reflexive E we have E ∈ (A) if and only if E ′ β ∈ (DN). Therefore, the spaces of germs of holomorphic functions H(K) for natural compact sets K ⊆ C have (A) as well as duals of all power series spaces or spaces of Whitney jets over ‘nice’ sets (see [29, 29.12; 32, 33]). If we can take p = 1 in (2), then E has even (A) and E ′ β ∈ (DN). The crucial step in the proof of sufficiency of Theorem 4.1 is given in Theorem 4.8 showing that H(D, l1(I)) is a ‘quotient universal’ space for the class of sequentially complete LB-spaces with the property (A). It is known that the analogous interpolation problem for holomorphic functions (that is, analytic of complex variables) has a positive solution for a complete DFspace E if and only if E has (A) [38, 4.2, 4.5], see also [6]. It is worth pointing out that the difference between (A) and (A) (or (DN) and (DN)) is quite essential. For H(U), U an arbitrary Stein manifold, H(U) has always (DN) but it has (DN) if and only if the strong Liouville property holds, that is, every plurisubharmonic function on U bounded from above is constant (cf. [30, Proposition 2.1; 36, p. 262; 40, 2.3.7]). Analogously, power series spaces Λr(α) have always (DN) but (DN) only if r = ∞ [29, 29.2, 29.12]. As we explained above, the most important motivation of the paper is provided by the problem of solving equations depending on parameters, see [4, 5, 21, 22, 26, 37, 38]. Our research is influenced by the recent extensive research on the space of scalar-valued real analytic functions, which is motivated mostly by its relevance to the theory of partial differential equations, see for example [2, 5, 7, 9, 13, 16, 18, 19, 21–25, 27, 28]. The paper is also motivated by the observation that our problem is a question on the lifting of weak*–weak continuous operators S :E ′ −→ CN with respect to the short exact sequence of the form 0−−−→ kerT −−−→A(ω) T −−−→CN−−−→ 0,

Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations μ n (f) = b n for n ∈ N has a real analytic solution f on ω C R d for every right-hand side (b n ) n ∈ N ⊆ C and give a complete characterization of such sequences of analytic functionals (μ n ). We also show that every open set w C R d has a complex neighbourhood Ω C C d such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on Ω.

A Note on the Spectrum of Composition Operators on Spaces of Real Analytic Functions

In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.