Extensions and derivatives of subsymmetric functions

1 Introduction

A Banach space X with a subsymmetric basis (e_*n*), n∈ℕ is isomorphic to every its subspace spanned on (e_{k_*n*}) for some increasing subsequence k_*n*∈ℕ. Such a self-similarity is a purely infinite-dimensional phenomena and the set of linear mappings e_*n*↦e_{k_*n*} form a semigroup of isometrical operators X. Functions on X that are invariant with respect to this semigroup of operators are called subsymmetric (see the main body of the paper for precise definitions). If a mapping on X is invariant with respect to all permutations of the basis vectors, then it is called symmetric. Clearly that every symmetric function is subsymmetric while the inverse is not so. Symmetric and subsymmetric functions on Banach spaces were considered first in [1, 2]. Further investigations of algebras generated by symmetric and subsymmetric can be found in [3–9] and references therein. For some generalizations of symmetric polynomials we refer the reader to [3, 10–13]. The problem of extensions of symmetric polynomials for various groups of symmetries has been considered in [14]. In [9] it was shown that every subsymmetric polynomial P on ℓ_*p*, 1 ≤ p < ∞, can be extended to a polynomial P𝔄 on ℓp(𝔄), for any infinite well ordered set 𝔄, so that the operator of extension P↦P𝔄 is an isometrical homomorphism of algebras of polynomials. Here we assume that ℓ_*p* is naturally embedded to ℓp(𝔄). Using this result, in [9] it was shown that points of ℓp(𝔄) form continuous complex homomorphisms of the Fréchet algebra of subsymmetric analytic functions on the complex ℓ_*p* that are bounded on bounded subsets. This approach and the subsymmetric “translation” x↦x⊲y introduced in [9] was used for approximations of subsymmetric analytic functions on open subsymmetric subsets of

ℓ_*p* by subsymmetric polynomials in [15]. Here, the map x↦x⊲y is well-defined on X if x is of the form x = (x₁, …, x_*n*, 0, …) by x⊲y = (x₁, …, x_*n*, y₁, y₂, …). In the general case, x⊲y can not be defined as a vector in X but, in [9], it was proved that the map

P↦P(x⊲y):=limn→∞P(x1,…,xn,y1,y2,…)(1)

is a well-defined complex homomorphism on the algebra of subsymmetric polynomials on ℓ_*p* for every x∈X.

In this paper, we considered larger algebras subsymmetric functions on a Banach space X that admit a simultaneous extension to X(𝔄) and found corresponding complex homomorphisms of these algebras. Also, we use the subsymmetric “translation” to define some derivatives of subsymmetric functions that preserve the subsymmetricity, and studied their properties. In Section 2, we give basic definitions and preliminary results concerning algebras of analytic functions on Banach spaces, their spectra and properties of symmetric and subsymmetric polynomials. In Section 3, we considered the question about extendibility of subsymmetric function on a Banach space X with a subsymmetric basis to functions on a larger Banach space which is naturally associated with X. We introduced a Banach space X(𝔄) for any well-ordered infinite subspace 𝔄 and considered a class of functions that are extendible to X(𝔄) so that the operator of extension is linear and multiplicative. Using this extension, we considered some new complex homomorphisms of the algebra of all subsymmetric continuous functions of bounded type on X in Section 4. I particular, we introduced a translation operator preserving the subsymmetry of functions. In Section 5 we introduced derivatives of subsymmetric functions associated with the subsymmetric translation and examined their properties and proved that the constructed derivatives are mixing on the algebra of subsymmetric analytic functions of bounded type on X. This result generalizes the classical results on hypercyclicity of the differentiation operators on spaces of entire functions [16].

2 Definitions and preliminary results

Let X and Y be Banach spaces over field K of real ℝ or complex ℂ numbers. A Y-valued mapping f_*n* on X is said to be n-homogeneous polynomial if there is an n-linear mapping A_{f_*n*} on the Cartesian power X^*n* = X×⋯ × X such that f_*n*(x) = A_{f_*n*}(x, …, x), x∈X. For n = 0, f₀ is a constant in Y. A finite sum of homogeneous polynomials is a polynomial. It is well-known that a polynomial is continuous if and only if it is bounded on bounded subsets of X. Throughout the paper we consider continuous polynomials. The algebra of all K-valued continuous polynomials on X is denoted by P(X).

Let K = ℂ and U be an open subset of a complex Banach space X. A mapping f:U→Y is analytic (or holomorphic) on U if for every x∈U there exists ε>0 such that for every z∈X, ‖z‖ < ε the vector x+z is in U and

f(x+z)=∑n=0∞fn(z)

for some n-homogeneous Y-valued polynomials on X. Let

df(x)(z)=limt→0f(x+tz)-f(x)t

be the Gâteaux derivative of f at x in direction z, and d^*n*f(x)(z) = d(⋯(df(x)(z))(z)⋯ ) be the nth Gâteaux derivative. Then (see [17], p. 152) fn(z)=dnf(x)(z)/n! and so we have the following Taylor series representation for f,

f(x+z)=∑n=0∞dnf(x)(z)n!.

A function f on a complex Banach space X is entire if it is analytic on X. An entire function is of bounded type if it is bounded on each bounded subset of X. The set of all entire complex-valued functions of bounded type on X forms a Fréchet algebra (the notation H_*b*(X)) with respect to the following countable family of norms

‖f‖r=sup‖x‖≤r|f(x)|, r∈ℚ+.(2)

For every x∈X the functional δ_*x*:f↦f(x) is a continuous complex homomorphism (character) of H_*b*(X). The set of (the so-called point evaluation) functionals δ_*x*, x∈X does not exhaust the set of all characters of H_*b*(X) in the general case [18]. More details about polynomials and analytic functions on Banach spaces can be found in [17].

A sequence of vectors (e_*n*), n∈ℕ in a real or complex Banach space X is a Schauder basis of X if it is linearly independent and every x∈X can be uniquely represented by the following convergent series in X

x=∑n=1∞xnen, xn∈ℂ.(3)

We will use notation supp x for the support of x∈X, that is, supp x = {n∈ℕ:x_*n*≠0}. A Schauder basis (e_*n*) is unconditional if Equation 3 converges unconditionally for all x∈X. A Schauder basis (e_*n*) is symmetric if for every injective map (permutation) σ:ℕ → ℕ it is equivalent the basis (e_σ(*n*)), that is, ∑n=1∞xnen converges if and only if ∑n=1∞xneσ(n) converges. A Schauder basis (e_*n*) is called spreading invariant if for every increasing mapping σ:ℕ → ℕ it is equivalent the basis (e_σ(*n*)). Finally, a Schauder basis (e_*n*) is subsymmetric if it is spreading invariant and unconditional. Every symmetric basis is subsymmetric, while the inverse statement is not true (see [7, 19] for details).

Let X be a Banach space with a subsymmetric basis (e_*n*), and n = (n₁, …, n_*k*, …) be an infinite subsequence of natural numbers ℕ. The mapping

𝒞n:x=∑n=1∞xnen↦∑k=1∞xkenk

is a linear isometric operator on X. A subset U⊂X is called subsymmetric if for every x∈U each vector z∈X such that 𝒞n(x)=𝒞m(z) for some n, m⊂ℕ belongs U. For example, an open ball, centered at zero of radius r>0, B(0, r) is a subsymmetric set while the ball B(x₀, r) centered at x₀≠0 is not.

We denote by S¯ the semigroup of operators generated by 𝒞n, n⊂ℕ and by S its sub-semigroup generated by operators 𝒞j,

𝒞j:x=(x1,x2,…)↦(x1,…xj-1,0,xj,xj+1,…).

A function f on a subsymmetric subset U⊂X is subsymmetric if

f(𝒞j(x))=f(x) for every j∈ℕ.

It is known [15] that a continuous function on X is subsymmetric if and only it is S¯-invariant. The algebra of entire subsymmetric functions of bounded type on a complex Banach space X with a subsymmetric Schauder basis is denoted by HbS(X). It is a closed subalgebra of H_*b*(X). If the basis (e_*n*) of X is symmetric, then function f on X is symmetric if

f(∑​n=1∞xneσ(n))=f(∑​n=1∞xnen) for every x=∑​n=1∞xnen∈X.

It is known [5] that the following (so-called standard) subsymmetric polynomials

Pα1,…,αn(x)=∑i1<⋯<inxi1α1⋯xinαn, αj≥⌈p⌉

form a linear basis in the linear space of subsymmetric polynomials on ℓ_*p*, 1 ≤ p < ∞, where ⌈p⌉ is the ceiling of p. That is, every subsymmetric polynomial on ℓ_*p* can be uniquely represented as a finite linear combination of standard polynomials. Also, in [2] it is proved that the power polynomials

Pk(x)=∑n=1∞xnk, k≥⌈p⌉

form an algebraic basis in the algebra of symmetric polynomials on ℓ_*p*, 1 ≤ p < ∞. The Banach space c₀ does not admit any nonconstant subsymmetric (and symmetric) polynomial [5].

For a given complex topological algebra 𝔄, its spectrum, M𝔄 comprises of continuous complex homomorphisms of 𝔄. The spectrum of the algebra of all symmetric analytic functions of bounded types on ℓ₁ was described in [4] combining point evaluation homomorphisms δ_*x*:f↦f(x), x∈X and so-called exceptional homomorphisms ψ_λ that have the following property: ψ_λ(P₁) = λ and ψ_λ(P_*k*) = 0 for k>1. Some complex homomorphisms of the algebra HbS(ℓp) of all subsymmetric analytic functions of bounded types on ℓ_*p* were constructed in [9]. In particular, in [9] it was introduced a complex homomorphism f↦f(x⊲y) of HbS(ℓp), 1 ≤ p < ∞ by the following way

f(x⊲y)=limn→∞f(x1,…,xn,y1,y2,…)

for given x = (x₁, x₂, …), and y = (y₁, y₂, …, ) in X. It is known [9] that the limit exist and

Pα1,…,αm(x⊲y)=∑j=0mPα1,…,αj(x)Pαj+1,…,αm(y), x,y∈ℓp.(4)

3 Extendible subsymmetric functions

Let us recall that a set 𝔄 is well-ordered with respect to an order “≺” if it is linearly ordered and each subset of 𝔄 has a minimal element. For every 1 ≤ p < ∞ we denote by ℓp(𝔄) the Banach space of functions u:𝔄→K, u(γ) = u_γ, such that

‖u‖:=∑γ∈𝔄|u(γ)|p=∑γ∈𝔄|uγ|p<∞.

For a given β∈𝔄 we denote by eβ the following vector in ℓp(𝔄)

𝔢β(γ)={ 1if γ=β,0otherwise.

Using this notations, we can represent any element u∈ℓp(𝔄) as

u=∑γ∈𝔄u(γ)𝔢γ.

Thus, vectors eγ, γ∈𝔄 form a Schauder basis (not necessary countable) in ℓp(𝔄) and numbers u_γ = u(γ) are coordinates of u in this basis (see [9] for details). Also, we will use notation c00(𝔄) for the linear space of finite sequences indexed by 𝔄

c00(𝔄)={ u=∑​j=1nuγj𝔢γj:γ1≺⋯≺γn,n∈ℕ }.

Clearly, c00(𝔄) is dense in ℓp(𝔄).

In [9] it was proved that every function f∈HbS(ℓp) can be extended to a function f𝔄∈Hb(ℓp(𝔄)) so that the extension operator f↦f𝔄 is a continuous homomorphism. More precisely, if

ι:ℕ→𝔄

is an injection of the set of natural numbers with the natural order ≤ to 𝔄 such that n₁ ≤ n₂ implies ι(n₁)≼ι(n₂), then ℓ_*p* can be considered as a closed subspace of ℓp(𝔄) consisting of vectors

(xι(1),xι(2),…,xι(n),…).

For a given standard subsymmetric polynomial P_{α₁, …, α_*n*}, its extension Pα1,…,αn𝔄 is defined as

Pα1,…,αn𝔄(x)=∑γ1≺⋯≺γnxγ1α1⋯xγnαn.(5)

We extend the map P↦P𝔄 by the linearity to an operator on the space of subsymmetric polynomials on ℓ_*p* with values in the space of polynomials on ℓp(𝔄). In [9] it was observed that the operator P↦P𝔄 is isometric, does not depend on ι and can be extended to a topological injective homomorphism of HbS(ℓp) with values in Hb(ℓp(𝔄)). In this section we construct such extension for more general case.

Let X be a Banach space over K with a subsymmetric basis (e_*n*) and 𝔄 be a well-ordered infinite set. We denote by X(𝔄) the completion of c00(𝔄) with respect to the norm

‖ u ‖X(𝔄)=‖ ∑​j∈supp(u)uγj𝔢γj ‖X(𝔄):=‖ ∑​j∈supp(u)uγjej ‖X.

In particular, we define c0(𝔄) as the completion of c00(𝔄) with respect to the norm

‖u‖:=supγ∈𝔄|u(γ)|<∞.

Example 3.1. Let Nm=ℕ1∪…∪ℕm, where each ℕ_*k* is a copy of the set of natural numbers ℕ. We denote by n^(*k*) a typical element in Nm that is a number n in the subset ℕ_*k*. Nm is a well-ordered set with respect to the order ≺ defined as follows: if k<j, then n^(*k*)≺m^(*j*) for any n and m, and if k = j then n^(*k*)≺m^(*j*) if and only if n<m. Similarly, we define

𝔑∞=⋃m=1∞ℕm

with the same order. If u∈𝔑2, then

‖u‖X(N2)=limm→∞limn→∞‖u1(1)e1+⋯+un(1)en+u1(2)en+1+⋯+ um(2)en+m‖X.

The Banach space X can be naturally embedded into X(𝔄) by the following way. Let (γ_*i*), i∈ℕ be a subsequence in 𝔄 such that γ_*i*≺γ_*j* if i<j. Then

J(γi):x=∑n=1∞xnen∋X↦∑n=1∞xneγn∋X(𝔄)

is an isometric embedding.

Let f be a subsymmetric function on X with values in a Banach space Y and f₀ its restriction to the dense subspace c₀₀⊂X. Then f₀ can be extended to a function f0𝔄 on c00(𝔄) by

f0𝔄(∑​γi∈supp(u)uγi𝔢γi)=f0(∑​γi∈supp(u)uγiei).

Clearly that f0𝔄 is an extension of f₀ and it does not depend on the embedding J.

Definition 3.2. A subsymmetric function f:X→Y is called 𝔄-extendible if for every u∈X(𝔄) there is a unique vector a∈Y such that for every sequence of vectors vn∈c00(𝔄) converging to u we have that

limn→∞f0𝔄(vn)=a.

In this case we set f𝔄(u)=a. We say that a subsymmetric map on X is extendible if it is Nm-extendible for every m∈ℕ.

Evidently, the extension of standard subsymmetric polynomials by Formula 5 agrees with Definition 3.2.

The following example shows that any Banach space with a subsymmetric basis admits many subsymmetric extendible functions.

Example 3.3. Let ξ be a scalar-valued continuous function with a compact support on the set of real nonnegative numbers and P_{α₁, …, α_*m*} a standard subsymmetric polynomial on ℓ₁. We define the following function g on a Banach space X with a subsymmetric basis (e_*n*) as

g(x)=limn→∞ξ(Pα1,…,αm(|x1|,…,|xn|,0,…))=limn→∞ξ∘Pα1,…,αm∑​nk=1|xk|ek.

Since for every fixed n, the vector (|x₁|, …, |x_*n*|, 0, …) is in ℓ₁, the value P_{α₁, …, α_*m*}(|x₁|, …, |x_*n*|, 0, …) is well-defined. The sequence

Pα1,…,αm(|x1|,…,|xn|,0,…)

is nonnegative and increasing. Thus, either the sequence has a limit in the support of ξ or only a finite number of elements of this sequence are in the support of ξ. For the both cases g(x) is well-defined. Since P_{α₁, …, α_*m*} is subsymmetric and extendible, the function g is subsymmetric and extendible.

The space c₀ does not admit nonconstant subsymmetric polynomials but we can construct a “tropical” version of such polynomials (replacing the addition by max and the multiplication by the addition cf. [20]).

Example 3.4. For some fixed real numbers n₁, …, n_*m* we define the following map on the real Banach space c₀:

Kn1,…,nm(x)=maxi1<⋯<im(n1xi1+⋯+nmxim,0).

Clearly that K_{n₁, …, n_*m*} is a subsymmetric function. Moreover, it is easy to verify that K_{n₁, …, n_*m*} is a Lipschitz function. Also, we can define another example of a Lipschitz subsymmetric function that works in both real and complex cases:

K~n1,…,nm(x)=maxi1<⋯<im(n1|xi1|+⋯+nm|xim|,0).

Here n₁, …, n_*m* are real and x is in real or complex c₀.

Theorem 3.5.

• (i) Every subsymmetric function in HbS(X) is 𝔄-extendible for every well-ordered set 𝔄.
• (ii) Functions K_{n₁, …, n_*m*} and K~n1,…,nm defined in Example 3.4 are 𝔄-extendible for every well-ordered set 𝔄.

Proof: Item (i) was proved in [9] using for extension Formula 5. To prove (ii), we define

Kn1,…,nm𝔄(u)=Kn1,…,nm𝔄(∑​γ∈𝔄uγ𝔢γ)=maxγ1≺⋯≺γm(n1uγ1+⋯+nmuγm,0).

Since ‖u‖=maxγ∈𝔄|uγ|,

and so Kn1,…,nm𝔄 is well-defined. The same works for K~n1,…,nm.

Note that if f is symmetric, then its value does not depend on the order of basis vectors and we can construct the extension f𝔄 without of using the ordering of 𝔄.

Proposition 3.6. Let X be a Banach space with a symmetric basis. Then every symmetric function on X is 𝔄-extendible for every set 𝔄.

Proof: For every u=∑i=1∞uγieγi∈𝔄(X) we set

f𝔄(u)=f(∑​i=1∞uγiei).

Theorem 3.5 and Proposition 3.6 show that the set of 𝔄-extendible functions is large. On the other hand, Proposition 3.6 gives a general result for symmetric functions, while the 𝔄-extendibility of subsymmetric functions is proved in Theorem 3.5 for special cases of functions using their explicit representation. In the next section, we show how to define the transformation f(·)↦f(·⊲y) for 𝔄-extendible functions. But it leads to the existence of the limit (Equation 1) and suggests to us that the condition of continuity can be important for the 𝔄-extendibility. However, we do not know how to prove it in the general case, even for spaces ℓ_*p* and c₀. Thus, we have the following open question: Is every continuous subsymmetric function on X 𝔄-extendible for some well-ordered set 𝔄?

4 Characters of algebras of subsymmetric functions

Let CbS(X) be the algebra of all subsymmetric continuous functions on X that are bounded on bounded subsets of X and 𝔄(X) any its closed subalgebra. Clearly, every point x∈X generates the point evaluation character δ_*x* on 𝔄(X) by formula δ_*x*(f) = f(x).

In [9] it was introduced a complex homomorphism f↦f(x⊲y), x, y∈ℓ_*p* on HbS(ℓp) as

f(x⊲y)=limn→∞f(x1,…,xn,y1,y2,…).

We consider how to define such a complex homomorphism for more general cases. If x∈X is of the form x = (x₁, …, x_*m*, 0, 0, …) and any y∈X there is a vector z∈X such that

f(z)=f(x⊲y):=limn→∞f(x1,…,xn,y1,y2,…)

for every subsymmetric function f. It is enough to set z = (x₁, …, x_*m*, y₁, y₂, …). But in the general case, when the support of x is infinite, such a vector z does not exist, that is, the functional f↦f(x⊲y) is not a point evaluation functional. However, for many cases, we can find a vector u∈X(𝔄) such that f𝔄(u)=f(x⊲y).

Proposition 4.1. Let f be an 𝔑2-extendible function on a Banach space X with a subsymmetric basis. For every x and y in X there exists a vector u∈X(𝔑2) such that f𝔑2(u)=f(x⊲y).

Proof: Since 𝔑2=ℕ1∪ℕ2, where ℕ₁ and ℕ₂ are two copies ℕ, the restriction of the order “≺” to each ℕ₁ and ℕ₂ coincides with the natural order “ < ” on ℕ, and if k∈ℕ₁ and m∈ℕ₂, then k≺m. Then we can put

u=∑i∈ℕ1,i=1∞xiei+∑j∈ℕ2,j=1∞yjej.

From the extendibility of f and the definition of f𝔄 it follows that

f(x⊲y)=limn→∞f(x1,…,xn,y1,y2,…)=limn→∞f𝔄(∑​i∈ℕ1,i=1nxi𝔢i+∑​j∈ℕ2,j=1∞yj𝔢j)=f𝔄(u).

By the similar way, we can get a generalization of Proposition 4.1 for multiply applications of “⊲”.

Proposition 4.2. Let f be an 𝔑∞-extendible function on a Banach space X with a subsymmetric basis. For every finite sequence of vectors x⁽¹⁾, x⁽²⁾, …, x^(*m*) in X there exists a vector u∈X(𝔑∞) such that

f𝔑∞(u)=f(x(1)⊲⋯⊲x(m))=limn1,…,nm→∞f(x1(1),…,xn1(1),…,x1(m),…,xnm(m)).

Proof: Since 𝔑∞=⋃k=1∞ℕk, it is enough to set

u=∑i∈ℕ1,i=1∞xi(1)ei+⋯+∑j∈ℕm,j=1∞xj(m)ej.

Let us denote x•y = (x₁, y₁, x₂, y₂, …). The following theorem shows that δ_*x*⊲*y* is not a point evaluation functional in the general case.

Theorem 4.3. Let X be the complex ℓ_*p* for some 1 ≤ p < ∞, vectors x and y in ℓ_*p* such that x has an infinite support and y≠0. Then there is no z∈ℓ_*p* such that δ_*z* = δ_*x*⊲*y*.

Proof: If such a vector z∈ℓ_*p* exists, then P(z) = P(x⊲y) for every subsymmetric polynomial P. In particular, for every power symmetric polynomial P_*k*,

Pk(z)=Pk(x⊲y)=Pk(x•y)=Pk(x1,y1,x2,y2,…).

From [21] it follows that z is equal to x•y up to permutations of nonzero coordinates. That is, nonzero coordinates of z and x•y form equal multisets. Without loss of generality, we may assume that y₁≠0. There is no more than a finite number N of coordinates x_*i* of x such that x_i₁ = ⋯ = x_{i_*N*} = y₁. Also, there is a finite number N′≥1 of coordinates y_*j* of y such that yj1=⋯=yjN′. Thus, the number of coordinates z_*l* of z that are equal to y₁ is N+N′>N and we denote by K the maximal number such that z_*K* = y₁. Let a be a number which is equal to a coordinate of x and satisfy the following condition: the minimal number k such that either x_*k* = a or x_*k* = −a is greater than max(N, K). Such a number exists because x has infinitely many nonzero coordinates and each of them has a finite multiplicity. Let m_*a* be the multiplicity of a in z. We consider the following multisets:

M1,2(x⊲y)={xiyj:i<j}∪{xixj:i<j}∪{yiyj:i<j},M1,2(z)={zizj:i<j}.

We claim that M_{1, 2}(x⊲y)≠M_{1, 2}(z). Indeed, the multiplicity of y1a2 in M_{1, 2}(x⊲y) is equal to m_*a*N while the multiplicity of y1a2 in M_{1, 2}(z) is equal to ma(N+N′)>maN. Therefore there is a number m∈ℕ such that

Pm(M1,2(x⊲y))≠Pm(M1,2(z)).

But

Pm(M1,2(x⊲y))=Pm,2m(x⊲y)

and

Pm(M1,2(z))=Pm,2m(z).

A contradiction.

Let CbSext(X) be the subalgebra of CbS(X) consisting of extendible functions.

Proposition 4.4. CbSext(X) is a closed subalgebra of CbS(X).

Proof: Let f_*n* be a sequence of functions in CbSext(X) such that fn→f∈CbS(X) uniformly on every bounded subset of X. Let u∈ℓp(𝔄) and (v_*n*) be a sequence of vectors in c00(𝔄) approaching u. The sequence (v_*n*) is bounded and we may assume that ‖u‖ < r and for every n, ‖v_*n*‖ < r for some r>0. Let f0𝔄 be the extension of f to c00(𝔄). For every ε>0 there is a number n₀ such that for every n>n₀, |f(y)−f_*n*(y)| < ε for all y∈X with ‖y‖ ≤ r. Thus, for every n>n₀,

because for each vm∈c00(𝔄) there is y_*m*∈X such that ‖v_*m*‖ = ‖y_*m*‖ and g(y_*m*) = g(v_*m*) for every subsymmetric function g. Also, for every ε_*n*>0 there is m₀ such that for every m>m₀ and k>m₀,

Therefore, the limit

limm→∞f0𝔄(vm)

exists and so f is extendible.

Corollary 4.5. For every closed subalgebra 𝔄(ℓp) of CbSext(ℓp) containing PS(ℓp) and for any point u∈ℓp(𝔄) there is a character δu:f↦f𝔄(u) on 𝔄(ℓp) which is not a point evaluation functional.

Remark 4.6. For a given h∈X let

vn(h)=‖ h ‖(h⊲h⊲⋯⊲h)︷n‖ h⊲h⊲⋯⊲h︸n ‖.

Clearly, ‖v_*n*(h)‖ = ‖h‖ and so δ_{v_*n*(h)} is a character on the Banach algebra CbS(BX(0,r)¯) for every r≥h. Since the spectrum of a Banach algebra is a compact topological space with respect to the Gelfand topology, the sequence δ_{v_*n*(h)} has cluster points in the spectrum. For a given free ultrafilter U in the set of natural numbers ℕ we denote by ψ(h)U the following character on CbS(X)

ψ(h)U(f)=limn→∞f(vn(h)).

Example 4.7. Let X = ℓ_*p* for some 1 ≤ p < ∞ and h = (λ, 0, 0, …) for some λ∈ℂ. Then

vn(h)=|λ|(λ,…,λ︷n,0,…)|λ|np=(λnp,…,λnp︸n,0,0,…).

Thus, we can see that the restriction ψ_(*h*) of ψ(h)U to the algebra HbS(ℓp) does not depend on U and coincides with the character ψ_λ.

If X = c₀, then ‖ h⊲h⊲⋯⊲h︸n ‖=||h|| and so vn(h)=h⊲h⊲⋯⊲h︸n. Thus, for every subsymmetric extendible functions f on c₀, we have

ψ(h)U(f)=limUf(h⊲h⊲⋯⊲h︸n).

Note that the concept of subsymmetric functions and the idea of 𝔄-extension can not be generalized to abstract Banach spaces in general. Indeed, classical spaces L₁[0, 1] and C[0, 1] do not admit unconditional (and so subsymmetric) bases [19, p. 24]. For these cases, we cannot guarantee the continuity of operators 𝒞n. Spaces L_*p*, 1 < p < ∞, have subsymmetric bases.

5 Derivatives

Let us introduce the following derivatives associated with the subsymmetric translation operators.

Definition 5.1. A subsymmetric function f on X with values in some Banach space is called

• left subsymmetrically differentiable (or ∂_*L*-differentiable) in direction h∈X at x∈X if there exists the value

∂Lf(x)(h):=limt→0f(th⊲x)-f(x)t;

• right subsymmetrically differentiable (or ∂_*R*-differentiable) in direction h∈X at x∈X if there exists the value

∂Rf(x)(h):=limt→0f(x⊲th)-f(x)t;

• and middle subsymmetrically differentiable (or ∂_*M*-differentiable) in direction h∈X (or ∂_*M*-differentiable) at x, y∈X if there exists the value

∂Mf(x,y)(h):=limt→0f(x⊲th⊲y)-f(x⊲y)t.

Here t is in the field of scalars K.

Theorem 5.2. Let f be N3 extendible. There are well-ordered sets 𝔄R, 𝔄L, and 𝔄M such that for all x, y and h in X there are x_*R*, hR∈𝔄R, x_*L*, hL∈𝔄L, and x_*M*, y_*M*, hM∈𝔄M, such that for every subsymmetric function f with values in some Banach space

∂Rf(x)(h)=df𝔄R(xR)(hR),

∂Lf(x)(h)=df𝔄L(xL)(hL),

∂Mf(x,y)(h)=df𝔄M(xM+yM)(hM)

in the means that if the derivatives ∂_*R*f(x)(h), ∂_*L*f(x)(h) and ∂_*M*f(x, y)(h) exists then the Gâteaux derivatives df𝔄R(xR)(hR), df𝔄L(xL)(hL), and df𝔄M(xM+yM)(hM) exist.

Proof: Let 𝔄R=N2=ℕ∪ℕ with the natural order. Thus, X(𝔄R)=X×X. We set x_*R* = (x, 0)∈X×X and h_*R* = (0, h)∈X×X. Then

f(x⊲h)=f𝔄R(xR+hR)

for every subsymmetric function f. Thus,

limt→0f(x⊲th)-f(x)t=limt→0f𝔄R(xR+thR)-f𝔄R(xR)t

providing that the first limit exists. So, in this case, ∂Rf(x)(h)=df𝔄R(xR)(hR).

Similarly, setting 𝔄L=ℕ∪ℕ, x_*L* = (0, x), and h_*l* = (h, 0) we have

f(h⊲x)=f𝔄L(xL+hL).

By the same reason as above, ∂Lf(x)(h)=df𝔄L(xL)(hL).

Finally, let 𝔄M=N3 with the natural order. Then X(𝔄M)=X×X×X. We set x_*M* = (x, 0, 0), y_*M* = (0, 0, y), and h_*M* = (0, h, 0). Clearly,

f(x⊲h⊲y)=f𝔄M(xM+yM+hM).

Thinking as above, we can deduce ∂Mf(x,y)(h)=df𝔄M(xM+yM)(hM).

Let ∂ be the common symbol for ∂_*R* and ∂_*L*.

Corollary 5.3. If f and g are ∂-differentiable K-valued functions, then the Leibniz rule holds:

∂(fg)(x)(h)=g(x)∂(f)(x)(h)+f(x)∂(g)(x)(h),

if f and g are ∂_*M*-differentiable K-valued functions, then

∂(fg)(x,y)(h)=g(x⊲y)∂(f)(x,y)(h)+f(x⊲y)∂(g)(x,y)(h), x,y∈X.

Proof: From the Leibniz rule for the Gâteaux derivative and Theorem 5.2 it follows that

∂R(fg)(x)(h)=d(f𝔄Rg𝔄R)(xR)(hR)=g𝔄R(xR)d(f𝔄R)(xR)(hR)+f𝔄R(xR)d(g𝔄R)(xR)(hR)=g(x)∂R(f)(x)(h)+f(x)∂R(g)(x)(h).

The same works for ∂_*R* and for ∂_*M*.

Corollary 5.4. Let K = ℂ. For any fixed h∈X the operators f(·)↦∂f(·)(h) is continuous on the space HbS(X).

Let us consider the case when X = ℓ₁.

Proposition 5.5. For every standard polynomial P_{α₁, …, α_*m*} on ℓ₁ we have

∂LPα1,…,αm(x)(h)={ P1(h)Pα2,…,αm(x)if α1=1,0otherwise;

∂RPα1,…,αm(x)(h)={ P1(h)Pα1,…,αm−1(x)if αm=1,0otherwise

for every x and h in ℓ₁.

Proof: Applying Formula 4 we have

Pα1,…,αm(th⊲x)-Pα1,…,αm(x)=∑j=1mPα1,…,αj(th)Pαj+1,…,αn(x)=∑j=1mtα1+⋯+αjPα1,…,αj(h)Pαj+1,…,αn(x).

Thus,

limt→0Pα1,…,αm(th⊲x)-Pα1,…,αm(x)t≠0

only if α₁+⋯+α_*j* = 1. It is possible only if j = 1 and α₁ = 1. For this case we have the required equality. The formula for ∂_*R*P_{α₁, …, α_*m*}(x)(h) can be computed similarly.

Proposition 5.6. For every standard polynomial P_{α₁, …, α_*m*} on ℓ₁

∂MPα1,…,αm(x,y)(h)=P1(h)∑{k:αk=1}Pα1,…,αk-1(x)Pα1,…,αk+1(y).

Proof: Applying Formula 4 twice we obtain

Pα1,…,αm(x⊲th⊲y)-Pα1,…,αm(x⊲y)=∑j=0n-1Pα1,…,αj(x)∑i=1m-jtαj+1+⋯+αj+iPαj+1,…,αj+i(h)Pαj+i,…,αm(y).

Thus,

limt→0Pα1,…,αm(x⊲th⊲y)-Pα1,…,αm(x⊲y)t≠0

only if α_*j*+1+⋯+α_*j*+*i* = 1 for some i and j. It is possible only if α_*k* = 1 for some k = j+1 and i = 1. Thus,

∂MPα1,…,αm(x,y)(h)=∑{k:αk=1}P1(h)Pα1,…,αk-1(x)Pα1,…,αk+1(y).

Theorem 5.7. Let f∈HbS(ℓ1) and h∈ℓ₁. Then the following identities hold

∂Lf(x)(h)=limt→0(tψ(h)⊲δx)(f)-f(x)t(6)

∂Rf(x)(h)=limt→0(δx⊲tψ(h))(f)-f(x)t(7)

∂Mf(x,y)(h)=limt→0(δx⊲tψ(h)⊲δy)(f)-f(x)t.(8)

Proof: To prove equality (Equation 6) we observe that for every standard polynomial P_{α₁, …, α_*m*},

(tψ(h)⊲δx)(Pα1,…,αm)−Pα1,…,αm(x)=∑j=1mtψ(h)(Pα1,…,αj)Pαj+1,…,αm(x) { tP1(h)Pα2,…,αm(x)if α1=1,0otherwise.

Comparing the last formula with Proposition 5.5 we can see that Equation 6 holds for standard polynomials. But standard polynomials form a Schauder basis in HbS(ℓ1). Thus identity Equation 6 holds for every functions in HbS(ℓ1). The same works for Equations 7 and 8.

Let us recall that a continuous operator T on a Fréchet space X is hypercyclic if there exists a vector x∈X such that the set {x, T(x), …, T^*n*(x), …} is dense in X. A continuous operator X is said to be topologically transitive if for every pair of nonempty open subsets U and V in X there is a number n such that

Tn(U)⋂V≠∅.

Finally, T is mixing if for every pair U, V of nonempty open subsets of X there exists a number N such that

Tn(U)⋂V≠∅ for all n≥N.

Clearly that if T is mixing then it is topologically transitive [22, p. 10]. It is known that the hypercyclicity of T is equivalent to its topological transitivity providing that X is separable. Also, the following known result (see e.g. [22, p. 71]) gives an important conditions on T to be mixing on X.

Theorem 5.8. (Kitai's Criterion). If there are dense subsets X₀, Y₀⊂X and a map S:Y₀→Y₀ such that, for any x∈X₀ and y∈Y₀,

• (i) T^*n*(x) → 0 as n → ∞,
• (ii) S^*n*(y) → 0 as n → ∞,
• (iii) T°S(y) = y,

then T is mixing.

Theorem 5.9. Let h∈ℓ₁ be such that P₁(h)>1. Then operators

∂L·(h):f↦∂Lf(·)(h) and ∂R·(h):f↦∂Rf(·)(h)

are mixing on HbS(ℓ1).

Proof: Consider the following mappings on standard subsymmetric polynomials

SL(Pα1,…,αm)=1|P1(h)|P1,α1,…,αm

and

SR(Pα1,…,αm)=1|P1(h)|Pα1,…,αm,1.

Since standard polynomials form a linear basis in the algebra of subsymmetric polynomials PS(ℓ1), we can extend mappings S_*L* and S_*R* to linear operators on PS(ℓ1) and we will use the same notations for the extensions.

The algebra PS(ℓ1) is a dense subspace in HbS(ℓ1). Clearly,

∂LnP(·)(h)→0 and ∂RnP(·)(h)→0 as n→∞,

and

(∂L(·)(h)∘SL)(P)=P and (∂R(·)(h)∘SR)(P)=P

for every P∈PS(ℓ1). Also, It is easy to check that ‖P_{1,_α1, …, α_*m*}‖ ≤ ‖P_{α₁, …, α_*m*}‖ and ‖P_{α₁, …, α_*m*, 1}‖ ≤ ‖P_{α₁, …, α_*m*}‖. Thus, for every r>0,

‖SLn(P)‖r≤1|P1(h)|n‖P‖r and ‖SRn(P)‖r≤1|P1(h)|n‖P‖r,

that is,

SLn(P)→0 and SRn(P)→0 as n→∞

for every P∈PS(ℓ1). Therefore, operators ∂_*L*(·)(h) and ∂_*R*(·)(h) satisfy Kitai's Criterium.

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Footnotes

Edited by: Alvise Sommariva, University of Padua, Italy

Reviewed by: Erdem Toksoy, Ondokuz Mayis University, Türkiye

Mohamed Morjane, Ibn Tofail University, Morocco