Domains of convergence in infinite dimensional holomorphy Von der Fakult¨at f¨urMathematik und Naturwissenschaften der Carl von Ossietzky Universit¨atOldenburg zur Erlangung des Grades und Titels eines Doktors der Naturwissenschaften (Dr. rer. nat.) angenommene Dissertation von Christopher Prengel geboren am 28.10.1975 in Hannover Gutachter: apl. Prof. Dr. A. Defant Zweitgutachter: Prof. Dr. P. Pflug Tag der Disputation: 26.10.2005 Abstract For certain subsets F(R) of holomorphic functions on a Reinhardt domain m R in a Banach sequence space X, mainly for H∞(R), P (X) and P(X), we give precise descriptions of the domain of convergence dom F(R), i.e. the set of all points of R where the monomial expansion X ∂αf(0) zα α! (N) α∈N0 of every function f ∈ F(R) is (unconditionally) convergent. The results are obtained by an interplay of complex analysis and local Banach space theory, improving work of Ryan and Lempert about monomial expan- sions of holomorphic functions on `1 and using methods from the theory of multidimensional Bohr radii. The problem of conditional convergence is solved for polynomials. It is closely connected with bases for full and symmetric tensor products and convergence preserving permutations. Zusammenfassung F¨urgewisse Teilmengen F(R) holomorpher Funktionen auf einem Reinhardt- m gebiet R in einem Banachfolgenraum X, vor allem f¨ur H∞(R), P (X) und P(X), geben wir eine genaue Beschreibung des Konvergenzbereichs dom F(R), d.h. der Menge aller Punkte aus R, in denen die Monomial- entwicklung X ∂αf(0) zα α! (N) α∈N0 einer jeden Funktion f ∈ F(R) (unbedingt) konvergiert. Die Resultate werden durch ein Zusammenspiel von komplexer Analysis und lokaler Banachraumtheorie erzielt, setzen Arbeiten von Ryan und Lempert ¨uber Monomialentwicklungen holomorpher Funktionen auf `1 fort und bedie- nen sich der Theorie mehrdimensionaler Bohrradien. Das Problem bedingter Konvergenz wird f¨urPolynome gel¨ost. Es ist eng verkn¨upft mit Basen f¨urvolle und symmetrische Tensorprodukte und kon- vergenzerhaltenden Permutationen. I thank my supervisor Andreas Defant for his guidance during the preparation of this thesis. Gracias a Manuel Maestre por las discusi´ones ´utilesy por su invitaci´ona Va- lencia. Tambi´ena todas las personas que hac´ıan que mi estancia in Valencia fuera muy agradable, especialmente Pilar y su familia, Pablo y Marta, y por supuesto Manolo. Thank you Melissa for correcting my English. Contents Introduction 9 Preliminaries 15 1. Norm estimates of identities between symmetric tensor products of Banach se- quence spaces 25 Identities between symmetric and full tensor products ............. 26 Volume estimates for unit balls of spaces of m-homogeneous polynomials ... 31 `-norm estimates for the injective and projective tensor product ........ 33 Volume ratio of m-fold tensor products .................... 44 2. Domains of convergence 51 Domains of convergence of holomorphic functions ............... 51 Lower inclusions for dom H∞(R) ....................... 65 The arithmetic Bohr radius .......................... 75 Upper inclusions for dom H∞(R) ...................... 84 A description of dom Pm(X) ........................ 89 A conjecture ................................. 93 The domain of convergence of the polynomials ................. 105 The exceptional role of `1 ........................... 107 Domains of convergence, Bohr radii and unconditionality ........... 113 3. Tensor and s-tensor orders 119 Orders on Nm ................................. 121 Orders for tensor products ........................... 130 Examples ................................... 134 Bases for symmetric tensor products and spaces of polynomials ........ 167 Pointwise convergence ............................. 170 Bibliography 174 Introduction A holomorphic function f on a Reinhardt domain R in Cn has a power series expansion X α f(z) = cα(f)z n α∈N0 ∂αf(0) with coefficients cα(f) = α! that converges to f in every point of R. In infinite dimensions, if f is holomorphic on a Reinhardt domain R in a ∂αf(0) Banach sequence space X and we define cα(f) := α! then X α f(z) = cα(f)z (N) α∈N0 holds for finite sequences z = (z1, . , zn, 0,...) ∈ R and we ask: Where on R does this monomial expansion series converge? We call the set dom(f) of all those points the domain of convergence of f. In this thesis we give precise descriptions of the domain of convergence \ dom F(R) = dom(f) f∈F(R) of certain subsets F(R) ⊂ H(R) of holomorphic functions. Historically the problem that we consider here is connected with the defini- tion of holomorphy in infinite dimensions. Holomorphy in finite dimensions can be characterized by convergence of monomial expansion series in the neighbourhood of every point and it was this approach that Hilbert tried at the beginning of the 20th century. In [46] he proposed a theory of analytic functions in infinitely many variables based on representations by power series X α cαz (0.1) (N) α∈N0 (absolutely) convergent on infinite dimensional polydiscs |z1| ≤ ε1, |z2| ≤ ε2, |z3| ≤ ε3,... (0.2) 10 Introduction where (εn) is a sequence of positive numbers. In the following years it became clear by the work of Fr´echet and Gˆateaux that expansions in power series of homogeneous polynomials are more suit- able, which formally can be obtained from (0.1) by building blocks X α X X α cαz = cαz (N) m |α|=m α∈N0 This theory was continued by students of Fr´echet. The current form for functions between normed spaces can be found in papers of Graves [43] and Taylor [75, 76] from the mid 1930’s, where the equivalence of Fr´echet differ- entiability, Gˆateaux differentiability plus continuity, and representation by a power series of homogeneous polynomials in the neighbourhood of every point is proved. A natural environment where it is possible to work with monomial expansions are fully nuclear locally convex spaces with basis, as was shown in 1978 by Boland and Dineen [15], see [35], [37] and [36, sec. 3.3, 4.5] for more results in this direction. Matos [60] characterized those holomorphic functions on an open subset of a Banach space with unconditional basis having monomial series representa- tions in the neighbourhood of every point to be functions of a certain holo- morphy type (see also [61]). He proved that in `1 these are all holomorphic functions. As we will see, the space `1 plays a special role in our theory. In some sense the results we give in this thesis justify the historical devel- opment. One problem with the Hilbert approach is that his definition is more restrictive than he actually thought. He gives us a criterion to de- cide if a power series (0.1) defines an analytic function in the domain (0.2). He claims that it is sufficient for the series to be bounded in the follow- P α ing sense: Every finite dimensional section n cαz of (0.1) (putting α∈N0 the variables zn+1, zn+2, zn+3,... equal to zero and identifying (α1, . , αn) = (N) (α1, . , αn, 0, 0,...) ∈ N0 ) is convergent for |z1| ≤ ε1,..., |zn| ≤ εn and X α sup sup cαz < ∞ (0.3) n∈N |zi|≤εi n α∈N0 It can be seen from an example of Bohnenblust and Hille [12] that this is not true (see example 2.4 and also p. 12 below). But our results even show, extending Hilbert’s definition of boundedness in an appropriate way (see remark 2.9), that `1 is almost the only Banach sequence space where the criterion is true. Nevertheless it is remarkable here that a power series (0.1) satisfying (0.3) with all εn equal, say, to r > 0 defines a bounded holomorphic function on rBc0 in the current sense (remark 2.9). Introduction 11 The roots of our results can be found in the work of Harald Bohr at the beginning of the 20th century. He studied Dirichlet series X an (0.4) ns n where an are complex coefficients and s is a complex variable. The domains where such a series converges absolutely, uniformly or conditionally are half planes [Re s > σ0] where σ0 = a, u or c is called the abscissa of absolute, uniform or conditional convergence respectively. More precisely σ0 = inf σ taken over all σ ∈ R such that on [Re s > σ] we have convergence of the requested type. Bohr wanted to know the greatest possible width T of the strip of uniform but non absolute convergence over all Dirichlet series (0.4): 6 uniform convergence - absolute convergence - - T = sup a − u P an ns u a He attacked this problem using the nowadays so called Bohr method (see [13]). Using the fundamental theorem of arithmetic he established a one to one correspondence between Dirichlet series and power series in infinitely many variables, given by X an X α c z , c = a α (0.5) ns ! α α p n (N) α∈N0 where p = (pn) is the sequence of prime numbers p1 < p2 < p3 < . Then he defined S to be the least upper bound of all q > 0 such that P α P α |cα| ε < ∞ for every power series cαz bounded in Hilbert sense on |zn| ≤ 1, i.e. with X α sup sup cα z < ∞ (0.6) n |zi|≤1 n α∈N0 and every ε ∈ `q ∩ ] 0 , 1[ N and used the prime number theorem to prove 1 T = (0.7) S Thus Bohr shifted the problem to the setting of bounded power series in infinitely many variables and he was able to prove S ≥ 2 ([13, Satz III]): 12 Introduction P α Theorem 0.1. Let cαz be a power series bounded in the domain [|zi| ≤ 1] and ε ∈ `2 ∩ ] 0 , 1[ N. Then X α |cα| ε < ∞ (N) α∈N0 Bohr did not know if 2 was the exact value of S, not even if S < ∞ (i.e.