An infinite dimensional umbral calculus

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by  Dmitri Finkelshtein

An infinite dimensional umbral calculus
Dmitri Finkelshtein
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP,
U.K.; e-mail: [email protected]
Yuri Kondratiev
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1 Fakult¨at fu¨r Mathematik, Universita¨t Bielefeld, 33615 Bielefeld, Germany;
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e-mail: [email protected]
2
n Eugene Lytvynov
a
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP,
J
6 U.K.; e-mail: [email protected]
1
Maria Jo˜ao Oliveira

] Departamento de Ciˆencias e Tecnologia, Universidade Aberta, 1269-001 Lisbon, Por-
A
tugal; CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal;
F
. e-mail: [email protected]
h
t
a
Abstract
m
[ The aim of this paper is to develop umbral calculus on the space D(cid:48) of distributions on Rd,

1 which leads to a general theory of Sheffer sequences on D(cid:48). We define a sequence of monic
v
polynomials on D(cid:48), a polynomial sequence of binomial type, and a Sheffer sequence. We
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2 present equivalent conditions for a sequence of monic polynomials on D(cid:48) to be of binomial
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typeoraSheffersequence,respectively. Ourtheoryhasremarkablesimilaritiestotheclassical
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0 setting of polynomials on R. For example, the form of the generating function of a Sheffer
1. sequence on D(cid:48) is similar to the generating function of a Sheffer sequence on R, albeit the
0 constants appearing in the latter function are replaced in the former function by appropriate
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linear continuous operators. We construct a lifting of a sequence of monic polynomials on R
1
: of binomial type to a polynomial sequence of binomial type on D(cid:48), and a lifting of a Sheffer
v
i sequence on R to a Sheffer sequence on D(cid:48). Examples of lifted polynomial sequences include
X
the falling and rising factorials on D(cid:48), Abel, Hermite, Charlier, and Laguerre polynomials
r
a on D(cid:48). Some of these polynomials have already appeared in different branches of infinite
dimensional analysis and played there a fundamental role.
Keywords: Generating function; polynomial sequence of binomial type; Sheffer
sequence; shift-invariance; umbral calculus.
2010 MSC. Primary: 05A40, 46E50. Secondary: 60H40, 60G55.
1 Introduction
In its modern form, umbral calculus is a study of shift-invariant linear operators acting
on polynomials, their associated polynomial sequences of binomial type, and Sheffer
1
sequences (including Appell sequences). We refer to the seminal papers [27,34,35],
see also the monographs [22,33]. Umbral calculus has applications in combinatorics,
theory of special functions, approximation theory, probability and statistics, topology,
and physics, see e.g. the survey paper [11] for a long list of references.
Many extensions of umbral calculus to the case of polynomials of several, or even
infinitelymanyvariableswerediscussede.g.in[5,10,13,26,30–32,37,38], foralongerlist
ofsuchpapersseetheintroductionto[12]. AppellandSheffersequencesofpolynomials
of several noncommutative variables arising in the context of free probability, Bollean
probability, and conditionally free probability were discussed in [2–4], see also the
references therein.
The paper [12] was a pioneering (and seemingly unique) work in which elements
of basis-free umbral calculus were developed on an infinite dimensional space, more
precisely, onarealseparableHilbertspaceH. Thispaperdiscussed, inparticular, shift-
invariant linear operators acting on the space of polynomials on H, Appell sequences,
and examples of polynomial sequences of binomial type.
In fact, examples of Sheffer sequences, i.e., polynomial sequences with generating
functionofacertainexponentialtype, haveappearedininfinitedimensionalanalysison
numerous occasions. Some of these polynomial sequences are orthogonal with respect
toagivenprobabilitymeasureonaninfinitedimensionalspace, whileothersarerelated
to analytical structures on such spaces. Typically, these polynomials are either defined
on a co-nuclear space Φ(cid:48) (i.e, the dual of a nuclear space Φ), or on an appropriate
subset of Φ(cid:48). Furthermore, in majority of examples, the nuclear space Φ consists of
(smooth) functions on an underlying space X. For simplicity, we choose to work in
this paper with the Gel’fand triple
Φ = D ⊂ L2(Rd,dx) ⊂ D(cid:48) = Φ(cid:48).
Here D is the nuclear space of smooth compactly supported functions on Rd, d ∈ N,
and D(cid:48) is the dual space of D, where the dual pairing between D(cid:48) and D is obtained
by continuously extending the inner product in L2(Rd,dx).
Let us mention several known examples of Sheffer sequences on D(cid:48) or its subsets:
(i) In infinite dimensional Gaussian analysis, also called white noise analysis, Her-
mite polynomial sequences on D(cid:48) (or rather on S(cid:48) ⊂ D(cid:48), the Schwartz space of
tempered distributions) appear as polynomials orthogonal with respect to Gaus-
sian white noise measure, see e.g. [6,14,15,29].
(ii) CharlierpolynomialsequencesontheconfigurationspaceofcountingRadonmea-
sures on Rd, Γ ⊂ D(cid:48), appear as polynomials orthogonal with respect to Poisson
point process on Rd, see [17,20,23].
(iii) Laguerre polynomial sequences on the cone of discrete Radon measures on Rd,
K ⊂ D(cid:48), appear as polynomials orthogonal with respect to the gamma random
measure, see [19,20].
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(iv) Meixner polynomial sequences on D(cid:48) appear as polynomials orthogonal with re-
spect to the Meixner white noise measure, see [24,25].
(v) Special polynomials on the configuration space Γ ⊂ D(cid:48) are used to construct
the K-transform, see e.g. [7,16,18]. Recall that the K-transform determines the
duality between point processes on Rd and their correlation measures. These
polynomials will be identified in this paper as the infinite dimensional analog of
the falling factorials (a special case of the Newton polynomials).
(vi) PolynomialsequencesonD(cid:48) withgeneratingfunctionofacertainexponentialtype
are used in biorthogonal analysis related to general measures on D(cid:48), see [1,21].
Note, however, that even the very notion of a general polynomial sequence on an
infinite dimensional space has never been discussed!
The classical umbral calculus on the real line gives a general theory of Sheffer
sequences. So our aim in this paper is to develop umbral calculus on the space D(cid:48),
which will eventually lead to a general theory of Sheffer sequences on D(cid:48). In fact,
our theory will have remarkable similarities to the classical setting of polynomials on
R. For example, the form of the generating function of a Sheffer sequence on D(cid:48) will
be similar to the generating function of a Sheffer sequence on R, albeit the constants
appearing in the latter function will be replaced in the former function by appropriate
linear continuous operators.
There is a principal point in our approach that we would like to stress. The paper
[12] deals with polynomials on a general Hilbert space H, while the monograph [6]
develops Gaussian analysis on a general co-nuclear space Φ(cid:48), without the assumption
that Φ(cid:48) consists of generalized functions on Rd (or on a general underlying space).
In fact, we will discuss in Remark 7.5 below that the case of the infinite dimensional
Hermitepolynomialsis, inasense, exceptionalanddoesnotrequirefromtheco-nuclear
space Φ(cid:48) any special structure. In all other cases, the choice Φ(cid:48) = D(cid:48) is crucial. Having
said this, let us note that our ansatz can still be applied to a rather general co-nuclear
space of generalized functions over a topological space X, equipped with a reference
measure.
Theoriginsoftheclassicalumbralcalculusareincombinatorics. So,byanalogy,one
canthinkofumbralcalculusonD(cid:48) asakindofspatialcombinatorics. Togivethereader
a better feeling of this, let us consider the following example. Let γ = (cid:80)∞ δ ∈ Γ
i=1 xi
be a configuration. Here δ denotes the Dirac measure with mass at x . We will
xi i
construct (the kernel of) the falling factorial, denoted by (γ) , as a function from Γ to
n
D(cid:48)(cid:12)n. (Here and below (cid:12) denotes the symmetric tensor product.) This will allow us
to define ‘γ choose n’ by (cid:0)γ(cid:1) := 1(γ) . And we will get the following explicit formula
n n! n
which supports this term:
(cid:18) (cid:19)
γ (cid:88)
= δ (cid:12)δ (cid:12)···(cid:12)δ , (1.1)
n xi1 xi2 xin
{i1,...,in}⊂N
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i.e., thesumisobtainedbychoosingallpossiblen-pointsubsetsfromthe(locallyfinite)
set {x } . The latter set can be obviously identified with the configuration γ.
i i∈N
The paper is organized as follows. In Section 2 we discuss preliminaries. In partic-
ular, we recall the construction of a general Gel’fand triple Φ ⊂ H ⊂ Φ(cid:48), where Φ is
0
a nuclear space and Φ(cid:48) is the dual of Φ (a co-nuclear space) with respect to the center
Hilbert space H . We discuss continuous linear operators acting on Φ and Φ(cid:48). We
0
further define the space P(Φ(cid:48)) of polynomials on Φ(cid:48). We equip P(Φ(cid:48)) with a nuclear
space topology and consider the dual space of P(Φ(cid:48)), denoted by F(Φ(cid:48)). The space
F(Φ(cid:48)) has a natural (commutative) algebraic structure with respect to the symmetric
tensor product, (cid:12). We also realize F(Φ(cid:48)) as a space of formal series in tensor powers
of ξ ∈ Φ. We define differentiation on Φ(cid:48) and a family of shift operators, (E(ζ)) .
ζ∈Φ(cid:48)
The shift operators are related to the differentiation operators by Boole’s formula. We
finally define shift-invariant continuous linear operators on P(Φ(cid:48)). We denote the space
of such operators by S(P(Φ(cid:48))).
In Section 3, we give the definitions of a polynomial sequence on Φ(cid:48), a monic
polynomial sequence on Φ(cid:48), and a monic polynomial sequence on Φ(cid:48) of binomial type.
Starting from Section 4, we choose the Gel’fand triple as D ⊂ L2(Rd,dx) ⊂ D(cid:48). We
formulateandprovethefirstmainresultofthepaper, Theorem4.1. Inthistheorem, we
present three equivalent conditions for a monic polynomial sequence to be of binomial
type. The first equivalent condition is that the corresponding lowering operators are
shift-invariant. The second condition gives a representation of each lowering operator
through directional derivatives in directions of delta functions, δ (x ∈ Rd). The third
x
conditiongivestheformofthegeneratingfunctionofapolynomialsequenceofbinomial
type.
To prove Theorem 4.1, we derive two essential results. The first one is an operator
expansion theorem (Theorem 4.7), which gives a description of any operator T ∈
S(P(D(cid:48))) in terms of the lowering operators in directions δ . The second result is an
x
isomorphism theorem (Theorem 4.9): we construct a bijection J : S(P(D(cid:48))) → F(D(cid:48))
such that, for any two operators S,T ∈ S(P(D(cid:48))), we have J(ST) = J(S)(cid:12)J(T). This
implies, in particular, that any two operators from S(P(D(cid:48))) commute.
Next, we define a family of delta operators on D(cid:48) and prove that, for each such
family, there exists a unique monic polynomial sequence of binomial type for which
these delta operators are the lowering operators.
In Section 5, we identify a procedure of the lifting of a polynomial sequence of bino-
mial type on R to a polynomial sequence of binomial type on D(cid:48). Using this procedure,
we identify, on D(cid:48), the falling factorials, the rising factorials, the Abel polynomials, and
the Laguerre polynomials of binomial type. We stress that the polynomial sequences
lifted from R to D(cid:48) form a subset of a (much larger) set of all polynomial sequences of
binomial type on D(cid:48).
In Section 6, we define a Sheffer sequence on D(cid:48) as a monic polynomial sequence
on D(cid:48) whose lowering operators are delta operators. Thus, to every Sheffer sequence,
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therecorrespondsa(unique)polynomialsequenceofbinomialtype. Inparticular, ifthe
corresponding binomial sequence is just the set of monomials (i.e., their delta operators
are differential operators), we call such a Sheffer sequence an Appell sequence. The
second main result of the paper, Theorem 6.2, gives several equivalent conditions for
a monic polynomial sequence to be a Sheffer sequence. In particular, we find the
generating function of a Sheffer sequence on D(cid:48).
In Section 7, we extend the procedure of the lifting described in Section 5 to Sheffer
sequences. Thus, for each Sheffer sequence on R, we define a Sheffer sequence on
D(cid:48). Using this procedure, we recover, on D(cid:48), the Hermite polynomials, the Charlier
polynomials, and the orthogonal Laguerre polynomials.
Finally, in Appendix, we discuss several properties of formal tensor power series.
We also introduce and discuss there the space of ‘Φ-valued’ formal series in tensor
powers of ξ ∈ D.
2 Preliminaries
2.1 Nuclear and co-nuclear spaces
Let us first recall the definition of a nuclear space, for details see e.g. [8, Chapter 14,
Section 2.2]. Consider a family of real separable Hilbert spaces (H ) , where T is
τ τ∈T
(cid:84)
an arbitrary indexing set. Assume that the set Φ := H is dense in each Hilbert
τ∈T τ
space H and the family (H ) is directed by embedding, i.e., for any τ ,τ ∈ T
τ τ τ∈T 1 2
there exists a τ ∈ T such that H ⊂ H and H ⊂ H and both embeddings are
3 τ3 τ1 τ3 τ2
continuous. We introduce in Φ the projective limit topology of the H spaces:
τ
Φ = projlimH .
τ
τ∈T
By definition, the sets {ϕ ∈ Φ | (cid:107)ϕ−ψ(cid:107) < ε} with ψ ∈ Φ, τ ∈ T, and ε > 0 form a
Hτ
system of base neighborhoods in this topology. Here (cid:107)·(cid:107) denotes the norm in H .
Hτ τ
Assume that, for each τ ∈ T, there exists a τ ∈ T such that H ⊂ H , and the
1 2 τ2 τ1
operator of embedding of H into H is of the Hilbert–Schmidt class. Then the linear
τ2 τ1
topological space Φ is called nuclear.
Next, let us assume that, for some τ ∈ T, each Hilbert space H with τ ∈ T is
0 τ
continuously embedded into H := H . We will call H the center space.
0 τ0 0
Let Φ(cid:48) denote the dual space of Φ with respect to the center space H , i.e., the dual
0
pairing between Φ(cid:48) and Φ is obtained by continuously extending the inner product in
H , see e.g. [8, Chapter 14, Section 2.3]. The space Φ(cid:48) is often called co-nuclear.
0
By the Schwartz theorem (e.g. [8, Chapter 14, Theorem 2.1]), Φ(cid:48) = (cid:83) H ,
τ∈T −τ
where H denotes the dual space of H with respect to the center space H . We
−τ τ 0
endow Φ(cid:48) with the Mackey topology—the strongest topology in Φ(cid:48) consistent with the
duality between Φ and Φ(cid:48) (i.e., the set of continuous linear functionals on Φ(cid:48) coincides
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with Φ). The Mackey topology in Φ(cid:48) coincides with the topology of the inductive limit
of the H spaces, see e.g. [36, Chapter IV, Proposition 4.4] or [6, Chapter 1, Section
−τ
1]. Thus, we obtain the Gel’fand triple (also called the standard triple)
Φ = projlimH ⊂ H ⊂ indlimH = Φ(cid:48). (2.1)
τ 0 −τ
τ∈T τ∈T
Let X and Y be linear topological spaces that are locally convex and Hausdorff.
(Both Φ and Φ(cid:48) are such spaces.) We denote by L(X,Y) the space of continuous linear
operators acting from X into Y. We will also denote L(X) := L(X,X). We denote by
X(cid:48) and Y(cid:48) the dual space of X and Y, respectively. We endow X(cid:48) with the Mackey
topology with respect to the duality between X and X(cid:48). We similarly endow Y(cid:48) with
the Mackey topology.
Each operator A ∈ L(X,Y) has the adjoint operator A∗ ∈ L(Y(cid:48),X(cid:48)) (also called
the transpose of A or the dual of A), see e.g. [28, Theorem 8.11.3].
Remark 2.1. Note that, since we chose the Mackey topology on Φ(cid:48), for an operator
A ∈ L(Φ(cid:48)), we have A∗ ∈ L(Φ).
Proposition 2.2. Consider the Gel’fand triple (2.1). Let A : Φ → Φ and B : Φ(cid:48) → Φ(cid:48)
be linear operators.
(i) We have A ∈ L(Φ) if and only if, for each τ ∈ T, there exists a τ ∈ T such
1 2
ˆ
that the operator A can be extended by continuity to an operator A ∈ L(H ,H ).
τ2 τ1
(ii) We have B ∈ L(Φ(cid:48)) if and only if, for each τ ∈ T, there exists a τ ∈ T such
1 2
that the operator Bˆ := B (cid:22) H takes on values in H and Bˆ ∈ L(H ,H ).
−τ1 −τ2 −τ1 −τ2
Remark 2.3. Proposition 2.2 admits a sraightforward generalization to the case of two
Gel’fand triples, Φ ⊂ H ⊂ Φ(cid:48) and Ψ ⊂ G ⊂ Ψ(cid:48), and linear operators A : Φ → Ψ and
0 0
B : Φ(cid:48) → Ψ(cid:48).
Remark 2.4. Part (ii) of Proposition 2.2 is related to the universal property of an
inductive limit, which states that any linear operator from an inductive limit of a
family of locally convex spaces to another locally convex space is continuous if and
only if the restriction of the operator to any member of the family is continuous, see
e.g. [9, II.29].
Proof of Proposition 2.2. (i) By the definition of the topology in Φ, the linear operator
A : Φ → Φ is continuous if and only if, for any τ ∈ T and ε > 0, there exist τ ∈ T
1 1 2
and ε > 0 such that the pre-image of the set
2
{ϕ ∈ Φ | (cid:107)ϕ(cid:107) < ε }
Hτ1 1
contains the set
{ϕ ∈ Φ | (cid:107)ϕ(cid:107) < ε }.
Hτ2 2
But this implies the statement.
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(ii) Assume B ∈ L(Φ(cid:48)). Then, by Remark 2.1, we have B∗ ∈ L(Φ). Hence, for each
τ ∈ T, there exists a τ ∈ T such that the operator B∗ can be extended by continuity
1 2
to an operator Bˆ∗ ∈ L(H ,H ). But the adjoint of the operator Bˆ∗ is Bˆ := B (cid:22) H .
τ2 τ1 −τ1
ˆ
Hence B ∈ L(H ,H ).
−τ1 −τ2
Conversely, assume that, for each τ ∈ T, there exists a τ ∈ T such that the
1 2
operator Bˆ := B (cid:22) H takes on values in H and Bˆ ∈ L(H ,H ). Therefore,
−τ1 −τ2 −τ1 −τ2
Bˆ∗ ∈ L(H ,H ). Denote A := Bˆ∗ (cid:22) Φ. As easily seen, the definition of the operator
τ2 τ1
A does not depend on the choice of τ ,τ ∈ T. Hence, A : Φ → Φ, and by part (i) we
1 2
conclude that A ∈ L(Φ). But B = A∗ and hence B ∈ L(Φ(cid:48)).
In what follows, ⊗ will denote the tensor product. In particular, for a real separable
Hilbert space H, H⊗n denotes the nth tensor power of H. We will denote by Sym ∈
n
L(H⊗n) the symmetrization operator, i.e., the orthogonal projection satisfying
1 (cid:88)
Sym f ⊗f ⊗···⊗f = f ⊗f ⊗···⊗f (2.2)
n 1 2 n n! σ(1) σ(2) σ(n)
σ∈S(n)
for f ,f ,...,f ∈ H. Here S(n) denotes the symmetric group acting on {1,...,n}.
1 2 n
We will denote the symmetric tensor product by (cid:12). In particular,
f (cid:12)f (cid:12)···(cid:12)f := Sym f ⊗f ⊗···⊗f , f ,f ,...,f ∈ H,
1 2 n n 1 2 n 1 2 n
and H(cid:12)n := Sym H⊗n is the nth symmetric tensor power of H. Note that, for each
n
f ∈ H, we have f(cid:12)n = f⊗n.
Starting with Gel’fand triple (2.1), one constructs its nth symmetric tensor power
as follows:
Φ(cid:12)n := projlimH(cid:12)n ⊂ H(cid:12)n ⊂ indlimH(cid:12)n =: Φ(cid:48)(cid:12)n,
τ 0 −τ
τ∈T τ∈T
see e.g. [6, Section 2.1] for details. In particular, Φ(cid:12)n is a nuclear space and Φ(cid:48)(cid:12)n is its
dualwithrespecttothecenterspaceH(cid:12)n. WewillalsodenoteΦ(cid:12)0 = H(cid:12)0 = Φ(cid:48)(cid:12)0 := R.
0 0
The dual pairing between F(n) ∈ Φ(cid:48)(cid:12)n and g(n) ∈ Φ(cid:12)n will be denoted by (cid:104)F(n),g(n)(cid:105).
Remark 2.5. Consider the set {ξ⊗n | ξ ∈ Φ}. By the polarization identity, the linear
span of this set is dense in every space H(cid:12)n, τ ∈ T.
τ
The following lemma will be very important for our considerations.
Lemma 2.6. (i) Let F(n),G(n) ∈ Φ(cid:48)(cid:12)n be such that
(cid:104)F(n),ξ⊗n(cid:105) = (cid:104)G(n),ξ⊗n(cid:105) for all ξ ∈ Φ,
then F(n) = G(n).
(ii) Let Φ and Ψ be nuclear spaces and let A,B ∈ L(Φ(cid:12)n,Ψ). Assume that
Aξ⊗n = Bξ⊗n for all ξ ∈ Φ.
Then A = B.
Proof. Statement (i) follows from Remark 2.5, statement (ii) follows from Proposi-
tion 2.2, (i) and Remarks 2.3 and 2.5.
7
2.2 Polynomials on a co-nuclear space
Below we fix the Gel’fand triple (2.1).
Definition 2.7. A function P : Φ(cid:48) → R is called a polynomial on Φ(cid:48) if
n
(cid:88)
P(ω) = (cid:104)ω⊗k,f(k)(cid:105), ω ∈ Φ(cid:48), (2.3)
k=0
where f(k) ∈ Φ(cid:12)k, k = 0,1,...,n, n ∈ N := {0,1,2,...}, and ω⊗0 := 1. If f(n) (cid:54)= 0,
0
one says that the polynomial P is of degree n. We denote by P(Φ(cid:48)) the set of all
polynomials on Φ(cid:48).
Remark 2.8. For each P ∈ P(Φ(cid:48)), its representation in form (2.3) is evidently unique.
For any f(k) ∈ Φ(cid:12)k and g(n) ∈ Φ(cid:12)n, k,n ∈ N , we have
0
(cid:104)ω⊗k,f(k)(cid:105)(cid:104)ω⊗n,g(n)(cid:105) = (cid:104)ω⊗(k+n),f(k) (cid:12)g(n)(cid:105), ω ∈ Φ(cid:48). (2.4)
Hence P(Φ(cid:48)) is an algebra under point-wise multiplication of polynomials on Φ(cid:48).
We will now define a topology on P(Φ(cid:48)). Let F (Φ) denote the topological direct
fin
sum of the nuclear spaces Φ(cid:12)n, n ∈ N . Hence, F (Φ) is a nuclear space, see e.g. [6,
0 fin
Section5.1]. Thisspaceconsistsofallfinitesequencesf = (f(0),f(1),...,f(n),0,0,...),
where f(k) ∈ Φ(cid:12)k, k = 0,1,...,n, n ∈ N . The convergence in F (Φ) means the
0 fin
uniform finiteness of non-zero elements and the coordinate-wise convergence in each
Φ(cid:12)k.
Remark 2.9. Below we will often identify f(n) ∈ Φ(cid:12)n with
(0,...,0,f(n),0,0,...) ∈ F (Φ).
fin
We define a natural bijective mapping I : F (Φ) → P(Φ(cid:48)) by
fin
n
(cid:88)
(If)(ω) := (cid:104)ω⊗k,f(k)(cid:105), (2.5)
k=0
for f = (f(0),f(1),...,f(n),0,0,...) ∈ F (Φ). We define a nuclear space topology on
fin
P(Φ(cid:48)) as the image of the topology on F (Φ) under the mapping I.
fin
The space F (Φ) may be endowed with the structure of an algebra with respect
fin
to the symmetric tensor product
(cid:32) (cid:33)∞
k
(cid:88)
f (cid:12)g = f(i) (cid:12)g(k−i) , (2.6)
i=0 k=0
where f = (f(k))∞ , g = (g(k))∞ ∈ F (Φ). The unit element of this (commutative)
k=0 k=0 fin
algebra is the vacuum vector Ω := (1,0,0...).
8
By (2.4)–(2.6), the bijective mapping I provides an isomorphism between the alge-
bras F (Φ) and P(Φ(cid:48)), namely, for any f,g ∈ F (Φ),
fin fin
(cid:0) (cid:1)
I(f (cid:12)g) (ω) = (If)(ω)(Ig)(ω), ω ∈ Φ(cid:48).
Let
∞
(cid:89)
F(Φ(cid:48)) := Φ(cid:48)(cid:12)k
k=0
denote the topological product of the spaces Φ(cid:48)(cid:12)k. The space F(Φ(cid:48)) consists of all
sequences F = (F(k))∞ , where F(k) ∈ Φ(cid:48)(cid:12)k, k ∈ N . Note that the convergence in
k=0 0
this space means the coordinate-wise convergence in each space Φ(cid:48)(cid:12)k.
Each element F = (F(k))∞ ∈ F(Φ(cid:48)) determines a continuous linear functional on
k=0
F (Φ) by
fin
∞
(cid:88)
(cid:104)F,f(cid:105) := (cid:104)F(k),f(k)(cid:105), f = (f(k))∞ ∈ F (Φ) (2.7)
k=0 fin
k=0
(note that the sum in (2.7) is, in fact, finite). The dual of F (Φ) is equal to F(Φ(cid:48)), and
fin
the topology on F(Φ(cid:48)) coincides with the Mackey topology on F(Φ(cid:48)) that is consistent
with the duality between F (Φ) and F(Φ(cid:48)), see e.g. [6]. In view of the definition of the
fin
topology on P(Φ(cid:48)), we may also think of F(Φ(cid:48)) as the dual space of P(Φ(cid:48)), equipped
with the Mackey topology on F(Φ(cid:48)) that is consistent with the duality between P(Φ(cid:48))
and F(Φ(cid:48)).
Similarly to (2.6), one can introduce the symmetric tensor product on F(Φ(cid:48)):
(cid:32) (cid:33)∞
k
(cid:88)
F (cid:12)G = F(i) (cid:12)G(k−i) , (2.8)
i=0 k=0
where F = (F(k))∞ , G = (G(k))∞ ∈ F(Φ(cid:48)). The unit element of this algebra is again
k=0 k=0
Ω = (1,0,0,...).
We will now discuss another realization of the space F(Φ(cid:48)). We denote by S(R,R)
the vector space of formal series R(t) = (cid:80)∞ r tn in powers of t ∈ R, where r ∈ R for
n=0 n n
n ∈ N . The S(R,R) is an algebra under the product of formal power series. Similarly
0
to S(R,R), we give the following
Definition 2.10. Each (F(n))∞ ∈ F(Φ(cid:48)) identifies a ‘real-valued’ formal series
n=0
(cid:80)∞ (cid:104)F(n),ξ⊗n(cid:105) in tensor powers of ξ ∈ Φ. We denote by S(Φ,R) the vector space of
n=0
such formal series with natural operations. We define a product on S(Φ,R) by
(cid:32) (cid:33)(cid:32) (cid:33) (cid:42) (cid:43)
∞ ∞ ∞ n
(cid:88) (cid:88) (cid:88) (cid:88)
(cid:104)F(n),ξ⊗n(cid:105) (cid:104)G(n),ξ⊗n(cid:105) = F(i) (cid:12)G(n−i),ξ⊗n , (2.9)
n=0 n=0 n=0 i=0
where (F(n))∞ ,(G(n))∞ ∈ F(Φ(cid:48)).
n=0 n=0
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Remark 2.11. Assume that, for some (F(n))∞ ,(G(n))∞ ∈ F(Φ(cid:48)) and ξ ∈ Φ, both
n=0 n=0
series (cid:80)∞ (cid:104)F(n),ξ⊗n(cid:105) and (cid:80)∞ (cid:104)G(n),ξ⊗n(cid:105) converge absolutely. Then also the series
n=0 n=0
on the right hand side of (2.9) converges absolutely and (2.9) holds as an equality of
two real numbers.
Remark 2.12. Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for (F(n))∞ ∈ F(Φ(cid:48)),
n=0
∞ ∞
(cid:88) (cid:88)
(cid:104)F(n),(tξ)⊗n(cid:105) = tn(cid:104)F(n),ξ⊗n(cid:105), (2.10)
n=0 n=0
the expression on the right hand side of equality (2.10) being the formal power series
in t that has coefficient (cid:104)F(n),ξ⊗n(cid:105) by tn.
According to the definition of S(Φ,R), there exists a natural bijective mapping
I : F(Φ(cid:48)) → S(Φ,R) given by
∞
(cid:88)
(IF)(ξ) := (cid:104)F(n),ξ⊗n(cid:105), F = (F(n))∞ ∈ F(Φ(cid:48)), ξ ∈ Φ. (2.11)
n=0
n=0
The mapping I provides an isomorphism between the algebras F(Φ(cid:48)) and S(Φ,R),
namely, for any F,G ∈ F(Φ(cid:48)),
(cid:0) (cid:1)
I(F (cid:12)G) (ξ) = (IF)(ξ)(IG)(ξ) (2.12)
see (2.8) and (2.9).
Remark 2.13. In view of the isomorphism I, we may think of S(Φ,R) as the dual space
of P(Φ(cid:48)).
Analogously to Definition 2.10 and Remark 2.12, we can introduce a space of Φ-
valued tensor power series.
Definition 2.14. Let (A )∞ be a sequence of operators A ∈ L(Φ(cid:12)n,Φ). Then the
n n=1 n
operators (A )∞ identify a ‘Φ-valued’ formal series (cid:80)∞ A ξ⊗n in tensor powers of
n n=1 n=1 n
ξ ∈ Φ. We denote by S(Φ,Φ) the vector space of such formal series.
Remark 2.15. Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for a sequence (A )∞ as in
n n=1
Definition 2.14
∞ ∞
(cid:88) (cid:88)
A (tξ)⊗n = tnA ξ⊗n
n n
n=1 n=1
is the formal power series in t that has coefficient A ξ⊗n ∈ Φ by tn. Recall that, by
n
Lemma 2.6, (ii), the values of the operator A on the vectors ξ⊗n ∈ Φ(cid:12)n uniquely
n
identify the operator A .
n
In Appendix, we discuss several properties of formal tensor power series.
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