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

Abstract

In many previous papers, an integral transform ℱ_𝛾,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 ℱ_𝛾,𝛽. Finally we give an application for the spectral representation of a self-adjoint operator which plays a key role in quantum mechanics.

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.

Let 𝐶₀[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 _{𝐶0[]0,𝑇}𝐹(𝑥)𝑑𝑚(𝑥). (1.1)

A subset 𝐵 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 ℱ_𝛾,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 ℱ_𝛾,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 ℱ_{𝛾,𝛽,ℎ} via a Gaussian process of functionals in a class 𝒮_𝛼 and then obtained various relationships involving the convolution product and the first variation of them.

However, in all previous works, the authors have considered ℱ_𝛾,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 ℱ_𝛾,𝛽 as an operator on a Banach space. We then apply various operator theories to ℱ_𝛾,𝛽. Furthermore, we obtain various theorems of the spectral theory for ℱ_𝛾,𝛽 involving the spectral representation.

Now we are ready to define the integral transform  (IT) of a functional on 𝐾≡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. Let 𝐹 be a functional defined on 𝐾. For each pair of nonzero complex numbers 𝛾 and 𝛽, the IT ℱ_𝛾,𝛽(𝐹) of 𝐹 is defined by ℱ_𝛾,𝛽(𝐹)𝑦≡_{𝐶0[0,𝑇]}𝐹𝛾𝑥+𝛽𝑦𝑑𝑚(𝑥),𝑦∈𝐾, (1.2) if it exists.

Remark 1.2. When 𝛾=1 and 𝛽=𝑖, ℱ_1,𝑖 is the Fourier-Wiener transform introduced by Cameron and Martin in [3], and when 𝛾=√2 and 𝛽=𝑖, ℱ_√2,𝑖 is the modified Fourier-Wiener transform used by Cameron and Martin in [4].

For 𝑣∈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 each 𝑣∈𝐿₂[0,𝑇], ⟨𝑣,𝑥⟩ exists for a.e. 𝑥∈𝐶₀[0,𝑇]. (2)If 𝑣∈𝐿₂[0,𝑇] is a function of bounded variation on [0,𝑇], ⟨𝑣,𝑥⟩ equals the Riemann-Stieltjes integral ∫^𝑇₀𝑣(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∶ℝ^𝑛→ℝ be Borel measurable and let 𝐻(𝑥)=ℎ(⟨𝛼₁,𝑥⟩,…,⟨𝛼_𝑛,𝑥⟩). Then _{𝐶0[0,𝑇]}𝐻(𝑥)𝑑𝑚(𝑥)=(2𝜋)^−𝑛/2_ℝ𝑛ℎ𝑢₁,…,𝑢_𝑛⋅exp−^𝑛_𝑗=1𝑢²_𝑗2𝑑𝑢₁⋯𝑑𝑢_𝑛 (1.3) 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 class 𝒜_𝑛⁽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 𝒜_𝑛⁽²⁾ be the space of all functionals 𝐹∶𝐾→ℂ of the form 𝐹(𝑥)=𝑓𝛼₁,𝑥,…,𝛼_𝑛,𝑥 (2.1) for some positive integer 𝑛 (throughout this paper, 𝑛 is fixed), where 𝑓(𝑢₁,…,𝑢_𝑛) is an entire function of the 𝑛 complex variables 𝑢₁,…,𝑢_𝑛 and _ℝ𝑛𝑓^→𝑢²𝑑^→𝑢<∞. (2.2) To simplify the expressions, we use the following notation: 𝑓^→𝛼,𝑥≡𝑓𝛼₁,𝑥,…,𝛼_𝑛,𝑥. (2.3)

Remark 2.1. For any 𝐹 and 𝐺 in 𝒜_𝑛⁽²⁾, we can always express 𝐹 by (2.1) and 𝐺 by 𝐺(𝑥)=𝑔𝛼₁,𝑥,…,𝛼_𝑛,𝑥≡𝑔^→𝛼,𝑥 (2.4) using the same positive integer 𝑛, where 𝑔 is an entire function and _ℝ𝑛𝑔^→𝑢²𝑑^→𝑢<∞. (2.5)

Note that 𝒜_𝑛⁽²⁾ is a very rich class of functionals because 𝒜_𝑛⁽²⁾ contains the Schwartz class 𝒮(ℝ^𝑛). 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 functions 𝑓 and 𝑔 on ℝ^𝑛, if lim_{𝐴→∞ℝ𝑛}^𝐴(𝑛)_−𝐴⋯^𝐴_−𝐴𝑓^→𝑢;^→𝑣𝑑^→𝑢−𝑔^→𝑣²𝑑^→𝑣=0, (2.6) then we say that the integral _ℝ𝑛𝑓^→𝑢;^→𝑣𝑑^→𝑢=𝑔^→𝑣 (2.7) 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. Let 𝛾 be nonzero complex number with Re(1/𝛾²)≥0. For 𝑓∈𝐿₂(ℝ^𝑛), let 𝑔^→𝑣=2𝜋𝛾^2−𝑛/2_ℝ𝑛𝑓^→𝑢exp−^𝑛_𝑗=1𝑢_𝑗−𝑣_𝑗²2𝛾²𝑑^→𝑢. (2.8) Then 𝑔∈𝐿₂(ℝ^𝑛), and 𝑔₂≤𝑓₂. (2.9) If Re(1/𝛾²)=0, the integral is to be interpreted as an 𝐿₂-limiting integral; moreover, in this case 𝑔₂=𝑓₂. (2.10)

The following lemma is very useful in establishing the existence of the IT.

Lemma 2.3. Let 𝑓,𝑔, and 𝛾 be as in Lemma 2.2 and let 𝛽 be a nonzero complex number with |𝛽|≥1. Let ℎ(^→𝑢)=𝑔(𝛽^→𝑢). Then ℎ∈𝐿₂(ℝ^𝑛) and ‖ℎ‖₂≤𝑔₂≤𝑓₂. (2.11) If Re(1/𝛾²)=0 and |𝛽|=1, the integral is to be interpreted as an 𝐿₂-limiting integral; moreover, in this case ‖ℎ‖₂=𝑔₂=𝑓₂. (2.12)

Proof. First note that for all nonzero real numbers 𝛽 with |𝛽|≥1, it follows that ‖ℎ‖²₂=_ℝ𝑛ℎ^→𝑢²𝑑^→𝑢=_ℝ𝑛𝑔𝛽^→𝑢²𝑑^→𝑢=1𝛽^𝑛_ℝ^𝑛𝑔^→𝑢²𝑑^→𝑢. (2.13) 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 ‖ℎ‖²₂≤𝑔²₂≤𝑓²₂. (2.14) Furthermore, this means that ℎ is an element of 𝐿₂(ℝ^𝑛) 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 functional 𝐹 in 𝒜_𝑛⁽²⁾.

Theorem 2.4. Let 𝛾 and 𝛽 be as in Lemma 2.3 and let 𝐹 be given by (2.1). Then the IT ℱ_𝛾,𝛽(𝐹) of 𝐹 exists, belongs to 𝒜_𝑛⁽²⁾, and is given by the formula ℱ_𝛾,𝛽𝐹𝑦=Γ_{ℱ𝛾,𝛽𝐹}^→𝛼,𝑦 (2.15) for 𝑦∈𝐾, where Γ_{ℱ𝛾,𝛽𝐹}^→𝑣=(2𝜋)^−𝑛/2_ℝ𝑛𝑓𝛾^→𝑢+𝛽^→𝑣exp−^𝑛_𝑗=1𝑢²_𝑗2𝑑^→𝑢. (2.16)

Proof. We first note that (2.15) follows from (1.2) and (1.3). Clearly the function Γ_{ℱ𝛾,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 𝒜_𝑛⁽²⁾. Now, we note that for all nonzero real values of 𝛾 and 𝛽, Γ_{ℱ𝛾,𝛽𝐹}^→𝑣=(2𝜋)^−𝑛/2_ℝ𝑛𝑓𝛾^→𝑢+𝛽^→𝑣exp−^𝑛_𝑗=1𝑢²_𝑗2𝑑^→𝑢=2𝜋𝛾^2−𝑛/2_ℝ𝑛𝑓^→𝑢exp−^𝑛_𝑗=1𝑢_𝑗−𝛽𝑣_𝑗²2𝛾²𝑑^→𝑢. (2.17) 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 Γ_{ℱ𝛾,𝛽𝐹} is an element of 𝐿₂(ℝ^𝑛). In fact, Γ_{ℱ𝛾,𝛽𝐹2}≤𝑓₂. (2.18) Moreover, if Re(1/𝛾²)=0 and |𝛽|=1, Γ_{ℱ𝛾,𝛽𝐹2}=𝑓₂, (2.19) which completes the proof of Theorem 2.4 as desired.

Remark 2.5. (1) Under the appropriate conditions for 𝛾 and 𝛽, we could establish the existences of the convolution product (CP) and the first variation of functionals in 𝒜_𝑛⁽²⁾ as in Theorem 2.4. We will state just formulas without proof because the main purpose of this paper is to concern ℱ_𝛾,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 𝒜_𝑛⁽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, ℱ_𝛾,𝛽(𝐹∗𝐺)_𝛾𝑦=ℱ_𝛾,𝛽𝐹𝑦√2ℱ_𝛾,𝛽𝐺𝑦√2. (2.20)(ii)A relationship among the CP, the IT, and the Inverse IT, (𝐹∗𝐺)_𝛾𝑦=ℱ_{𝑖(𝛾/𝛽),1/𝛽}ℱ_𝛾,𝛽𝐹⋅√2ℱ_𝛾,𝛽𝐺⋅√2𝑦. (2.21)(iii)A relationship between the IT, and the first variation, 𝛽ℱ_𝛾,𝛽𝛿𝐹(⋅∣𝑤)𝑦=𝛿ℱ_𝛾,𝛽𝐹𝑦∣𝑤. (2.22)(iv)A relationship among the CP, the IT, and the first variation, 𝛽²ℱ_𝛾,𝛽(𝛿𝐹(⋅∣𝑤)∗𝛿𝐺(⋅∣𝑤))_𝛾(𝑧)=𝛿ℱ_𝛾,𝛽𝐹𝑦√2∣𝑤𝛿ℱ_𝛾,𝛽𝐺𝑦√2∣𝑤. (2.23)(v)A relationship among the CP, the inverse IT, and the first variation, ℱ_{𝑖(𝛾/𝛽),1/𝛽}(𝛿𝐹(⋅∣𝑤)∗𝛿𝐺(⋅∣𝑤))_𝛾𝑦=𝛽²𝛿ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹𝑦√2∣𝑤𝛿ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐺𝑦√2∣𝑤 (2.24)for 𝑦,𝑤∈𝐾, where the CP (𝐹∗𝐺)_𝛾 of 𝐹 and 𝐺 is defined by (𝐹∗𝐺)_𝛾𝑦=_{𝐶0[0,𝑇]}𝐹𝑦+𝛾𝑥√2𝐺𝑦−𝑥√2𝑑𝑚(𝑥),𝑦∈𝐾, (2.25) and the first variation is defined by formula ()=𝜕𝛿𝐹𝑥∣𝑤𝐹(𝜕𝑘𝑥+𝑘𝑤)_𝑘=0,𝑥,𝑤∈𝐾, (2.26) if they exist.

In our next theorem, we establish the inverse IT of our IT of functionals in 𝒜_𝑛⁽²⁾.

Theorem 2.6. Let 𝛾 and 𝛽 be as in Theorem 2.4 with Re(𝛽²/𝛾²)≤0 and |𝛽|=1, and let 𝐹∈𝒜_𝑛⁽²⁾ be given by (2.1). Then ℱ_{𝑖(𝛾/𝛽),1/𝛽}ℱ_𝛾,𝛽𝐹𝑦=𝐹𝑦=ℱ_𝛾,𝛽ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹𝑦 (2.27) for all 𝑦∈𝐾. That is to say, ℱ_{𝑖(𝛾/𝛽),1/𝛽} is the inverse IT of the IT.

Proof. Since Re(1/𝛾²)≥0 and |𝛽|=1, ℱ_𝛾,𝛽∈𝒜_𝑛⁽²⁾ for all 𝐹∈𝒜_𝑛⁽²⁾ by Theorem 2.4. Also, since Re(1/(𝑖𝛾²/𝛽))=Re(−𝛽²/𝛾²)≥0 and |1/𝛽|=1, ℱ_{𝑖(𝛾/𝛽),1/𝛽}(ℱ_𝛾,𝛽𝐹)∈𝒜_𝑛⁽²⁾ for all 𝐹∈𝒜_𝑛⁽²⁾. By using the similar method, we can show that ℱ_𝛾,𝛽(ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹)∈𝒜_𝑛⁽²⁾ for all 𝐹∈𝒜_𝑛⁽²⁾. In [2], the author showed that for a integrable functional 𝐹, _{𝐶0[0,𝑇]𝐶0[0,𝑇]}𝐹𝑝𝑥₁+𝑞𝑥₂+𝑦𝑑𝑚𝑥₁𝑑𝑚𝑥₂=_{𝐶0[0,𝑇]}𝐹𝑝²+𝑞²𝑧+𝑦𝑑𝑚(𝑧) (2.28) for all nonzero complex numbers 𝑝 and 𝑞. Using this formula and (1.2), we have ℱ_{𝑖(𝛾/𝛽),1/𝛽}ℱ_𝛾,𝛽𝐹𝑦=_{𝐶0[0,𝑇]𝐶0[0,𝑇]}𝐹𝛾𝑥₂+𝑖𝛾𝑥₁+𝑦𝑑𝑚𝑥₁𝑑𝑚𝑥₂𝑦,ℱ=𝐹_𝛾,𝛽ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹𝑦=_{𝐶0[0,𝑇]𝐶0[0,𝑇]}𝐹𝑖𝛾𝛽𝑥₂+𝛾𝛽𝑥₁+𝑦𝑑𝑚𝑥₁𝑑𝑚𝑥₂=𝐹𝑦. (2.29) Hence we complete the proof of Theorem 2.6.

3. A Bounded Linear Operator ℱ_𝛾,𝛽

In previous Section 2, we have considered ℱ_𝛾,𝛽 as a transform of functionals in 𝒜_𝑛⁽²⁾. From now on we will consider ℱ_𝛾,𝛽 as an operator from 𝒜_𝑛⁽²⁾ into 𝒜_𝑛⁽²⁾ and then apply various operator theories to the IT. In particular, we obtain various spectral theorems for an operator ℱ_𝛾,𝛽.

For 𝐹 and 𝐺 in 𝒜_𝑛⁽²⁾, let (𝐹,𝐺)_𝒜𝑛⁽²⁾=_ℝ𝑛𝑓^→𝑢𝑔^→𝑢𝑑^→𝑢 (3.1) denote the inner product on 𝒜_𝑛⁽²⁾ and ‖𝐹‖_𝒜𝑛⁽²⁾=(𝐹,𝐹)_𝒜^1/2_𝑛(2).

Remark 3.1. One can show that (𝒜_𝑛⁽²⁾,‖⋅‖_𝒜𝑛⁽²⁾) is a complex normed linear space. Also, from the fact that 𝐿₂(ℝ^𝑛) is complete, one can easily show that the space (𝒜_𝑛⁽²⁾,‖⋅‖_𝒜𝑛⁽²⁾) is also complete.

In our first theorem in this section, we show that the operator ℱ_𝛾,𝛽 is well defined on 𝒜_𝑛⁽²⁾.

Theorem 3.2. Let 𝛾 and 𝛽 be as in Lemma 2.3. Then the ℱ_𝛾,𝛽 is a well defined operator from 𝒜_𝑛⁽²⁾ into 𝒜_𝑛⁽²⁾ and ℱ_𝛾,𝛽(𝐹)_𝒜𝑛(2)≤‖𝐹‖_𝒜𝑛⁽²⁾. (3.2) Moreover, if Re(1/𝛾²)=0 and |𝛽|=1, then the operator preserves the norm, namely, ℱ_𝛾,𝛽(𝐹)_𝒜𝑛(2)=‖𝐹‖_𝒜𝑛⁽²⁾. (3.3)

Proof. In Section 2, we showed that for each 𝐹∈𝒜_𝑛⁽²⁾, ℱ_𝛾,𝛽(𝐹) exists, belongs to 𝒜_𝑛⁽²⁾ and ℱ_𝛾,𝛽(𝐹)𝑦=Γ_{ℱ𝛾,𝛽𝐹}^→𝛼,𝑦, (3.4) where Γ_{ℱ𝛾,𝛽𝐹}^→𝑣=2𝜋𝛾^2−𝑛/2_ℝ𝑛𝑓^→𝑢exp−^𝑛_𝑗=1𝑢_𝑗−𝛽𝑣_𝑗²2𝛾²𝑑^→𝑢. (3.5) Furthermore, we obtained that Γ_{ℱ𝛾,𝛽𝐹2}≤𝑓₂. (3.6) This tells us that ℱ_𝛾,𝛽(𝐹)_𝒜𝑛(2)≤‖𝐹‖_𝒜𝑛⁽²⁾ (3.7) for all 𝛾 and 𝛽 satisfy the conditions in Lemma 2.3. Moreover, if Re(1/𝛾²)=0 and |𝛽|=1, ℱ_𝛾,𝛽(𝐹)_𝒜𝑛(2)=‖𝐹‖_𝒜𝑛⁽²⁾, (3.8) which completes the proof of Theorem 3.2 as desired.

Next, we give a simple example to illustrate our results and formulas in Theorem 3.2.

Example 3.3. Let 𝛾 and 𝛽 be are nonzero real numbers with |𝛽|≥1. Let 𝒮(ℝ^𝑛) 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 𝑓^→𝑢=exp−^𝑛_𝑗=1𝑢²_𝑗2𝛾². (3.9) Then 𝑓∈𝒮(ℝ^𝑛)⊂𝐿₂(ℝ^𝑛) and hence 𝐹(𝑥)=𝑓(⟨^→𝛼,𝑥⟩) is an element of 𝒜_𝑛⁽²⁾. Also, using (2.15), (2.16), and (1.3), we have ℱ_𝛾,𝛽(𝐹)𝑦=2^−𝑛/2exp−^𝑛_𝑗=1𝛽²4𝛾²𝛼_𝑗,𝑦². (3.10) Furthermore, we note that ‖𝐹‖²_𝒜𝑛(2)=𝛾^𝑛𝜋^𝑛/2,ℱ_𝛾,𝛽(𝐹)²_𝒜𝑛(2)=2^−𝑛/2𝛾^𝑛𝜋^𝑛/2𝛽^𝑛. (3.11) In fact, ℱ_𝛾,𝛽(𝐹)²_𝒜𝑛(2)=2^−𝑛/2𝛾^𝑛(𝜋)^𝑛/2𝛽^𝑛≤𝛾^𝑛(2𝜋)^𝑛=‖𝐹‖²_𝒜𝑛(2). (3.12) All expressions in Example 3.3 are valid for nonzero real values of 𝛾 and 𝛽. But 𝑓∈𝒮(ℝ^𝑛) 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 operator ℱ_𝛾,𝛽. First, in our next theorem, we show that the operator ℱ_𝛾,𝛽 is a bounded linear operator on 𝒜_𝑛⁽²⁾.

Theorem 3.4. Let 𝛾 and 𝛽 be as in Lemma 2.3. Then ℱ_𝛾,𝛽 is a bounded operator on 𝒜_𝑛⁽²⁾ and hence it is continuous on 𝒜_𝑛⁽²⁾. Furthermore, if Re(1/𝛾²)=0 and |𝛽|=1, then ℱ_𝛾,𝛽 is injective from 𝒜_𝑛⁽²⁾ into 𝒜_𝑛⁽²⁾.

Proof. We first note that ℱ_𝛾,𝛽0=sup_{‖𝐹‖𝒜𝑛(2)=1}ℱ_𝛾,𝛽(𝐹)_𝒜𝑛(2)≤sup_{‖𝐹‖𝒜𝑛(2)=1}‖𝐹‖_𝒜𝑛⁽²⁾=1, (3.13) where ‖𝑇‖₀ is the operator norm of an operator 𝑇. Hence ℱ_𝛾,𝛽 is bounded and so it is continuous. Furthermore if Re(1/𝛾²)=0 and |𝛽|=1, then (3.3) tells us that ℱ_𝛾,𝛽 preserves the norm and hence it is injective from 𝒜_𝑛⁽²⁾ into 𝒜_𝑛⁽²⁾. So we complete the proof of Theorem 3.4.

The following corollary follows from Theorem 3.4 and some basic properties for bounded linear operators on Hilbert space.

Corollary 3.5. Let 𝛾 and 𝛽 be as in Theorem 3.4. Then we have the following assertions. (1)The null space 𝒩(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is closed. (2)Let {𝐹_𝑛}^∞_𝑛=1 be a sequence in 𝒜_𝑛⁽²⁾ with 𝐹_𝑛→𝐹 as 𝑛→∞ for some 𝐹∈𝒜_𝑛⁽²⁾. Then ℱ_𝛾,𝛽(𝐹_𝑛)→ℱ_𝛾,𝛽(𝐹) as 𝑛→∞. That is to say, it is closed. (3)For all 𝐹 and 𝐺 in 𝒜_𝑛⁽²⁾, we have the Cauchy-Schwartz inequality ℱ_𝛾,𝛽𝐹,ℱ_𝛾,𝛽𝐺_𝒜𝑛(2)≤ℱ_𝛾,𝛽𝐹_𝒜𝑛(2)ℱ_𝛾,𝛽𝐺_𝒜𝑛(2)≤‖𝐹‖_𝒜𝑛⁽²⁾‖𝐺‖_𝒜𝑛⁽²⁾. (3.14)

Remark 3.6. As mentioned in Section 1, the Fourier-Wiener transform, the modified Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are also well-defined operators on 𝒜_𝑛⁽²⁾. In particular, from the definition of analytic Fourier-Feynman transform, it is an injective operator on 𝒜_𝑛⁽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 ℱ_𝛾,𝛽𝐹 is an element of 𝐿₂(𝐶₀[0,𝑇]) and ℱ_𝛾,𝛽𝐹₂=‖𝐹‖₂. (3.15) 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 operator ℱ_𝛾,𝛽 is invertible.

Theorem 3.7. Let 𝛾 and 𝛽 be as in Theorem 3.4 with Re(𝛽²/𝛾²)≤0 and |𝛽|=1. Then the inverse operator of the IT ℱ_𝛾,𝛽 exists and is given by ℱ⁻¹_𝛾,𝛽=ℱ_{𝑖(𝛾/𝛽),1/𝛽}. (3.16) Furthermore, the null space 𝒩(ℱ_𝛾,𝛽) consists of the zero vector only.

Proof. From Theorem 3.4, the operator is continuous from 𝒜_𝑛⁽²⁾ into 𝒜_𝑛⁽²⁾. Since Re(𝛽²/𝛾²)≤0 and ℱ_𝛾,𝛽(𝐹) is in 𝒜_𝑛⁽²⁾, ℱ_{𝑖(𝛾/𝛽),1/𝛽}(ℱ_𝛾,𝛽(𝐹)) exists and is in 𝒜_𝑛⁽²⁾. Also, since Re(1/𝛾²)≥0 and ℱ_{𝑖(𝛾/𝛽),1/𝛽}(𝐹) is in 𝒜_𝑛⁽²⁾, ℱ_𝛾,𝛽(ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹) exists and is in 𝒜_𝑛⁽²⁾. Now, using (2.27), for 𝐹∈𝒜_𝑛⁽²⁾, ℱ_{𝑖(𝛾/𝛽),1/𝛽}ℱ_𝛾,𝛽(𝐹)(𝑥)=𝐹(𝑥)=ℱ_𝛾,𝛽ℱ_{𝑖(𝛾/𝛽),1/𝛽}𝐹(𝑥), (3.17) which completes the proof of Theorem 3.7 as desired.

We have some observations for the inverse operator ℱ_{𝑖(𝛾/𝛽),1/𝛽} of ℱ_𝛾,𝛽.

Remark 3.8. (1) If 𝛾=√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 operator ℱ_𝛾,𝛽 might not be bijective. Hence we should consider that the domain of the inverse operator ℱ_{𝑖(𝛾/𝛽),1/𝛽} is the range of ℱ_𝛾,𝛽.
(3) The operator ℱ_𝛾,𝛽 is an homeomorphism from 𝒜_𝑛⁽²⁾ into ℛ(ℱ_𝛾,𝛽), where ℛ(ℱ_𝛾,𝛽) is the range of ℱ_𝛾,𝛽.

4. Some Spectral Theorems for the Bounded Linear Operator ℱ_𝛾,𝛽

In this section we will apply some spectral theories to the ℱ_𝛾,𝛽. To do this, we need some concepts related to the spectral theory on a Banach space.

With ℱ_𝛾,𝛽 we associate the operator ℱ^𝜆_𝛾,𝛽=ℱ_𝛾,𝛽−𝜆𝐼, (4.1) where 𝜆 is a complex number and 𝐼 is the identity operator on 𝒜_𝑛⁽²⁾. If ℱ^𝜆_𝛾,𝛽 has an inverse, we denote it by 𝑅_𝜆(ℱ_𝛾,𝛽); that is, 𝑅_𝜆ℱ_𝛾,𝛽=ℱ^𝜆_𝛾,𝛽⁻¹=ℱ_𝛾,𝛽−𝜆𝐼⁻¹, (4.2) and call it the resolvent operator of ℱ_𝛾,𝛽 or, simply, the resolvent of ℱ_𝛾,𝛽.

Definition 4.1. A regular value 𝜆 of ℱ_𝛾,𝛽 is a complex number such that (1)𝑅_𝜆(ℱ_𝛾,𝛽) exists, (2)𝑅_𝜆(ℱ_𝛾,𝛽) is bounded, (3)𝑅_𝜆(ℱ_𝛾,𝛽) is defined on a set which is dense in 𝒜_𝑛⁽²⁾. The resolvent set 𝜌(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is the set of all regular values 𝜆 of ℱ_𝛾,𝛽. Its complement 𝜎(ℱ_𝛾,𝛽)=ℂ−𝜌(ℱ_𝛾,𝛽) is called the spectrum of ℱ_𝛾,𝛽, and a 𝜆∈𝜎(ℱ_𝛾,𝛽) is called a spectral value of ℱ_𝛾,𝛽. For more details, see [16].

Remark 4.2. (1) The spectrum 𝜎(ℱ_𝛾,𝛽) is partitioned into three disjoint sets as follows. (i)The point spectrum or discrete spectrum 𝜎_𝑝(ℱ_𝛾,𝛽) is the set such that 𝑅_𝜆(ℱ_𝛾,𝛽) does not exist. A 𝜆∈𝜎_𝑝(ℱ_𝛾,𝛽) is called an eigenvalue of ℱ_𝛾,𝛽. (ii)The continuous spectrum 𝜎_𝑐(ℱ_𝛾,𝛽) is the set such that 𝑅_𝜆(ℱ_𝛾,𝛽) exists and satisfies (3) but not (2) in Definition 4.1; that is to say, 𝑅_𝜆(ℱ_𝛾,𝛽) is unbounded. (iii)The residual spectrum 𝜎_𝑟(ℱ_𝛾,𝛽) is the set such that 𝑅_𝜆(ℱ_𝛾,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 𝑅_𝜆(ℱ_𝛾,𝛽) is not dense in 𝒜_𝑛⁽²⁾.
(2) We know that ℂ=𝜌(ℱ_𝛾,𝛽)∪𝜎(ℱ_𝛾,𝛽)=𝜌(ℱ_𝛾,𝛽)∪𝜎_𝑝(ℱ_𝛾,𝛽)∪𝜎_𝑐(ℱ_𝛾,𝛽)∪𝜎_𝑟(ℱ_𝛾,𝛽).

From now on, if what operator ℱ_𝛾,𝛽 refers to is clear, we will write 𝑅_𝜆 instead of 𝑅_𝜆(ℱ_𝛾,𝛽).

In our next theorem, we apply the spectral theory to the operator ℱ_𝛾,𝛽.

Theorem 4.3. Let 𝛾 and 𝛽 be as in Theorem 3.4. Then the resolvent set 𝜌(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is open and hence the spectrum 𝜎(ℱ_𝛾,𝛽) is closed. Furthermore, for every 𝜆₀∈𝜌(ℱ_𝛾,𝛽), the resolvent 𝑅_𝜆 has the representation 𝑅_𝜆=^∞_𝑗=1𝜆−𝜆₀^𝑗𝑅_𝜆^𝑗+1₀, (4.3) where the series is absolutely convergent for every 𝜆 in the open disk given by 𝜆−𝜆₀<1𝑅_𝜆00 (4.4) in the complex plane. This disk is a subset of 𝜌(ℱ_𝛾,𝛽).

Proof. Theorem 4.3 immediately follows the fact that the ℱ_𝛾,𝛽 is a bounded linear operator on 𝒜_𝑛⁽²⁾.

Next, we note that the spectral radius 𝑟_𝜎(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is the radius 𝑟_𝜎ℱ_𝛾,𝛽=sup_{ℱ𝜆∈𝜎𝛾,𝛽}|𝜆| (4.5) of the smallest closed disk centered at the origin of the complex 𝜆-plane and containing 𝜎(ℱ_𝛾,𝛽).

Theorem 4.4. Let 𝛾 and 𝛽 be as in Theorem 3.4. Then the spectrum 𝜎(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is compact and lies in the disk given by |𝜆|≤ℱ_𝛾,𝛽0. (4.6) Hence the resolvent set 𝜌(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is not empty. Furthermore, the spectral radius 𝑟_𝜎(ℱ_𝛾,𝛽)≤‖ℱ_𝛾,𝛽‖₀ and 𝑟_𝜎ℱ_𝛾,𝛽=lim_𝑛→∞ℱ_𝛾,𝛽^𝑛₀^1/𝑛. (4.7)

Proof. From Theorem 3.4, the ℱ_𝛾,𝛽 is a bounded linear operator on 𝒜_𝑛⁽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 ℱ_𝛾,𝛽 we have 𝑟_𝜎ℱ_𝛾,𝛽≤ℱ_𝛾,𝛽0. (4.8) Also, we can easily obtain (4.7) as desired.

In our next theorem, we give a spectral mapping theorem for polynomials of ℱ_𝛾,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. Let 𝛾 and 𝛽 be as in Theorem 3.4. Let 𝑃(𝜆)=𝑎_𝑛𝜆^𝑛+𝑎_𝑛−1𝜆^𝑛−1+⋯+𝑎₀,𝑎_𝑛≠0,𝑛=1,2,…. (4.9) Then 𝜎𝑃ℱ_𝛾,𝛽=𝑃𝜎ℱ_𝛾,𝛽. (4.10) This implies that the spectrum 𝜎(𝑃(ℱ_𝛾,𝛽)) of the operator 𝑃(𝜆)=𝑎_𝑛𝑇^𝑛+𝑎_𝑛−1𝑇^𝑛−1+⋯+𝑎₀𝐼 (4.11) consists precisely of all those values which the polynomial 𝑃 assumes on the spectrum 𝜎(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽.

Next, we will explain that our study is meaningful to obtain the solution to a differential equation.

Let 𝐻 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 functional 𝑢(𝑥) on 𝐵, let 𝑁_𝑐 be an operator defined by the formula 𝑁_𝑐𝑢(𝑥)=−Tr_𝐻𝐷²𝑢(𝑥)+𝑐(𝑥,𝐷𝑢(𝑥))^∼,𝑥∈𝐵,𝑐∈ℂ/{0}, (4.12) where 𝐷 denotes the second Fréchet derivative and Tr_𝐻 denotes the trace of an operator. In [2], Lee showed that the integral transform ℱ_1/𝑐,𝑖, 𝑐∈ℂ/{0} forms the solution of a differential equation which is called a Cauchy problem 𝑢_𝑡(𝑥,𝑡)=𝒫𝑁_𝑐𝑢()𝑢()(𝑥),𝑥,𝑡,𝑥∈𝐵,𝑡>0𝑥,0=𝐹 (4.13) where 𝒫(𝜂)=𝑎_𝑚𝜂^𝑚+⋯+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 𝑢(𝑥,𝑡)=_𝐵𝐹𝑒^−𝑡𝑥+√1−𝑒^−2𝑡𝑦𝑑𝜈𝑦, (4.14) or, equivalently, 𝑢(𝑥,𝑡)=_𝐵𝐹𝑦𝑜_𝑡𝑥,𝑑𝑦, (4.15) where 𝑜_𝑡(𝑥,𝑑𝑦)=𝜈_{1−𝑒−2𝑡}(𝑒^−𝑡𝑥,𝑑𝑦). This showed that the family of measures {𝑜_𝑡(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 ℱ_1/𝑐,𝑖 is a well-defined bounded linear operator on 𝒜_𝑛⁽²⁾ and could be applied to all the results and formulas in this paper.

5. Applications for the Spectral Theory

In Sections 3 and 4, we treated the IT ℱ_𝛾,𝛽 as a bounded linear operator on 𝒜_𝑛⁽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 ℱ_𝛾,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 ℱ_𝛾,𝛽.

In our next theorem, we show that the operator ℱ_𝛾,𝛽 is a self-adjoint operator on 𝒜_𝑛⁽²⁾ under an appropriate condition for 𝛾 and 𝛽.

Theorem 5.1. Let 𝛾 be as in Theorem 3.4 with 𝛾²=𝛾² and let 𝛽=1. Then the operator ℱ_𝛾,𝛽 is a self-adjoint operator on 𝒜_𝑛⁽²⁾.

Proof. Let 𝑇^∗ denote the adjoint operator on an operator 𝑇. For all 𝐹 and 𝐺 in 𝒜_𝑛⁽²⁾, we note that 𝐹,ℱ^∗_𝛾,𝛽𝐺_𝒜𝑛(2)=ℱ_𝛾,𝛽𝐹,𝐺_𝒜𝑛(2)=_ℝ𝑛Γ_{ℱ𝛾,𝛽𝐹}^→𝑣𝑔^→𝑣𝑑^→𝑣=^𝑛_𝑗=12𝜋𝛾²⁻¹_{ℝ𝑛ℝ𝑛}𝑓^→𝑢exp−^𝑛_𝑗=1𝑢_𝑗−𝑣_𝑗²2𝛾²𝑑^→𝑢𝑔^→𝑣𝑑^→𝑣=^𝑛_𝑗=12𝜋𝛾²⁻¹_ℝ𝑛𝑓^→𝑢_ℝ𝑛𝑔^→𝑣exp−^𝑛_𝑗=1𝑣_𝑗−𝑢_𝑗²2𝛾²𝑑^→𝑣𝑑^→𝑢=^𝑛_𝑗=12𝜋𝛾²⁻¹_ℝ𝑛𝑓^→𝑢_ℝ𝑛𝑔^→𝑣exp−^𝑛_𝑗=1𝑣_𝑗−𝑢_𝑗²2𝛾²𝑑^→𝑣𝑑^→𝑢=𝐹,ℱ_𝛾,𝛽𝐺_𝒜𝑛(2), (5.1) which completes the proof of Theorem 5.1 as desired.

Remark 5.2. We gave the conditions for 𝛾 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 operator ℱ_𝛾,𝛽.

Corollary 5.3. Let 𝛾 and 𝛽 be as in Theorem 5.1. Then we have the following assertions.(1)For 𝜆∈𝜎(ℱ_𝛾,𝛽), there exists a positive real number 𝑘₀ such that ℱ_𝛾,𝛽−𝜆𝐼𝐹_𝒜𝑛(2)≥𝑘₀‖𝐹‖_𝒜𝑛⁽²⁾ (5.2) and so there exists a sequence (𝐹_𝑛)^∞_𝑛=1 in 𝒜_𝑛⁽²⁾ with ‖𝐹_𝑛‖_𝒜𝑛⁽²⁾=1, 𝑛=1,2,… such that lim_𝑛→∞ℱ_𝛾,𝛽−𝜆𝐼𝐹_{𝑛𝒜𝑛(2)}=0. (5.3)(2)For all 𝐹∈𝒜_𝑛⁽²⁾ and 𝜆∈ℂ, ℱ_𝛾,𝛽−𝜆𝐼𝐹_𝒜𝑛(2)≥|Im𝜆|‖𝐹‖_𝒜𝑛⁽²⁾. (5.4)

In our next theorem, we apply the spectral theory to the operator ℱ_𝛾,𝛽 as a self-adjoint operator.

Theorem 5.4. Let 𝛾 and 𝛽 be as in Theorem 5.1. Then the spectrum 𝜎(ℱ_𝛾,𝛽) of ℱ_𝛾,𝛽 is real and it lies in the closed interval [𝑘₁,𝑘₂] where 𝑘₁=inf_{‖𝐹‖𝒜𝑛(2)=1}ℱ_𝛾,𝛽(𝐹),𝐹_𝒜𝑛(2),𝑘₂=sup_{‖𝐹‖𝒜𝑛(2)=1}ℱ_𝛾,𝛽(𝐹),𝐹_𝒜𝑛(2). (5.5)(5.6) Furthermore, 𝑘₁ and 𝑘₂ are spectral values of ℱ_𝛾,𝛽 and max{|𝑘₁|,|𝑘₂|}=1.

Proof. First, since ℱ_𝛾,𝛽 is a self-adjoint operator on 𝒜_𝑛⁽²⁾, the spectrum 𝜎(ℱ_𝛾,𝛽) of ℱ_𝛾,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 𝑚=inf_{‖ℎ‖𝐻=1}(𝑇ℎ,ℎ)_𝐻,𝑀=sup_{‖ℎ‖𝐻=1}(𝑇ℎ,ℎ)_𝐻, (5.7) and (⋅,⋅)_𝐻 is an inner product on 𝐻. Hence the spectrum 𝜎(ℱ_𝛾,𝛽) lies in the closed interval [𝑘₁,𝑘₂]. Furthermore, 𝑘₁ and 𝑘₂ are spectral values of ℱ_𝛾,𝛽 and max{|𝑘₁|,|𝑘₂|}=‖ℱ_𝛾,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 (𝜆_𝑛)^∞_𝑛=1 be the set of eigenvalues of ℱ_𝛾,𝛽 with 𝜆_𝑛<𝜆_𝑚 for 𝑚<𝑛 and let (𝑒_𝑛)^∞_𝑛=1 be the set of eigenfunctions corresponding to (𝜆_𝑛)^∞_𝑛=1. Then we note that ℱ_𝛾,𝛽𝐹(𝑥)=^∞_𝑛=1𝑎_𝑛𝜆_𝑛𝑒_𝑛(𝑥), (5.8) where 𝑎_𝑛=(𝐹,𝑒_𝑛)_𝒜𝑛⁽²⁾. For each 𝑗=1,2,…, define an (orthogonal) projection 𝑃_𝑗 on 𝒜_𝑛⁽²⁾ by 𝑃_𝑗𝐹(𝑥)=𝑎_𝑗𝑒_𝑗(𝑥). Then we also note that ℱ_𝛾,𝛽𝐹(𝑥)=^∞_𝑛=1𝜆_𝑛𝑃_𝑛𝐹(𝑥). (5.9) Now, for 𝜆∈ℝ, define an operator 𝐸_𝜆 on 𝒜_𝑛⁽²⁾ by 𝐸_𝜆𝐹(𝑥)=∑_{𝜆𝑗≤𝜆}𝑃_𝑗𝐹(𝑥). In this case, ℰ≡(𝐸_𝜆) is called the spectral family of ℱ_𝛾,𝛽.

Theorem 5.5. Let 𝛾 and 𝛽 be as in Theorem 5.1. Then we have the spectral representation for ℱ_𝛾,𝛽 as follows: ℱ_𝛾,𝛽=^𝑘2_𝑘1𝜆𝑑𝐸_𝜆, (5.10) where 𝑘₁ and 𝑘₂ are given by (5.5) and (5.6), and {𝐸_𝜆} is the spectral family of ℱ_𝛾,𝛽.

Proof. First, we note that for 𝛾 and 𝛽 in Theorem 5.1, ℱ_𝛾,𝛽 is a self-adjoint operator on 𝒜_𝑛⁽²⁾. Furthermore, 𝜎(ℱ_𝛾,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 Theorem 5.5, for all real-valued continuous functions 𝑓 on [𝑘₁,𝑘₂] and for 𝐹 and 𝐺 in 𝒜_𝑛⁽²⁾, 𝑓ℱ_𝛾,𝛽=^𝑘2_𝑘1𝑓(𝜆)𝑑𝐸_𝜆,ℱ_𝛾,𝛽𝐹,𝐺_𝒜𝑛(2)=^𝑘2_𝑘1𝜆𝑑𝑤(𝜆), (5.11) where 𝑤(𝜆)=(𝐸_𝜆𝐹,𝐺)_𝒜𝑛⁽²⁾ and the integral is an ordinary Riemann-Stieltjes integral.
(2) An alternative formulation of the spectral theorem expresses the operator ℱ_𝛾,𝛽 as an integral of the coordinate function over the operator's spectrum with respect to a projection-valued measure ℱ_𝛾,𝛽=∫_{𝜎(ℱ𝛾,𝛽)}𝜆𝑑𝐸_𝜆. 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 Sections 3, 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.

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions which have improved the original paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A1004774:2011-0014552).