# Some Applications of the Spectral Theory for the Integral Transform Involving the Spectral Representation

Authors: Hyun Soo Chung; Seung Jun Chang
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

In many previous papers, an integral transformℱγ,β was just considered as a transform on appropriate function spaces. In this paper we deal with the integral transform as an operator on a function space. We then apply various operator theories to ℱγ,β. Finally we give an application for the spectral representation of a self-adjoint operator which plays a key role in quantum mechanics.


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## Body

## 1. Introduction and Definitions

It is a well-known fact that the spectral theory is one of the main subjects of modern functional analysis and applications. It arises quite naturally in connection with the problems of solving equations. In particular, the spectrum of bounded linear operators on a normed and Banach space is the most important concept to understand the spectral theories. Furthermore, the spectral representation is used widely to apply theories in many fields.LetC0[0,T] denote one-parameter Wiener space, that is, the space of continuous real-valued functions x on [0,T] with x(0)=0. Let ℳ denote the class of all Wiener measurable subsets of C0[0,T], and let m denote Wiener measure. (C0[0,T]0,ℳ,m) is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional F by
(1.1)∫C0[0,T]F(x)dm(x).A subsetB of C0[0,T] is said to be scale-invariant measurable provided ρB is ℳ measurable for all ρ≥0, and a scale-invariant measurable set N is said to be a scale-invariant null set provided m(ρN)=0 for all ρ≥0. A property that holds except on a scale-invariant null set is said to hold scale invariant almost everywhere (s-a.e.) [1].In [2], for each pair of nonzero complex numbers γ and β, Lee introduced an integral transform ℱγ,β(F) of a functional F on abstract Wiener space. For certain values of the parameters γ and β and for certain classes of functionals, the Fourier-Wiener transform [3], the modified Fourier-Wiener transform [4], the Fourier-Feynman transform, and the Gauss transform are special cases of Lee's integral transform ℱγ,β. These transforms play an important role in studying stochastic processes and functional integrals on infinite dimensional spaces.In many papers [5–9], the authors studied the integral transform with related topics of functionals in several classes. Recently, in [10, 11], the authors established the existence of a generalized integral transform ℱγ,β,h via a Gaussian process of functionals in a class 𝒮α and then obtained various relationships involving the convolution product and the first variation of them.However, in all previous works, the authors have consideredℱγ,β just as a transform of a functional F on function space and then they obtained various relationships as seen in Remark 2.5 below. In this paper, we consider the integral transform ℱγ,β as an operator on a Banach space. We then apply various operator theories to ℱγ,β. Furthermore, we obtain various theorems of the spectral theory for ℱγ,β involving the spectral representation.Now we are ready to define the integral transform  (IT) of a functional onK≡K0[0,T], the space of all complex-valued continuous functions defined on [0,T] which vanish at t=0, used in [5–9, 12].Definition 1.1.
LetF be a functional defined on K. For each pair of nonzero complex numbers γ and β, the IT ℱγ,β(F) of F is defined by
(1.2)ℱγ,β(F)(y)≡∫C0[0,T]‍F(γx+βy)dm(x),y∈K,
if it exists.Remark 1.2.
Whenγ=1 and β=i, ℱ1,i is the Fourier-Wiener transform introduced by Cameron and Martin in [3], and when γ=2 and β=i, ℱ2,i is the modified Fourier-Wiener transform used by Cameron and Martin in [4].Forv∈L2[0,T] and x∈C0[0,T], let 〈v,x〉 denote the Paley-Wiener-Zygmund (PWZ) stochastic integral. Then we have the following assertions.  (1)
For eachv∈L2[0,T], 〈v,x〉 exists for a.e. x∈C0[0,T].(2)
Ifv∈L2[0,T] is a function of bounded variation on [0,T], 〈v,x〉 equals the Riemann-Stieltjes integral ∫0T‍v(t)dx(t) for s-a.e. x∈C0[0,T].(3)
The PWZ stochastic integral〈v,x〉 has the expected linearity property.(4)
The PWZ stochastic integral〈v,x〉 is a Gaussian process with mean 0 and variance ∥v∥22. For a more detailed study of the PWZ stochastic integral, see [5, 10, 12–14].Let{α1,α2,…} be an orthonormal set in L2[0,T]. The following formula is a well-known Wiener integration formula. Let h:ℝn→ℝ be Borel measurable and let H(x)=h(〈α1,x〉,…,〈αn,x〉). Then
(1.3)∫C0[0,T]H(x)dm(x)=(2π)-n/2∫ℝn‍h(u1,…,un)·exp{-∑j=1n‍uj22}du1⋯dun
in the sense that if either side of (1.3) exists, both sides exist and equality holds.


## 2. Some Results as a Transform

In this section, we establish the existence of the IT of functionals in a class𝒜n(2), as seen in Theorem 2.4 below. We then give the inverse IT of our IT. Finally we state possible relationships for the IT with related topics.We start this section by describing the class of functionals that we work with in this paper. Let{α1,α2,…} be an orthonormal set in L2[0,T]. Let 𝒜n(2) be the space of all functionals F:K→ℂ of the form
(2.1)F(x)=f(〈α1,x〉,…,〈αn,x〉)
for some positive integer n (throughout this paper, n is fixed), where f(u1,…,un) is an entire function of the n complex variables u1,…,un and
(2.2)∫ℝn‍|f(u→)|2du→<∞.
To simplify the expressions, we use the following notation:
(2.3)f(〈α→,x〉)≡f(〈α1,x〉,…,〈αn,x〉).Remark 2.1.
For anyF and G in 𝒜n(2), we can always express F by (2.1) and G by
(2.4)G(x)=g(〈α1,x〉,…,〈αn,x〉)≡g(〈α→,x〉)
using the same positive integer n, where g is an entire function and
(2.5)∫ℝn‍|g(u→)|2du→<∞.Note that𝒜n(2) is a very rich class of functionals because 𝒜n(2) contains the Schwartz class 𝒮(ℝn). These functionals are of interest in Feynman integration theories and quantum mechanics.Now, we will introduce a notation. It will be convenient to express for the type of limiting integral that occurs in our paper. For appropriate functionsf and g on ℝn, if
(2.6)limA→∞∫ℝn‍|∫-AA‍⋯(n)∫-AA‍f(u→;v→)du→-g(v→)|2dv→=0,
then we say that the integral
(2.7)∫ℝn‍f(u→;v→)du→=g(v→)
is to be interpreted as an L2-limiting integral, see [15].The following lemma is due to Cameron and Storvick in [15, Lemma H].Lemma 2.2.
Letγ be nonzero complex number with Re(1/γ2)≥0. For f∈L2(ℝn), let
(2.8)g(v→)=(2πγ2)-n/2∫ℝn‍f(u→)exp{-∑j=1n‍(uj-vj)22γ2}du→.
Then g∈L2(ℝn), and
(2.9)∥g∥2≤∥f∥2.
If Re(1/γ2)=0, the integral is to be interpreted as an L2-limiting integral; moreover, in this case
(2.10)∥g∥2=∥f∥2.The following lemma is very useful in establishing the existence of the IT.Lemma 2.3.
Letf,g, and γ be as in Lemma 2.2 and let β be a nonzero complex number with |β|≥1. Let h(u→)=g(βu→). Then h∈L2(ℝn) and
(2.11)∥h∥2≤∥g∥2≤∥f∥2.
If Re(1/γ2)=0 and |β|=1, the integral is to be interpreted as an L2-limiting integral; moreover, in this case
(2.12)∥h∥2=∥g∥2=∥f∥2.Proof.
First note that for all nonzero real numbersβ with |β|≥1, it follows that
(2.13)∥h∥22=∫ℝn‍|h(u→)|2du→=∫ℝn‍|g(βu→)|2du→=1|β|n∫ℝn‍|g(u→)|2du→.
But each side of the above expression is an analytic function of β throughout the region {β∈ℂ:|β|≥1}. Hence, by the uniqueness theorem for analytic functions, the above equality holds for all β with {β∈ℂ:|β|≥1}. Since |β|≥1, using Lemma 2.2 we have
(2.14)∥h∥22≤∥g∥22≤∥f∥22.
Furthermore, this means that h is an element of L2(ℝn) and so we complete the proof of Lemma 2.3 as desired.In our first theorem, we establish the existence of the IT of a functionalF in 𝒜n(2).Theorem 2.4.
Letγ and β be as in Lemma 2.3 and let F be given by (2.1). Then the IT ℱγ,β(F) of F exists, belongs to 𝒜n(2), and is given by the formula
(2.15)ℱγ,βF(y)=Γℱγ,βF(〈α→,y〉)
for y∈K, where
(2.16)Γℱγ,βF(v→)=(2π)-n/2∫ℝn‍f(γu→+βv→)exp{-∑j=1n‍uj22}du→.Proof.
We first note that (2.15) follows from (1.2) and (1.3). Clearly the function Γℱγ,βF(λ→) is an entire function since f is an entire function. What is left to show is that the left-hand side of (2.16) is an element of 𝒜n(2). Now, we note that for all nonzero real values of γ and β,
(2.17)Γℱγ,βF(v→)=(2π)-n/2∫ℝn‍f(γu→+βv→)exp{-∑j=1n‍uj22}du→=(2πγ2)-n/2∫ℝn‍f(u→)exp{-∑j=1n‍(uj-βvj)22γ2}du→.
As mentioned in the proof of Lemma 2.3, each side of the above expression is an analytic function of γ throughout the region {γ∈ℂ:Re(1/γ2)≥0} and β throughout the region {β∈ℂ:|β|≥1}. Hence, by the uniqueness theorem for analytic functions, the above equality holds for all γ and β with {γ∈ℂ:Re(1/γ2)≥0} and {β∈ℂ:|β|≥1}. Using Lemma 2.3, the function Γℱγ,βF is an element of L2(ℝn). In fact,
(2.18)∥Γℱγ,βF∥2≤∥f∥2.
Moreover, if Re(1/γ2)=0 and |β|=1,
(2.19)∥Γℱγ,βF∥2=∥f∥2,
which completes the proof of Theorem 2.4 as desired.Remark 2.5.
(1) Under the appropriate conditions forγ and β, we could establish the existences of the convolution product (CP) and the first variation of functionals in 𝒜n(2) as in Theorem 2.4. We will state just formulas without proof because the main purpose of this paper is to concern ℱγ,β as an operator on Hilbert space.
(2) In [5–8, 10], the authors established various basic formulas for the IT involving the CP and the first variation of functionals in various classes. Like these, we can obtain various basic relationships for the IT with related topic of functionals in 𝒜n(2) under appropriate conditions for γ and β. We list some relationships as follows.(i)
The IT of a CP is the product of ITs,(2.20)ℱγ,β(F*G)γ(y)=ℱγ,βF(y2)ℱγ,βG(y2).(ii)
A relationship among the CP, the IT, and the Inverse IT,(2.21)(F*G)γ(y)=ℱi(γ/β),1/β(ℱγ,βF(·2)ℱγ,βG(·2))(y).(iii)
A relationship between the IT, and the first variation,(2.22)βℱγ,βδF(·∣w)(y)=δℱγ,βF(y∣w).(iv)
A relationship among the CP, the IT, and the first variation,(2.23)β2ℱγ,β(δF(·∣w)*δG(·∣w))γ(z)=δℱγ,βF(y2∣w)δℱγ,βG(y2∣w).(v)
A relationship among the CP, the inverse IT, and the first variation,(2.24)ℱi(γ/β),1/β(δF(·∣w)*δG(·∣w))γ(y)=β2δℱi(γ/β),1/βF(y2∣w)δℱi(γ/β),1/βG(y2∣w)for y,w∈K, where the CP (F*G)γ of F and G is defined by
(2.25)(F*G)γ(y)=∫C0[0,T]‍F(y+γx2)G(y-x2)dm(x),y∈K,
and the first variation is defined by formula
(2.26)δF(x∣w)=∂∂kF(x+kw)|k=0,x,w∈K,
if they exist.In our next theorem, we establish the inverse IT of our IT of functionals in𝒜n(2).Theorem 2.6.
Letγ and β be as in Theorem 2.4 with Re(β2/γ2)≤0 and |β|=1, and let F∈𝒜n(2) be given by (2.1). Then
(2.27)ℱi(γ/β),1/β(ℱγ,βF)(y)=F(y)=ℱγ,β(ℱi(γ/β),1/βF)(y)
for all y∈K. That is to say, ℱi(γ/β),1/β is the inverse IT of the IT.Proof.
SinceRe(1/γ2)≥0 and |β|=1, ℱγ,β∈𝒜n(2) for all F∈𝒜n(2) by Theorem 2.4. Also, since Re(1/(iγ2/β))=Re(-β2/γ2)≥0 and |1/β|=1, ℱi(γ/β),1/β(ℱγ,βF)∈𝒜n(2) for all F∈𝒜n(2). By using the similar method, we can show that ℱγ,β(ℱi(γ/β),1/βF)∈𝒜n(2) for all F∈𝒜n(2). In [2], the author showed that for a integrable functional F,
(2.28)∫C0[0,T]‍∫C0[0,T]‍F(px1+qx2+y)dm(x1)dm(x2)=∫C0[0,T]‍F(p2+q2z+y)dm(z)
for all nonzero complex numbers p and q. Using this formula and (1.2), we have
(2.29)ℱi(γ/β),1/β(ℱγ,βF)(y)=∫C0[0,T]‍∫C0[0,T]‍F(γx2+iγx1+y)dm(x1)dm(x2)=F(y),ℱγ,β(ℱi(γ/β),1/βF)(y)=∫C0[0,T]‍∫C0[0,T]‍F(iγβx2+γβx1+y)dm(x1)dm(x2)=F(y).
Hence we complete the proof of Theorem 2.6.


## 3. A Bounded Linear Operatorℱγ,β

In previous Section2, we have considered ℱγ,β as a transform of functionals in 𝒜n(2). From now on we will consider ℱγ,β as an operator from 𝒜n(2) into 𝒜n(2) and then apply various operator theories to the IT. In particular, we obtain various spectral theorems for an operator ℱγ,β.ForF and G in 𝒜n(2), let
(3.1)(F,G)𝒜n(2)=∫ℝn‍f(u→)g(u→)¯du→
denote the inner product on 𝒜n(2) and ∥F∥𝒜n(2)=(F,F)𝒜n(2)1/2.Remark 3.1.
One can show that(𝒜n(2),∥·∥𝒜n(2)) is a complex normed linear space. Also, from the fact that L2(ℝn) is complete, one can easily show that the space (𝒜n(2),∥·∥𝒜n(2)) is also complete.In our first theorem in this section, we show that the operatorℱγ,β is well defined on 𝒜n(2).Theorem 3.2.
Letγ and β be as in Lemma 2.3. Then the ℱγ,β is a well defined operator from 𝒜n(2) into 𝒜n(2) and
(3.2)∥ℱγ,β(F)∥𝒜n(2)≤∥F∥𝒜n(2).
Moreover, if Re(1/γ2)=0 and |β|=1, then the operator preserves the norm, namely,
(3.3)∥ℱγ,β(F)∥𝒜n(2)=∥F∥𝒜n(2).Proof.
In Section2, we showed that for each F∈𝒜n(2), ℱγ,β(F) exists, belongs to 𝒜n(2) and
(3.4)ℱγ,β(F)(y)=Γℱγ,βF(〈α→,y〉),
where
(3.5)Γℱγ,βF(v→)=(2πγ2)-n/2∫ℝn‍f(u→)exp{-∑j=1n‍(uj-βvj)22γ2}du→.
Furthermore, we obtained that
(3.6)∥Γℱγ,βF∥2≤∥f∥2.
This tells us that
(3.7)∥ℱγ,β(F)∥𝒜n(2)≤∥F∥𝒜n(2)
for all γ and β satisfy the conditions in Lemma 2.3. Moreover, if Re(1/γ2)=0 and |β|=1,
(3.8)∥ℱγ,β(F)∥𝒜n(2)=∥F∥𝒜n(2),
which completes the proof of Theorem 3.2 as desired.Next, we give a simple example to illustrate our results and formulas in Theorem3.2.Example 3.3.
Letγ and β be are nonzero real numbers with |β|≥1. Let 𝒮(ℝn) be the Schwartz space of infinitely differentiable functions f(u→) decaying at infinity together with all its derivatives faster than any polynomial of |u→|-1. For nonzero real values of γ, let
(3.9)f(u→)=exp{-∑j=1n‍uj22γ2}.
Then f∈𝒮(ℝn)⊂L2(ℝn) and hence F(x)=f(〈α→,x〉) is an element of 𝒜n(2). Also, using (2.15), (2.16), and (1.3), we have
(3.10)ℱγ,β(F)(y)=2-n/2exp{-∑j=1n‍β24γ2〈αj,y〉2}.
Furthermore, we note that
(3.11)∥F∥𝒜n(2)2=|γ|nπn/2,∥ℱγ,β(F)∥𝒜n(2)2=2-n/2|γ|nπn/2|β|n.
In fact,
(3.12)∥ℱγ,β(F)∥𝒜n(2)2=2-n/2|γ|n(π)n/2|β|n≤|γ|n(2π)n=∥F∥𝒜n(2)2.
All expressions in Example 3.3 are valid for nonzero real values of γ and β. But f∈𝒮(ℝn) and hence they are still valid for nonzero complex values of γ and β which satisfy the conditions in Lemma 2.3.Now, we establish some basic operator theories for the operatorℱγ,β. First, in our next theorem, we show that the operator ℱγ,β is a bounded linear operator on 𝒜n(2).Theorem 3.4.
Letγ and β be as in Lemma 2.3. Then ℱγ,β is a bounded operator on 𝒜n(2) and hence it is continuous on 𝒜n(2). Furthermore, if Re(1/γ2)=0 and |β|=1, then ℱγ,β is injective from 𝒜n(2) into 𝒜n(2).Proof.
We first note that(3.13)∥ℱγ,β∥0=sup∥F∥𝒜n(2)=1∥ℱγ,β(F)∥𝒜n(2)≤sup∥F∥𝒜n(2)=1∥F∥𝒜n(2)=1,
where ∥T∥0 is the operator norm of an operator T. Hence ℱγ,β is bounded and so it is continuous. Furthermore if Re(1/γ2)=0 and |β|=1, then (3.3) tells us that ℱγ,β preserves the norm and hence it is injective from 𝒜n(2) into 𝒜n(2). So we complete the proof of Theorem 3.4.The following corollary follows from Theorem3.4 and some basic properties for bounded linear operators on Hilbert space.Corollary 3.5.
Letγ and β be as in Theorem 3.4. Then we have the following assertions. (1)
The null space𝒩(ℱγ,β) of ℱγ,β is closed.(2)
Let{Fn}n=1∞ be a sequence in 𝒜n(2) with Fn→F as n→∞ for some F∈𝒜n(2). Then ℱγ,β(Fn)→ℱγ,β(F) as n→∞. That is to say, it is closed.(3)
For allF and G in 𝒜n(2), we have the Cauchy-Schwartz inequality
(3.14)|(ℱγ,βF,ℱγ,βG)𝒜n(2)|≤∥ℱγ,βF∥𝒜n(2)∥ℱγ,βG∥𝒜n(2)≤∥F∥𝒜n(2)∥G∥𝒜n(2).Remark 3.6.
As mentioned in Section1, the Fourier-Wiener transform, the modified Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are also well-defined operators on 𝒜n(2). In particular, from the definition of analytic Fourier-Feynman transform, it is an injective operator on 𝒜n(2). Hence all those transforms can be applied to our main results and formulas in this paper. In particular, the authors studied that for |β|=1 and F∈L2(C0[0,T]), the IT ℱγ,βF is an element of L2(C0[0,T]) and
(3.15)∥ℱγ,βF∥2=∥F∥2.
That is to say, the IT is injective [7, 9]. This result is a special case of our result in this paper. In addition, in [3, 4], the authors showed that the Fourier Wiener transform acts as a unitary operator on L2(C0[0,T]).We finish this section by stating that the operatorℱγ,β is invertible.Theorem 3.7.
Letγ and β be as in Theorem 3.4 with Re(β2/γ2)≤0 and |β|=1. Then the inverse operator of the IT ℱγ,β exists and is given by
(3.16)ℱγ,β-1=ℱi(γ/β),1/β.
Furthermore, the null space 𝒩(ℱγ,β) consists of the zero vector only.Proof.
From Theorem3.4, the operator is continuous from 𝒜n(2) into 𝒜n(2). Since Re(β2/γ2)≤0 and ℱγ,β(F) is in 𝒜n(2), ℱi(γ/β),1/β(ℱγ,β(F)) exists and is in 𝒜n(2). Also, since Re(1/γ2)≥0 and ℱi(γ/β),1/β(F) is in 𝒜n(2), ℱγ,β(ℱi(γ/β),1/βF) exists and is in 𝒜n(2). Now, using (2.27), for F∈𝒜n(2),
(3.17)ℱi(γ/β),1/β(ℱγ,β(F))(x)=F(x)=ℱγ,β(ℱi(γ/β),1/βF)(x),
which completes the proof of Theorem 3.7 as desired.We have some observations for the inverse operatorℱi(γ/β),1/β of ℱγ,β.Remark 3.8.
(1) Ifγ=2 and β=i, then γ and β always satisfy the hypotheses of Theorems 3.2 and 3.4. In fact, there are many pairs (γ,β) satisfying the hypotheses of Theorems 3.2 and 3.4.
(2) The operatorℱγ,β might not be bijective. Hence we should consider that the domain of the inverse operator ℱi(γ/β),1/β is the range of ℱγ,β.
(3) The operatorℱγ,β is an homeomorphism from 𝒜n(2) into ℛ(ℱγ,β), where ℛ(ℱγ,β) is the range of ℱγ,β.


## 4.   Some Spectral Theorems for the Bounded Linear Operatorℱγ,β

In this section we will apply some spectral theories to theℱγ,β. To do this, we need some concepts related to the spectral theory on a Banach space.Withℱγ,β we associate the operator
(4.1)ℱγ,βλ=ℱγ,β-λI,
where λ is a complex number and I is the identity operator on 𝒜n(2). If ℱγ,βλ has an inverse, we denote it by Rλ(ℱγ,β); that is,
(4.2)Rλ(ℱγ,β)=ℱγ,βλ-1=(ℱγ,β-λI)-1,
and call it the resolvent operator of ℱγ,β or, simply, the resolvent of ℱγ,β.Definition 4.1.
A regular valueλ of ℱγ,β is a complex number such that  (1)
Rλ(ℱγ,β) exists,(2)
Rλ(ℱγ,β) is bounded,(3)
Rλ(ℱγ,β) is defined on a set which is dense in 𝒜n(2). The resolvent set ρ(ℱγ,β) of ℱγ,β is the set of all regular values λ of ℱγ,β. Its complement σ(ℱγ,β)=ℂ-ρ(ℱγ,β) is called the spectrum of ℱγ,β, and a λ∈σ(ℱγ,β) is called a spectral value of ℱγ,β. For more details, see [16].Remark 4.2.
(1) The spectrumσ(ℱγ,β) is partitioned into three disjoint sets as follows.  (i)
The point spectrum or discrete spectrumσp(ℱγ,β) is the set such that Rλ(ℱγ,β) does not exist. A λ∈σp(ℱγ,β) is called an eigenvalue of ℱγ,β.(ii)
The continuous spectrumσc(ℱγ,β) is the set such that Rλ(ℱγ,β) exists and satisfies (3) but not (2) in Definition 4.1; that is to say, Rλ(ℱγ,β) is unbounded.(iii)
The residual spectrumσr(ℱγ,β) is the set such that Rλ(ℱγ,β) exists (and it may be bounded or not) but does not satisfy (3) in Definition 4.1. That is to say, the domain of Rλ(ℱγ,β) is not dense in 𝒜n(2).
(2) We know thatℂ=ρ(ℱγ,β)∪σ(ℱγ,β)=ρ(ℱγ,β)∪σp(ℱγ,β)∪σc(ℱγ,β)∪σr(ℱγ,β).From now on, if what operatorℱγ,β refers to is clear, we will write Rλ instead of Rλ(ℱγ,β).In our next theorem, we apply the spectral theory to the operatorℱγ,β.Theorem 4.3.
Letγ and β be as in Theorem 3.4. Then the resolvent set ρ(ℱγ,β) of ℱγ,β is open and hence the spectrum σ(ℱγ,β) is closed. Furthermore, for every λ0∈ρ(ℱγ,β), the resolvent Rλ has the representation
(4.3)Rλ=∑j=1∞‍(λ-λ0)jRλ0j+1,
where the series is absolutely convergent for every λ in the open disk given by
(4.4)|λ-λ0|<1∥Rλ0∥0
in the complex plane. This disk is a subset of ρ(ℱγ,β).Proof.
Theorem4.3 immediately follows the fact that the ℱγ,β is a bounded linear operator on 𝒜n(2).Next, we note that the spectral radiusrσ(ℱγ,β) of ℱγ,β is the radius
(4.5)rσ(ℱγ,β)=supλ∈σ(ℱγ,β)|λ|
of the smallest closed disk centered at the origin of the complex λ-plane and containing σ(ℱγ,β).Theorem 4.4.
Letγ and β be as in Theorem 3.4. Then the spectrum σ(ℱγ,β) of ℱγ,β is compact and lies in the disk given by
(4.6)|λ|≤∥ℱγ,β∥0.
Hence the resolvent set ρ(ℱγ,β) of ℱγ,β is not empty. Furthermore, the spectral radius rσ(ℱγ,β)≤∥ℱγ,β∥0 and
(4.7)rσ(ℱγ,β)=limn→∞(∥ℱγ,β∥0n)1/n.Proof.
From Theorem3.4, the ℱγ,β is a bounded linear operator on 𝒜n(2). Using a basic property for the spectrum, we establish (4.6), and hence the resolvent set is not empty. Furthermore, using (4.6), it is obvious that for the spectral radius of a bounded linear operator ℱγ,β we have
(4.8)rσ(ℱγ,β)≤∥ℱγ,β∥0.
Also, we can easily obtain (4.7) as desired.In our next theorem, we give a spectral mapping theorem for polynomials ofℱγ,β. The proof of Theorem 4.5 is omitted because it immediately follows the spectral mapping theorem for polynomial on a Banach space.Theorem 4.5.
Letγ and β be as in Theorem 3.4. Let
(4.9)P(λ)=anλn+an-1λn-1+⋯+a0,an≠0,n=1,2,….
Then
(4.10)σ(P(ℱγ,β))=P(σ(ℱγ,β)).
This implies that the spectrum σ(P(ℱγ,β)) of the operator
(4.11)P(λ)=anTn+an-1Tn-1+⋯+a0I
consists precisely of all those values which the polynomial P assumes on the spectrum σ(ℱγ,β) of ℱγ,β.Next, we will explain that our study is meaningful to obtain the solution to a differential equation.LetH be a real separable infinite-dimensional Hilbert space with the inner product 〈·,·〉 and norm |·|=〈·,·〉. Let ∥·∥0 be a measurable norm on H with respect to the Gaussian cylinder set measure ν0 on H. Let B denote the completion of H with respect to ∥·∥0. Let i denote the natural injection from H to B. The adjoint operator i* of i is one to one and maps B* continuously onto a dense subset H*, where B* and H* are topological duals of B and H, respectively. By identifying H* with H and B* with i*B*, we have a triple B*⊂H*≈H⊂B with 〈x,y〉=(x,y)~ for all x in H and y in B*, where (·,·)~ denotes the natural dual pairing between B and B*. By a well-known result of Gross [17], ν0∘i-1 has a unique countably additive extension ν to the Borel σ-algebra ℬ(B) of B. The triple (B,H,ν) is called an abstract Wiener space. The classical Wiener space C0[0,T] is one of the examples of abstract Wiener space.For an appropriate functionalu(x) on B, let Nc be an operator defined by the formula
(4.12)Ncu(x)=-TrHD2u(x)+c(x,Du(x))~,x∈B,c∈ℂ/{0},
where D denotes the second Fréchet derivative and TrH denotes the trace of an operator. In [2], Lee showed that the integral transform ℱ1/c,i, c∈ℂ/{0} forms the solution of a differential equation which is called a Cauchy problem
(4.13)ut(x,t)=𝒫(Nc)u(x,t),x∈B,t>0u(x,0)=F(x),
where 𝒫(η)=amηm+⋯+a1η+a0 is an m-dimensional polynomial function with respect to η. In addition, let 𝒫=-η and c=1 in (4.13). Then the solution of the Cauchy problem is given by formula
(4.14)u(x,t)=∫B‍F(e-tx+1-e-2ty)dν(y),
or, equivalently,
(4.15)u(x,t)=∫B‍F(y)ot(x,dy),
where ot(x,dy)=ν1-e-2t(e-tx,dy). This showed that the family of measures {ot(x,dy)} serves as the “fundamental solution” of the operator ∂/∂t+N1. For more details see [2, 18]. Hence our discussions as a bounded linear operator in this paper have some meaningful subjects. That is to say, from Theorems 3.2 through 4.5, if we take a complex number c such that Re(c2)≥0, then the transform ℱ1/c,i is a well-defined bounded linear operator on 𝒜n(2) and could be applied to all the results and formulas in this paper.


## 5. Applications for the Spectral Theory

In Sections3 and 4, we treated the IT ℱγ,β as a bounded linear operator on 𝒜n(2). Also, we applied spectral theorems to the IT to obtain various useful formulas and results. In this section we will show that the operator ℱγ,β is self-adjoint under appropriate parameters γ and β. We then apply the spectral theory to a self-adjoint operator on a Banach space. In particular, we obtain the spectral representation for IT ℱγ,β.In our next theorem, we show that the operatorℱγ,β is a self-adjoint operator on 𝒜n(2) under an appropriate condition for γ and β.Theorem 5.1.
Letγ be as in Theorem 3.4 with γ2¯=γ2 and let β=1. Then the operator ℱγ,β is a self-adjoint operator on 𝒜n(2).Proof.
LetT* denote the adjoint operator on an operator T. For all F and G in 𝒜n(2), we note that
(5.1)(F,ℱγ,β*G)𝒜n(2)=(ℱγ,βF,G)𝒜n(2)=∫ℝn‍Γℱγ,βF(v→)g(v→)¯dv→=(∏j=1n‍2πγ2)-1∫ℝn‍∫ℝn‍f(u→)exp{-∑j=1n‍(uj-vj)22γ2}du→g(v→)¯dv→=(∏j=1n‍2πγ2)-1∫ℝn‍f(u→)∫ℝn‍g(v→)¯exp{-∑j=1n‍(vj-uj)22γ2}dv→du→=(∏j=1n‍2πγ2)-1¯∫ℝn‍f(u→)∫ℝn‍g(v→)exp{-∑j=1n‍(vj-uj)22γ2}dv→¯du→=(F,ℱγ,βG)𝒜n(2),
which completes the proof of Theorem 5.1 as desired.Remark 5.2.
We gave the conditions forγ and β in Sections 2 and 3, and Theorem 5.1. We note that these conditions imply that γ is real and β=1 only. But, a self-adjoint operator may not have eigenvalues. So if it has eigenvalues, then it must be real. Hence these are very natural conditions.Throughout the next corollary, we give some results of the spectral theories for self-adjoint operatorℱγ,β.Corollary 5.3.
Letγ and β be as in Theorem 5.1. Then we have the following assertions.(1)
Forλ∈σ(ℱγ,β), there exists a positive real number k0 such that
(5.2)∥(ℱγ,β-λI)F∥𝒜n(2)≥k0∥F∥𝒜n(2)
and so there exists a sequence (Fn)n=1∞ in 𝒜n(2) with ∥Fn∥𝒜n(2)=1, n=1,2,… such that
(5.3)limn→∞∥(ℱγ,β-λI)Fn∥𝒜n(2)=0.(2)
For allF∈𝒜n(2) and λ∈ℂ,
(5.4)∥(ℱγ,β-λI)F∥𝒜n(2)≥|Imλ|∥F∥𝒜n(2).In our next theorem, we apply the spectral theory to the operatorℱγ,β as a self-adjoint operator.Theorem 5.4.
Letγ and β be as in Theorem 5.1. Then the spectrum σ(ℱγ,β) of ℱγ,β is real and it lies in the closed interval [k1,k2] where
(5.5)k1=inf∥F∥𝒜n(2)=1(ℱγ,β(F),F)𝒜n(2),(5.6)k2=sup∥F∥𝒜n(2)=1(ℱγ,β(F),F)𝒜n(2).
Furthermore, k1 and k2 are spectral values of ℱγ,β and max{|k1|,|k2|}=1.Proof.
First, sinceℱγ,β is a self-adjoint operator on 𝒜n(2), the spectrum σ(ℱγ,β) of ℱγ,β must be real. Next we recall that for each bounded self-adjoint operator T:H→H on a complex Hilbert space H, σ(T)⊂[m,M] on the real axis and ∥T∥0=max{|m|,|M|}, where
(5.7)m=inf∥h∥H=1(Th,h)H,M=sup∥h∥H=1(Th,h)H,
and (·,·)H is an inner product on H. Hence the spectrum σ(ℱγ,β) lies in the closed interval [k1,k2]. Furthermore, k1 and k2 are spectral values of ℱγ,β and max{|k1|,|k2|}=∥ℱγ,β∥0=1. Hence we complete the proof of Theorem 5.4.We finish this paper by giving an application for the spectral representation of the self-adjoint operator which is one of very important subjects in the fields of the quantum mechanics and physical theories.In our last theorem, we give the spectral representation for the self-adjoint operator. To do this, we need some concepts for the spectral theory.Let(λn)n=1∞ be the set of eigenvalues of ℱγ,β with λn<λm for m<n and let (en)n=1∞ be the set of eigenfunctions corresponding to (λn)n=1∞. Then we note that
(5.8)ℱγ,βF(x)=∑n=1∞‍anλnen(x),
where an=(F,en)𝒜n(2). For each j=1,2,…, define an (orthogonal) projection Pj on 𝒜n(2) by PjF(x)=ajej(x). Then we also note that
(5.9)ℱγ,βF(x)=∑n=1∞‍λnPnF(x).
Now, for λ∈ℝ, define an operator Eλ on 𝒜n(2) by EλF(x)=∑λj≤λPjF(x). In this case, ℰ≡(Eλ) is called the spectral family of ℱγ,β.Theorem 5.5.
Letγ and β be as in Theorem 5.1. Then we have the spectral representation for ℱγ,β as follows:
(5.10)ℱγ,β=∫k1k2‍λdEλ,
where k1 and k2 are given by (5.5) and (5.6), and {Eλ} is the spectral family of ℱγ,β.Proof.
First, we note that forγ and β in Theorem 5.1, ℱγ,β is a self-adjoint operator on 𝒜n(2). Furthermore, σ(ℱγ,β)=[k1,k2]⊂[-1,1]. Using the spectral representation of the self-adjoint operator, we establish (5.10).Remark 5.6.
(1) In view of Theorem5.5, for all real-valued continuous functions f on [k1,k2] and for F and G in 𝒜n(2),
(5.11)f(ℱγ,β)=∫k1k2‍f(λ)dEλ,(ℱγ,βF,G)𝒜n(2)=∫k1k2‍λdw(λ),
where w(λ)=(EλF,G)𝒜n(2) and the integral is an ordinary Riemann-Stieltjes integral.
(2) An alternative formulation of the spectral theorem expresses the operatorℱγ,β as an integral of the coordinate function over the operator's spectrum with respect to a projection-valued measure ℱγ,β=∫σ(ℱγ,β)‍λdEλ. When the normal operator in question is compact, this version of the spectral theorem reduces to the finite-dimensional spectral theorem, except that the operator is expressed as a linear combination of possibly infinitely many projections.
(3) In Sections3, 4, and 5, we considered the IT as an operator. Like this, we expect that the convolution product could be dealt with as an operator. Furthermore, we could obtain various relations between the IT and the convolution product as a composition of operators.



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