Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

: In this paper we investigate analytic functions of unbounded type on a complex inﬁnite dimensional Banach space X . The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufﬁcient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.


Introduction and Preliminaries
Let X be an infinite dimensional complex Banach space. A function P : X → C is an n-homogeneous polynomial if there exists a symmetric n-linear map B P defined on the Cartesian power X n to C such that P(x) = B P (x, . . . , x). The mapping B P is called n-linear form associated with P and is necessary and unique because of the well-known polarization formula (see, e.g., p. 4 [1]). The Banach space of all continuous n-homogeneous polynomials on X with respect to the norm |P(x)| is denoted by P ( n X). For us the following version of the polarization inequality is important (p. 8 [1]). |B P (x n 1 1 , . . . , x n k k )| ≤ n 1 ! · · · n k !m m n n 1 1 · · · n n k k m! P x 1 n 1 · · · x k n k , where n 1 , . . . , n k are positive integers with n 1 + · · · + n k = m.
A continuous function f : X → C is said to be an entire analytic function (or just entire function) if its restriction on any finite dimensional subspace is analytic. The space of all entire analytic functions on X is denoted by H(X). For every entire function f there exists a sequence of continuous n-homogeneous polynomials (so-called Taylor polynomials) such that f (x) = ∞ ∑ n=0 f n (x) (2) and the series converges for every x ∈ X. Here f 0 = f (0) is a constant. The Taylor series expansion (2) uniformly converges on the open ball rB centered at zero with radius r = 0 ( f ), where 0 ( f ) = 1 lim sup n→∞ f n 1/n . The radius r = 0 ( f ) is called the radius of uniform convergence of f or the radius of boundedness of f because the ball rB is the largest open ball at zero such that f is bounded on it. If 0 ( f ) = ∞, then f is bounded on all bounded subsets of X and is called a function of bounded type. The set of all functions of bounded type on X is denoted by H b (X).
It is known that H b (X) is a Fréchet algebra with respect to the topology of uniform convergence on bounded sets. Algebras H b (X) were considered first in [2,3] and studied by many authors for various Banach spaces X.
A polynomial P on X is called a polynomial of finite type if it is a finite sum of finite products of linear functionals and constants. We denote by A n (X) the smallest closed subalgebra of H b (X) containing the space of all n-homogeneous polynomials P n (X). In particular, A 1 (X) is the closure of the space of finite type polynomials in H b (X). In the general case, A n (X) = A k (X) if n = k. For example, it is so for X = 1 (see [4] for details).
We say that a function f is an entire function of unbounded type if f ∈ H(X) \ H b (X). It is known that for given weak*-null sequences φ n ∈ X * , φ = 1, which always exists (see p. 157 [5]) the function is an entire function of unbounded type on X. It is easy to see that the Taylor polynomials f n = φ n n of f are polynomials of finite type. In this paper we consider the following natural question. Question 1. Let P 0 ( n X) be subspaces of P ( n X), n ∈ N. Under which conditions is there a function f = ∑ ∞ n=0 f n ∈ H(X) \ H b (X) such that f n ∈ P 0 ( n X)?
In Section 2 we obtain some general results on entire functions of unbounded type. In particular, we show that if X is such that A m (X) = A m−1 (X) for some m > 1, then there exists a function f ∈ H(X) \ H b (X) such that all its Taylor polynomials are in A m (X). Also, we establish some sufficient conditions for a function f to be in H(X) \ H b (X). In Section 3 the obtained results are applied to construct examples of symmetric analytic functions of unbounded type on p , 1 ≤ p ≤ ∞. These examples can be considered as generalizations of the example, constructed in [6] on 1 . In addition, the paper contains some discussions and open questions.

General Results
Evidently, the set of entire functions of unbounded type is not a linear space. Moreover, the following example shows that the product of two functions of unbounded type is not necessarily a function of unbounded type because there are invertible analytic functions of unbounded type. Example 1. Let X = p or c 0 for 1 ≤ p < ∞ and φ n be the coordinate functionals multiplied by −1 So the function f of the form (3) is of unbounded type. We set On the other hand, and so 1/g ∈ H(X) \ H b (X). However, the product of g and 1/g equals 1, which is bounded.
Note that H(X) \ H b (X) surprisedly contains infinite dimension linear subspaces and subalgebras without the zero vector [19].
The following proposition shows that the set of entire functions of unbounded type has some kind of ideal property.
and P n ∈ P ( n X), 0 < c ≤ P n ≤ C < ∞ for some constants c, C > 0, and n ∈ N. Then Proof. Let us show that g is well-defined on X. Note first that since f (x) is well-defined on X, the series is absolutely convergent. It is true, in particular, for t = 1.
For every To prove that g is analytic but not of bounded type we need to show that the radius of boundedness 0 (g) of g at zero satisfies 0 < 0 (g) < ∞. One can check that That is, 0 (g) = 0 ( f ) and so g ∈ H(X) \ H b (X).
Theorem 1. Let us suppose that there is a dense subset Ω ⊂ X and a sequence of polynomials P n ∈ P ( n X), lim sup n→∞ P n 1/n = c, 0 < c < ∞ such that for every z ∈ Ω there exists m ∈ N such that for every y ∈ X, for all k > m and n > k. Then Proof. Note first that 0 (g) = 1 lim sup n→∞ P n 1/n = 1 c and so 0 < 0 (g) < ∞. Thus g is locally bounded and if it is well-defined on X, then it belongs to H(X) \ H b (X).
Let us show that g is well-defined on X. Let x ∈ X and z ∈ Ω such that x − z < 1/2c. Let y = x − z. Then For the fixed z, g [k] (y) is an entire function of y and its j-homogeneous Taylor polynomial is j (y) = B P k+j (z, . . . , z k , y, . . . , y j ).
Taking into account inequality (1) we have So, using the Stirling asymptotic formula for j!/j j , we have It is easy to check (c.f. Lemma 2 [6]) that lim sup j→∞ P k+j 1/j = lim sup j→∞ P j 1/j .

So the radius of boundedness of g [k]
is equal to 1/c. Thus g [k] is defined at y because y < 1/2c. Since g is well-defined at x.

Symmetric Analytic Functions of Unbounded Type
Let X be a complex Banach space and S be a semigroup of isometric operators on X. A polynomial P ∈ P (X) is said to be S-symmetric if it is invariant with respect to the action of S, that is, P(σ(x)) = P(x) for every σ ∈ S. S-symmetric polynomials from the general point of view were considered in [12,13].
Symmetric polynomials on p , 1 ≤ p < ∞ can be defined as S-symmetric polynomials if S is the group of permutation of the standard basis vectors in p . Due to [20] we know that polynomials form an algebraic basis in the algebra of all symmetric polynomials on p (here p is the smallest integer that is greater than or equal to p), x = (x 1 , x 2 , . . . , x n , . . .) ∈ p . For the case 1 we can use, also, another algebraic basis G k (x) = ∑ n 1 <n 2 <···<n k x n 1 . . . x n k .
In [15] it is proved that G n = 1/n!. Let P n = n!G n . It is easy to check that polynomials P n satisfy the condition of Theorem 1 if Ω is the subspace c 00 of all finite sequences in 1 . In [6] there is a direct proof that the following function belongs to H( 1 ) \ H b ( 1 ). We will prove it for more general situation. For a given positive integer number s we denote G Proof. If s ≥ p, then for every x ∈ p , x ≤ 1 we have So G where 0 ≤ s 1 + · · · + s n ≤ s. So P (s) n satisfies the condition of Theorem 1 for Ω = c 00 .
From Theorem 3 it follows that there exists an entire function of unbounded type g(x) on p such that its Taylor polynomials P (s) n are symmetric. From [4] it follows, also that P (s) n ∈ A sn (X) for s ≥ p. Using similar arguments like in Theorem 3, it is possible to construct examples of separately symmetric and entire block-symmetric functions on a finite Cartesian power of p , 1 ≤ p < ∞. For the definition and properties of separately symmetric analytic functions we refer the reader to [21]. Block symmetric polynomials on a Cartesian power of p were studied in [22,23]. Here we consider the case of ∞ .
Each vector x in ∞ is a bounded number sequence x = (x 1 , x 2 , . . . , x n , . . .). So we can consider symmetric polynomials on ∞ , that is, invariant polynomials with respect to all permutations σ : In [17] it is proved that only constants are symmetric polynomials on ∞ . However, ∞ admits polynomials that are invariant with respect to the subgroup of finite permutations. Such polynomials are called finitely symmetric. A permutation σ : N → N is finite if there is m ∈ N such that σ(n) = n for every n > m. The following example shows that there exists a finitely symmetric entire function of unbounded type on ∞ .

Example 2. Let us fix a presentation of the set of positive integers as a disjoint union
Let U i be a free ultrafilter on N i for every i ∈ N. We denote by It is clear that Q k are nontrivial continuous n-homogeneous polynomials for Let us define otherwise.
Let In other words, if x ∈ Ω, then the function x(n) = x n has just a finite number of different values. By the definition of P n one can check that if x ∈ Ω m , then Since Ω is dense in ∞ , polynomials P n satisfy the condition of Theorem 1. In addition, polynomials P n , n ∈ N are finitely symmetric because all functionals are evidently so.

Discussion, Conclusions and Open Questions
As a result, we can say that various classes of polynomials on Banach spaces support entire analytic functions of unbounded type. However  (see [24]).
In a more general case, let P = (P n ), n ∈ N be a sequence of algebraically independent polynomials on a Banach space X. Let us denote by H bP (X) the minimal subalgebra of H b (X) containing all polynomials in P. Algebras H bP (X) were studied in [16,25]. It is natural to ask: Question 2. Under which conditions on P does there exist an entire functions f of unbounded type on X such that its Taylor polynomials are in H bP (X)?
Note that algebras of symmetric (block-symmetric, supersymmetric) analytic functions of bounded types are partial cases of H bP (X). Let us consider the following example. Example 3. Let X = p for 1 ≤ p ≤ ∞ and P n (x) = x n n , x = (x 1 , . . . , x n , . . .) ∈ p .
If p < ∞, then polynomials in H bP ( p ) support a function of unbounded type, for example, f (x) = ∑ ∞ n=1 P n (x). However, if p = ∞, the function f is no longer defined on the whole space. Let us make a note about products of functions of unbounded type. In [19] it is proved that if f is a function of unbounded type on X of the form (3) and P is a nonzero continuous polynomial on X, then P f ∈ H(X) \ H b (X). Example 1 shows that the product of two functions of unbounded type is not necessarily of unbounded type.

Question 4.
Does a nonzero function f ∈ H b (X) and g ∈ H(X) \ H b (X) exist such that f g ∈ H b (X)?
By the following example we can see that the answer to this question is positive for the real case but we do not know the answer for the complex case. Analytic functions of unbounded type on real Banach spaces were studied in [26].
Example 4. Let f be a function of unbounded type defined by (3) on a real Banach space X. We set g(x) = e ( f (x)) 2 .
Clearly g ∈ H(X) \ H b (X). However, has a bounded range and so is of bounded type.