Spaces of von Neumann Type

Abstract

This chapter develops the mathematical foundations for the time-evolution of a physical systems as a three-dimensional motion picture (time-ordering). Our objective is to construct the mathematical version of a physical film on which space-time events can evolve. We first construct the film using infinite tensor products of Hilbert spaces, which is natural for physics. Although von Neumann [VN2] did not develop his theory for our purpose, it will be clear that it is natural for our approach. This film, as a Hilbert space, will be used as the ambient space in Chap. 7 for the Feynman (time-ordered) operator calculus. In order to make the theory available for applications beyond physics, we extend von Neumann’s method to construct infinite tensor products of Banach spaces. (This approach makes it easy to transfer the operator calculus to the Banach space setting.) We assume that the reader has read Sect. 1.4 of Chap. 1 This section provides a fairly complete introduction to the finite tensor product theory for both Hilbert and Banach spaces.

Appendices

6.7 Appendix

The study of infinite tensor products of Banach spaces is an important but neglected area. It offers a natural arena for the constructive, but general study of analysis in infinitely many variables, including partial differential equations and path integrals. In this appendix, we introduce a few topics that have independent interest.

6.1.1 6.7.1 The Fourier Transform Again

In Chap. 2, we defined the Fourier transform as a mapping from a uniformly convex Banach space to its dual space. This approach exploits the strong relationship between a uniformly convex Banach space and a Hilbert space at the expense of a restricted Fourier transform.

In addition to the definition in Chap. 2, it is also possible to define the Fourier transform, $\mathfrak{F}$, as a mapping on $L^{1}[\mathbb{R}_{I}^{n}]$ to $\mathbb{C}_{0}[\mathbb{R}_{I}^{n}]$ for all n as one fixed linear operator that extends to a definition on $L^{1}[\mathbb{R}_{I}^{\infty }]$. To do this requires a closer look at our Banach spaces defined on $\mathbb{R}_{I}^{\infty }$. Recall that $I = [-\tfrac{1} {2}, \tfrac{1} {2}]$, $\bar{x} = (x_{k})_{k=1}^{n},\;\hat{x} = (x_{k})_{k=n+1}^{\infty }$ and $h_{n}(\hat{x}) = \otimes _{k=n+1}^{\infty }\chi _{I}(x_{k})$ with $I = [-\tfrac{1} {2}, \tfrac{1} {2}]$. The measurable functions on $\mathbb{R}_{I}^{n},\;\mathcal{M}_{I}^{n}$ are defined by $f_{n}(x) = f_{n}^{n}(\bar{x}) \otimes h_{n}(\hat{x})$, where $f_{n}^{n}(\bar{x})$ is measurable on $\mathbb{R}^{n}$, so that $\mathcal{M}_{I}^{n}$ is a partial tensor product subspace generated by the unit vector $h(x) = h_{0}(\hat{x})$. From this, we see that all of the spaces of functions considered in Chap. 2 are also partial tensor product spaces generated by h(x). In this section we show how the replacement of $L^{1}[\mathbb{R}_{I}^{n}],\; \mathbb{C}_{0}[\mathbb{R}_{I}^{n}]$ by $L^{1}[\mathbb{R}_{I}^{n}](h),\; \mathbb{C}_{0}[\mathbb{R}_{I}^{n}](h)$ allows us to offer a different approach to the Fourier transform.

We define $\mathfrak{F}(f_{n})(\mathbf{x})$, mapping $\mathbf{L}^{1}[\mathbb{R}_{I}^{n}](h)$ into $\mathbb{C}_{0}[\mathbb{R}_{I}^{n}](\hat{h})$ by

$$\displaystyle\begin{array}{rcl} \mathfrak{F}(f_{n})(\mathbf{x}) = \otimes _{k=1}^{n}\mathfrak{F}_{ k}(f_{n}) \otimes _{k=n+1}^{\infty }\hat{h}_{ n}(\hat{x}),& &{}\end{array}$$

(6.2)

where the product of Sinc functions $\hat{h}_{n}(\hat{x}) = \left [\otimes _{k=n+1}^{n}\tfrac{sin(\pi y_{k})} {\pi y_{k}} \right ]$ is the Fourier transform of the product $\prod \nolimits _{k=n+1}^{\infty }I$ of the interval I.

Theorem 6.43.

The operator $\mathfrak{F}$ extends to a bounded linear mapping of $L^{1}[\mathbb{R}_{I}^{\infty }](h)$ into $\mathbb{C}_{0}[\mathbb{R}_{I}^{\infty }](\hat{h})$ .

Proof.

Since

$$\displaystyle{\mathop{\lim }\limits _{n\rightarrow \infty }L^{1}[\mathbb{R}_{ I}^{n}](h) =\bigcup _{ n=1}^{\infty }L^{1}[\mathbb{R}_{ I}^{n}](h) = L^{1}[\mathbb{R}{'}_{ I}^{\infty }](h)}$$

and $L^{1}[\mathbb{R}_{I}^{\infty }](h)$ is the closure of $L^{1}[\mathbb{R}{'}_{I}^{\infty }](h)$ in the $L^{1}\text{-}norm =\varDelta _{1}$, it follows that $\mathfrak{F}$ is a bounded linear mapping of $L^{1}[\mathbb{R}{'}_{I}^{\infty }](h)$ into $\mathbb{C}_{0}[\mathbb{R}_{I}^{\infty }](\hat{h})$.

Suppose that $\{f_{n}\} \subset L^{1}[\mathbb{R}{'}_{I}^{\infty }](h)$, converges to $f \in L^{1}[\mathbb{R}_{I}^{\infty }](h)$. Since the sequence is Cauchy, $\left \|f_{n} - f_{m}\right \|_{1} \rightarrow 0$ as $m,\ n \rightarrow \infty$, it follows that

$$\displaystyle{\left \vert \mathfrak{F}\left (f_{n}(\mathbf{x}) - f_{m}(\mathbf{x})\right )\right \vert \leqslant \int _{\mathbb{R}_{I}^{\infty }}\left \vert f_{n}(\mathbf{y}) - f_{m}(\mathbf{y})\right \vert d\lambda _{\infty }(\mathbf{y}) = \left \|f_{n} - f_{m}\right \|_{1}.}$$

Thus, $\left \vert \mathfrak{F}\left (f_{n}(\mathbf{x}) - f_{m}(\mathbf{x})\right )\right \vert$ is also a Cauchy sequence in $\mathbb{C}_{0}[\mathbb{R}_{I}^{\infty }](\hat{h})$. Since $L^{1}[\mathbb{R}{'}_{I}^{\infty }](h)$ is dense in $L^{1}[\mathbb{R}_{I}^{\infty }](h)$, it follows that $\mathfrak{F}$ has a bounded extension, mapping $L^{1}[\mathbb{R}_{I}^{\infty }](h)$ into $\mathbb{C}_{0}[\mathbb{R}_{I}^{\infty }](\hat{h})$. □ 

Corollary 6.44.

The operator $\mathfrak{F}$ extends to a bounded linear mapping of $L^{1}[\mathcal{B}](h)$ into $\mathbb{C}_{0}[\mathcal{B}](\hat{h})$ .

Just as for L ², the Fourier transform is an isometric isomorphism from $KS^{2}[\mathbb{R}^{n}]$ onto $KS^{2}[\mathbb{R}^{n}]$.

Corollary 6.45.

The operator $\mathfrak{F}$ is an isometric isomorphism of $KS^{2}[\mathbb{R}_{I}^{\infty }](h)$ onto $KS^{2}[\mathbb{R}_{I}^{\infty }](\hat{h})$ and an isometric isomorphism of  $KS^{2}[\mathcal{B}](h)$ onto $KS^{2}[\mathcal{B}](\hat{h})$ .

Thus, unlike the theory in Chap. 2, the natural interpretation is that the Fourier transform induces a Pontryagin duality like theory that does not depend on the group structure of $\mathbb{R}_{I}^{\infty }$, (or $\mathcal{B}$) but depends on the pairing of different function spaces. This approach is direct, constructive, and applies to all separable Banach spaces (with an S-basis). Thus, the group structure of the underlying measure space plays no role.

6.1.2 6.7.2 Unbounded Operators on $\mathcal{B}_{\otimes }^{\alpha }$

In this section, we assume that I is countable. For each i ∈ I, let A _i be a closed densely defined linear operator on $\mathcal{B}_{i}$, with domain D(A _i ), and let A _i be its extension to $\mathcal{B}_{\otimes }^{\alpha }$, with domain $D(\mathbf{A}_{i}) \supset \tilde{ D}(\mathbf{A}_{i}) = D(A_{i}) \otimes (\otimes _{k\neq i}\mathcal{B}_{k})$. The next theorem follows directly from the definition of the tensor product of semigroups and the fact that α is a faithful relative tensor norm.

Theorem 6.46.

Let $A_{i},\;1\leqslant i\leqslant n$ be generators of a family of C ₀ -semigroups S _i (t) on $\mathcal{B}_{i}$ with $\left \|S_{i}(t)\right \|_{\mathcal{B}_{i}}\leqslant M_{i}e^{\omega _{i}t}$ . Then $\mathbf{S}_{n}(t) =\hat{ \otimes }_{i=1,n}^{\alpha }S_{i}(t)$ , defined on $\hat{\otimes }_{i=1,n}^{\alpha }\mathcal{B}_{i}$ , has a unique extension (also denoted by S _n (t)) to all of $\mathcal{B}_{\otimes }^{\alpha }$ , such that for all vectors $\sum _{k=1}^{K} \otimes _{i\in I}\varphi _{i}^{k}$ with $\varphi _{l}^{k} \in D(A_{l}),\;1\leqslant l\leqslant n$ , the infinitesimal generator for S _n (t) satisfies:

$$\displaystyle{\mathbf{A}^{n}\left [\sum _{ k=1}^{K} \otimes _{ i\in I}\varphi _{i}^{k}\right ] =\sum _{ l=1}^{n}\sum _{ k=1}^{K}A_{ l}\varphi _{l}^{k}(\otimes _{ i\in I}^{i\neq l}\varphi _{ i}^{k}).}$$

Definition 6.47.

Let {A _i }, i ∈ I be a family of closed densely defined linear operators on $\mathcal{B}_{i}$ and let $\varphi _{i} \in D(A_{i})$ (respectively ψ _i  ∈ D(A _i )), with $\left \|\varphi _{i}\right \|_{\mathcal{B}} = 1$ (respectively $\left \|\psi _{i}\right \|_{\mathcal{B}} = 1$), for all i ∈ I.

• (1) We say that $\varphi = \otimes _{i\in I}\varphi _{i}$ is a strong convergence sum (scs)-vector for the family {A _i } if $\mathop{\lim }\limits _{n\rightarrow \infty }\sum \nolimits _{k=1}^{n}\mathbf{A}_{k}\varphi =\sum _{ k=1}^{\infty }A_{k}\varphi _{k} (\otimes _{i\in I}^{i\neq k}\varphi _{i})$ exists.

• (2) We say that ψ = ⊗ _i ∈ I ψ _i is a strong convergence product (scp)-vector for the family {A _i } if $\mathop{\lim }\limits _{n\rightarrow \infty }\prod \nolimits _{k=1}^{n}\mathbf{A}_{k}\psi = \otimes _{i\in I}A_{i}\psi _{i}$ exists.

Let $\mathbf{D}_{\varphi }$ be the linear span of {χ = ⊗ _i ∈ I χ _i ,  χ _i  ∈ D(A _i )}, with $\;\chi _{i} =\varphi _{i}$ (and let D _η be the linear span of {η = ⊗ _i ∈ I η _i ,  η _i  ∈ D(A _i )}, with  η _i  = ψ _i ) for all i > L, where L is arbitrary but finite. Clearly, $\mathbf{D}_{\varphi }$ is dense in $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ (D _η is dense in $\mathcal{B}_{\otimes }^{\alpha }(\psi )^{s}$). If there is a possible chance for confusion, we let A _s , respectively A _p , denote the closure of $\sum \nolimits _{k=1}^{\infty }\mathbf{A}_{k}$ on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ (respectively $\prod \nolimits _{k=1}^{\infty }\mathbf{A}_{k}$ on $\mathcal{B}_{\otimes }^{\alpha }(\psi )^{s}$). It follows that $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ (respectively $\mathcal{B}_{\otimes }^{\alpha }(\psi )^{s}$) are natural spaces for the study of infinite sums or products of unbounded operators. The notion of a strong convergence sum vector first appeared in Reed [RE].

Definition 6.48.

We call $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ a RS-space (respectively a RP-space $\mathcal{B}_{\otimes }^{\alpha }(\psi )^{s}$) for the family {A _i }.

Let {U _k (t)} be a set of unitary groups on $\{\mathcal{H}_{k}\}$. It is easy to see that $U(t) =\hat{ \otimes }_{k=1}^{\infty }U_{k}(t)$ is a unitary group on $\mathcal{H}_{\otimes }^{2}$. However, it need not be reduced on any partial tensor product subspace. The following results are due to Streit [ST] and Reed [RE], as indicated.

Theorem 6.49 (Streit).

Suppose { A _k } is a set of self-adjoint linear operators on the space $\mathcal{H}_{\otimes }^{2}(\varphi )^{s}$ , with corresponding unitary groups {U _k (t)}. If $U(t) =\hat{ \otimes }_{k=1}^{\infty }U_{k}(t)$ , then $\mathbf{P}_{\varphi }^{s}U(t) = U(t)\mathbf{P}_{\varphi }^{s}$ (i.e., U(t) is reduced on $\mathcal{H}_{\otimes }^{2}(\varphi )^{s}$ ) and U(t) is a strongly continuous unitary group on $\mathcal{H}_{\otimes }^{2}(\varphi )^{s}$ if and only if, for each c > 0, the following three conditions are satisfied:

• (1) $\sum \nolimits _{k=1}^{\infty }\left \vert \left \langle \mathbf{A}_{k}E_{k}[-c,c]\varphi _{k},\varphi _{k}\right \rangle \right \vert <\infty$ ,

• (2) $\sum \nolimits _{k=1}^{\infty }\left \vert \left \langle \mathbf{A}_{k}^{2}E_{k}[-c,c]\varphi _{k},\varphi _{k}\right \rangle \right \vert <\infty$ ,

• (3) $\sum \nolimits _{k=1}^{\infty }\left \vert \left \langle (I_{k} - E_{k}[-c,c])\varphi _{k},\varphi _{k}\right \rangle \right \vert <\infty$ ,

where E _k [−c,c] are the spectral projectors of A _k and, in this case, $U(t) = s -\lim _{n\rightarrow \infty }\hat{\otimes }_{k=1}^{n}U_{k}(t)$ .

Corollary 6.50.

Conditions 1–3 are satisfied if and only if there exists a strong convergence vector $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ for the family {A _k } such that $\varphi _{k} \in D(A_{k})$ and

$$\displaystyle{\sum \nolimits _{k=1}^{\infty }\left \vert \left \langle \mathbf{A}_{ k}\varphi _{k},\varphi _{k}\right \rangle \right \vert <\infty,\text{ }\sum \nolimits _{k=1}^{\infty }\left \|\mathbf{A}_{ k}\varphi _{k}\right \|^{2} <\infty.}$$

Theorem 6.51 (Reed).

U(t) is reduced on $\mathcal{H}_{\otimes }^{2}(\varphi )^{s}$ and U(t) is a strongly continuous unitary group on $\mathcal{H}_{\otimes }^{2}(\varphi )^{s}$ if and only if $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ is a strong convergence vector for the family {A _k } and $\sum \nolimits _{k=1}^{\infty }\left \vert \left \langle \mathbf{A}_{k}\varphi _{k},\varphi _{k}\right \rangle \right \vert <\infty$ . If each A _k is positive, the statement is true without the absolute value in the above. In either case, A , the closure of $\sum \nolimits _{k=1}^{\infty }\mathbf{A}_{k}$ , is the generator of U(t).

The next result strengthens and extends Reed’s theorem to contraction semigroups on Banach spaces (e.g., the positivity requirement above can be dropped).

Theorem 6.52.

Let {S _k (t)} be a family of strongly continuous contraction semigroups with generators {A _k } defined on $\{\mathcal{B}_{k}\}$ , and let $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ be a strong convergence vector for the family {A _k }. Then $\mathbf{S}(t) =\hat{ \otimes }_{k=1}^{\infty }S_{k}(t)$ is reduced on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ and is a strongly continuous contraction semigroup. If $\mathbf{S}(t) =\hat{ \otimes }_{k=1}^{\infty }S_{k}(t)$ is reduced on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ and is a strongly continuous contraction semigroup on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ , then there exists a strong convergence vector $\psi = \otimes _{k=1}^{\infty }\psi _{k} \in \mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ for the family {A _k }.

Proof.

Let $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ be a strong convergence vector for the family {A _k }. Without loss, we can assume that $\left \|\varphi _{k}\right \| = 1$. Let $\mathbf{S}_{n}(t) =\hat{ \otimes }_{k=1}^{n}S_{k}(t)\hat{ \otimes } (\otimes _{k=n+1}^{\infty }I_{k})$ and observe that S _n (t) is a contraction semigroup on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ for all finite n. Furthermore, its generator is the closure of $\mathbf{A}^{n} =\sum \nolimits _{ k=1}^{n}\mathbf{A}_{k}$, where $\mathbf{A}_{k} = A_{k}\hat{ \otimes } (\otimes _{i\neq k}^{\infty }I_{i})$. If n and m are arbitrary, then

$$\displaystyle{\begin{array}{l} \left [\mathbf{S}_{n}(t) -\mathbf{S}_{m}(t)\right ]\varphi =\int _{ 0}^{1}\frac{d} {d\lambda }\left \{\mathbf{S}_{n}[\lambda t]\mathbf{S}_{m}[(1-\lambda )t]\right \}\varphi d\lambda \\ \text{ } = t\int _{0}^{1}\mathbf{S}_{n}[\lambda t]\mathbf{S}_{m}[(1-\lambda )t]\left [\mathbf{A}^{n} -\mathbf{A}^{m}\right ]\varphi d\lambda,\\ \end{array} }$$

where we have used the fact that if two semigroups commute, then their corresponding generators also commute. It follows that:

$$\displaystyle{\left \|\left [\mathbf{S}_{n}(t) -\mathbf{S}_{m}(t)\right ]\varphi \right \|\leqslant t\left \|\left [\mathbf{A}^{n} -\mathbf{A}^{m}\right ]\varphi \right \|.}$$

Since $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ is a strong convergence vector for the family {A _k }, it follows that $s\text{ - }\lim _{n\rightarrow \infty }\mathbf{S}_{n}(t) = \mathbf{S}(t)$ exists on a dense set in $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$. As $\left \|\mathbf{S}(t)\right \|\leqslant \overline{\lim }_{n\rightarrow \infty }\left \|\mathbf{S}_{n}(t)\right \| <\infty$, we see that S(t) is bounded. To see that it must be a contraction, choose n so large that $\left \|\left [\mathbf{S}_{n}(t) -\mathbf{S}(t)\right ]\varphi \right \|_{\otimes } <\varepsilon \left \|\varphi \right \|_{\otimes }$. It follows that

$$\displaystyle{\left \|\mathbf{S}(t)\varphi \right \|_{\otimes }\leqslant \left \|\mathbf{S}_{n}(t)\varphi \right \|_{\otimes } + \left \|\left [\mathbf{S}_{n}(t) -\mathbf{S}(t)\right ]\varphi \right \|_{\otimes } <\left \|\varphi \right \|_{\otimes }(1+\varepsilon ).}$$

Thus, S(t) is a contraction operator on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$. It is easy to check that it is a C ₀-semigroup.

Now suppose that $\mathbf{S}(t) =\hat{ \otimes }_{k=1}^{\infty }S_{k}(t)$ is a strongly continuous contraction semigroup which is reduced on $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$. It follows that the generator A of S(t) is m-dissipative, and hence defined on a dense domain D(A) in $\mathcal{B}_{\otimes }^{\alpha }(\varphi )^{s}$ with S′(t)ψ = S(t)A ψ = AS(t)ψ for all ψ ∈ D(A). Since any such ψ is of the form $\psi =\sum _{ l=1}^{\infty }\psi ^{l} =\sum _{ l=1}^{\infty }\otimes _{k=1}^{\infty }\psi _{k}^{l}$, where $\psi ^{l} = \otimes _{k=1}^{\infty }\psi _{k}^{l}$ is in D(A). A simple computation shows that $\mathbf{A}\psi ^{l} =\sum _{ k=1}^{\infty }\mathbf{A}_{k}\psi ^{l}$, so that any ψ ^l is a strong convergence vector for the family {A _k }. □ 

It is easy to see that, in the second part of the theorem, we cannot require that $\varphi = \otimes _{k=1}^{\infty }\varphi _{k}$ itself be a strong convergence vector for the family {A _k } since it need not be in the domain of A. For example, $\varphi _{1}\notin D(A_{1})$, while $\varphi _{k} \in D(A_{k}),\;k\neq 1$.

Example 6.53.

Let A _i be the generator of a C ₀ -contraction semigroup T _i (t) on $\mathbb{C}_{0}[X_{i}]$ for each i ∈ I, and assume that T _i (t) has the representation:

$$\displaystyle{T_{i}(t)\varphi _{i}(\mathbf{x}) =\int _{X_{i}}K_{i}[\mathbf{x},t;\mathbf{y},0]\varphi _{i}(\mathbf{y})dm_{i}(\mathbf{y}).}$$

Where m _i is an associated measure and K _i [ x ,t; z ,s] is a kernel function which satisfies

$$\displaystyle{\int _{X_{i}}K_{i}[\mathbf{x},t;\mathbf{z},s]K_{i}[\mathbf{z},s;\mathbf{y},0]dm_{i}(\mathbf{z}) = K_{i}[\mathbf{x},t + s;\mathbf{y},0].}$$

Let $\varphi _{i} \in \ker \{\mathbf{A}_{i}\}$ , with $\left \|\varphi _{i}\right \|_{X_{i}} = 1$ for each i ∈ I, and note that $\varphi _{i} \in \ker \{\mathbf{A}_{i}\} \Rightarrow T_{i}(t)\varphi _{i} =\varphi _{i}$ . With $\varphi = \otimes _{i\in I}\varphi _{i}$ , construct $\mathbb{C}_{\otimes }^{\lambda }[\varphi ]^{s}$ . It follows that, for any $\psi =\sum \nolimits _{ j=1}^{m}\otimes _{i\in I}\psi _{i}^{j}$ with ψ _i ^j ∈ D( A _i ) and $\psi _{i}^{j} =\varphi _{i}$ for all but a finite number of i for each j, we have that the operator

$$\displaystyle{\mathbf{A}^{n}\psi =\sum \nolimits _{ k=1}^{n}\mathbf{A}_{ k}\psi =\sum \nolimits _{ k=1}^{n}\sum \nolimits _{ j=1}^{m}A_{ k}\psi _{k}^{j} \otimes _{ i\neq k}(\otimes _{i\in I}\psi _{i}^{j})}$$

is finite and well defined on a dense set D in $\mathbb{C}_{\otimes }^{\lambda }[\varphi ]^{s}$ and hence has a closure, which we also denote by A ⁿ .

From Theorems 6.33 and 6.52, we have:

Theorem 6.54.

For each n, A ⁿ is the generator of a C ₀ -contraction semigroup T ⁿ (t) on $\mathbb{C}_{\otimes }^{\lambda }[\varphi ]^{s}$ and

• (1) $s -\lim _{n\rightarrow \infty }\mathbf{A}^{n} = \mathbf{A}$ has a closure which generates a C ₀ -contraction semigroup T (t),

• (2) $s -\lim _{n\rightarrow \infty }\mathbf{T}^{n}(t) = \mathbf{T}(t)$ ,

• (3) for all $F(\mathbf{x}) \in \mathbb{C}_{0}[X]$ ,

  $$\displaystyle{\mathbf{T}(t)F(\mathbf{x}) =\int _{X}\mathbf{K}[\mathbf{x},t\,;\,\mathfrak{D}\mathbf{y},0]F(\mathbf{y}),}$$

  where x = ( x ₁ ,  x ₂ , ⋯ ),   y = ( y ₁ ,  y ₂ , ⋯ ), and

  $$\displaystyle{\mathbf{K}\left [\mathbf{x},t: \mathfrak{D}\mathbf{y},0\right ] =\mathop{ \otimes }\limits _{i=1}^{\infty }K_{ i}\left [\mathbf{x}_{i},t: \mathbf{y}_{i},0\right ]dm_{i}\left (\mathbf{y}_{i}\right ).}$$

Example 6.55.

Let {m _i } be a family of probability measures on $\mathbb{R}_{I}$ , and let m be the induced version of the family on $\mathbb{R}_{I}^{\infty }$ . Let $\phi _{i} = a_{i}x_{i} \in \mathbf{L}^{p}[\mathbb{R}_{I},m_{i}]$ , with $0 <\prod _{i\in I}\left \vert a_{i}\right \vert <\infty$ , $\left \|\phi _{i}\right \|_{\mathbb{R}_{I}}^{p} = 1$ and construct $\mathbf{L}_{\otimes }^{\varDelta _{p}}[\phi ]^{s}\mathop{\cong}\mathbf{L}^{p}[\mathbb{R}_{I}^{\infty },m]$ . Let {δ _i (x _i )} be a family of functions such that $\sum _{i=1}^{\infty }\left \|a_{i}\delta _{i}\right \|^{p} <\infty$ and define $A_{i} = \tfrac{1} {2}\sigma _{ii}(x_{i}) \frac{\partial ^{2}} {\partial x_{i}^{2}} -\delta _{i}(x_{i}) \frac{\partial } {\partial x_{i}}$ , where $0 <\sigma _{ii}(x_{i})$ . Since $\frac{\partial \phi _{i}} {\partial x_{i}} = D_{i}\phi _{i} = a_{i}$ and $\varDelta _{i}\phi _{i} = \frac{\partial ^{2}\phi _{ i}} {\partial x_{i}^{2}} = 0$ , it is easy to see that ϕ _i ∈ D(A _i ) for each i. It follows that $\phi = \otimes _{i=1}^{\infty }\phi _{i} \in \mathbf{L}^{p}[\mathbb{R}_{I}^{\infty },m]$ is a strong convergence vector for the family {A _i } and a strong convergence product vector for the family {D _i }.

Theorem 6.56.

With the conventions as above:

• (1) The closure of the operator $\mathbf{A} = \sum _{i=1}^{\infty }[\tfrac{1} {2}\sigma _{ii}(x_{i}) \frac{\partial ^{2}} {\partial x_{i}^{2}} -\delta _{i}(x_{i}) \frac{\partial } {\partial x_{i}}]$ is a densely defined generator of a contraction semigroup on $\mathbf{L}^{p}[\mathbb{R}_{I}^{\infty },m]$ .

• (2) The closure of $D = \frac{\partial ^{\infty }} {\partial x_{1}\partial x_{2}\cdots }$ is a densely defined linear operator on $\mathbf{L}^{p}[\mathbb{R}_{I}^{\infty },m]$ .

Remark 6.57.

Theorem 6.56 can easily be shown to apply to any Banach space with an S-basis, with minor changes. Compare this with Theorem 2.102 of Chap. 2.

Discussion

The following special cases have appeared in the literature:

• (1) If, in our definition of A, we set δ(x _i ) = 0 and $\sigma _{ii}(x_{i}) = 2$, we get the natural infinite dimensional Laplacian:

  $$\displaystyle{\mathbf{A} =\varDelta _{\infty } =\sum \nolimits _{ i=1}^{\infty }\partial ^{2}/\partial x_{ i}^{2}.}$$

• (2) If δ(x _i ) = −b _i x _i and $\sigma _{ii}(x_{i}) = 1$, we get the nonterminating diffusion generator in infinitely many variables (also known as the Ornstein–Uhlenbeck operator):

  $$\displaystyle{\mathbf{A} = \tfrac{1} {2}\varDelta _{\infty }- B\mathbf{x} \cdot \nabla _{\infty } = \tfrac{1} {2}\sum \nolimits _{i=1}^{\infty }\partial ^{2}/\partial x_{ i}^{2} -\sum \nolimits _{ i=1}^{\infty }b_{ i}x_{i}\partial /\partial x_{i}.}$$

• (3) If $\delta (x_{i}) = \frac{-x_{i}} {c^{2}}$ and $\sigma _{ii}(x_{i}) = 2$, we get the infinite dimensional Laplacian of Umemura [UM]:

  $$\displaystyle{\mathbf{A} =\sum \limits _{ i=1}^{\infty }\left ( \frac{\partial ^{2}} {\partial x_{i}^{2}} -\frac{x_{i}} {c^{2}} \frac{\partial } {\partial x_{i}}\right ).}$$

Berezanskii and Kondratyev [BK, pp. 520–521] have also discussed operators analogous to (2) and (3).

Open Problem

In this section, we identify an interesting problem that we believe is worthy of further study.

From our definition of Δ:

$$\displaystyle{\varDelta = \left \{\{\phi _{\nu }\}\left \vert \right.0\neq \left \|\otimes _{\nu \in I}\phi _{\nu }\right \|_{\mathcal{H}_{\otimes }^{2}},\;\,\&\;\sum \nolimits _{\nu \in I}\left \vert 1 -\left \|\phi _{\nu }\right \|_{\mathcal{B}}\right \vert <\infty \right \},}$$

we see that every nonzero basic vector in $\mathcal{B}_{\otimes }^{\alpha }$ is nonzero in $\mathcal{H}_{\otimes }^{2}$. This raises an important question, but we first need a little background.

Recall that $\left (\mathbf{L}^{1}[X_{i}]\right )^{{\ast}{\ast}} = \mathcal{M}[X_{i}]$, where $\mathcal{M}[X_{i}]$ is the set of bounded, regular, complex-valued measures on X _i that are absolutely continuous with respect to m _i (see below). We define the (total) variation of μ in $\mathcal{M}[X_{i}]$ by:

$$\displaystyle{\left \vert \mu \right \vert (X_{i})\; =\mathop{ \sup }\limits _{ess.\sup \vert h(x)\vert \leqslant 1}\left \vert \int _{X_{i}}h(x)d\mu (x)\right \vert.}$$

The sup is over $h \in \mathbf{L}^{\infty }[X_{i}]$, and $\left \vert \;\cdot \;\right \vert$ is the induced norm on $\mathcal{M}[X_{i}]$. Since $\mathcal{M}[X_{i}]$ is a separable Banach space, construct $\mathcal{H}_{i}^{1} \subset \mathcal{M}[X_{i}] \subset \mathcal{H}_{i}^{2}$.

Definition 6.58.

If μ,  μ′ are any two measures in $\mathcal{M}$:

• (1) We say that μ′ is singular with respect to μ and write it as μ′ ⊥ μ if, for each $\varepsilon> 0$, there exists a set $\varOmega \subset X_{i}$ such that $\mu '(\varOmega ) <\varepsilon$ and $\,\mu (X_{i})\,\setminus \varOmega ) <\varepsilon$.

• (2) We say that μ′ is absolutely continuous with respect to μ and write it as μ′ ≪ μ if, for each set $\varOmega \subset X_{i}$ such that $\mu (\varOmega ) = 0,\;\Rightarrow \,\mu '(\varOmega ) = 0$.

• (3) If μ′ ≪ μ and μ ≪ μ′, we say that μ and μ′ are equivalent and write it as μ′ ≈ μ.

If we define the square root of a complex function using the principal branch, in the third case, by the Radon–Nikodym theorem there exist (unique) measurable complex-valued functions p′(x),  p(x) such that $p'(x) = d\mu '(x)/d\mu (x)^{c}$, and $p(x) = d\mu (x)/d\mu '(x)^{c}$, where a ^c is the complex conjugate of a. If we set

$$\displaystyle\begin{array}{rcl} \begin{array}{l} H_{i}(\mu,\mu ') =\int _{X_{i}}\sqrt{d\mu (x)}\sqrt{d\mu '(x)^{c}} =\int _{X_{i}}\sqrt{\left (d\mu (x) /d\mu '(x)^{c } \right )}\;d\mu '(x)^{c} \\ \text{ } =\int _{X_{i}}\sqrt{\left (d\mu '(x)^{c } /d\mu (x) \right )}\;d\mu (x) =\int _{X_{i}}\sqrt{\left (d\mu (x) /d\lambda \right ) \left (d\mu '(x)^{c } /d\lambda \right )}\;d\lambda,\\ \end{array} & & {}\\ \end{array}$$

we obtain a complex version of the Hellinger integral, which defines a complex inner product, where $\lambda$ is any positive measure with μ ≪ λ and μ′ ≪ λ (for example, $\lambda = m_{i} \vee \;\tfrac{1} {2}\left \vert \mu +\mu '\right \vert$). In this case, H _i (μ, μ′)^c = H _i (μ′, μ) and $\mu '\, \approx \mu \Rightarrow \; H_{i}\left (\mu,\mu '\right )\neq 0$. It is easy to see that $H_{i}(\mu,\mu )\leqslant ([\left \vert \mu \right \vert ]^{1/2})^{2}(X_{i}) = \left \|\mu \right \|_{i}$, so, without loss, we can assume that $H_{i}(\mu,\mu ') = \left (\mu,\mu '\right )_{2i}$ is the inner product for our Hilbert space $\mathcal{H}_{2i}$.

If γ is the natural tensor norm for the space of measures, so that $\mathcal{M}[X_{i}]\mathop{\hat{\otimes }^{\gamma }}\mathcal{M}[X_{j}] = \mathcal{M}[X_{i} \times X_{j}]$, we can construct $\mathop{\hat{\otimes }^{\gamma }}\limits _{i\in \mathbb{N}}\mathcal{M}_{i} = \mathcal{M}_{\otimes }^{\gamma }$ so that $\mathcal{H}_{\otimes }^{1} \subset \mathcal{M}_{\otimes }^{\gamma }\subset \mathcal{H}_{\otimes }^{2}$. For each $\lambda _{i},\;\mu _{i} \in \mathcal{M}_{i}$, let $\lambda _{i}^{{\ast}},\;\mu _{i}^{{\ast}}$ be the Steadman duality maps, where $\left \langle \mu _{i},\lambda _{i}^{{\ast}}\right \rangle _{i} = \left (\mu _{i},\lambda _{i}\right )_{2i}\left (\left \|\lambda _{i}\right \|_{\mathcal{M}}^{2}/\left \|\lambda _{i}\right \|_{\mathcal{H}_{2}}^{2}\right )$ and $\left \langle \lambda _{i},\mu _{i}^{{\ast}}\right \rangle _{i} = \left (\lambda _{i},\mu _{i}\right )_{2i}\left (\left \|\mu _{i}\right \|_{\mathcal{M}}^{2}/\left \|\mu _{i}\right \|_{\mathcal{H}_{2}}^{2}\right )$. We now have the following problem:

• (I) Is it true that for $\mu = \otimes _{i\in \mathbb{N}}\mu _{i},\;\lambda = \otimes _{i\in \mathbb{N}}\lambda _{i}$ in $\mathcal{M}_{\otimes }^{\gamma }$ with $\mu _{i} \approx \lambda _{i}$ for each $i \in \mathbb{N}$, we have that $\mu \equiv ^{s}\,\lambda \, \Leftrightarrow \,\mu \approx \lambda$ (so that $\mu \in \mathcal{M}_{\otimes }^{\gamma }(\lambda )^{s}$) and $\mu \,\perp \,\lambda \,\Leftrightarrow \,\mu \notin \mathcal{M}_{\otimes }^{\gamma }(\lambda )^{s}$?

von Neumann [VN2] first mentioned this problem, in a restricted sense, in relation to the decomposition of $\mathcal{H}_{\otimes }^{2}$ into orthogonal subspaces and the theory of probability measures on infinite product spaces. (Note that his incomplete direct product is our partial tensor product.) He stated that: “Another application of our theory could be made to the theory of measures in infinite product spaces, which is the basis for the modern theory of probabilities. Here a certain incomplete direct product of $\mathcal{H}_{\otimes }^{2}$ is fundamental.”

Ten years later, Kakutani [KA], in Chap. 5, published his now famous paper on the equivalence and orthogonality of infinite product measures. In the second paragraph of the introduction to his paper, Kakutani states: “In particular, the introduction of the inner product and isometric embedding of $\mathfrak{M}(\varOmega,\mathfrak{B},m)$ (set of all probability measures on $(\varOmega,\mathfrak{B},m)$) into a general Euclidean space (Hilbert space), as well as the indication of the relationship of this paper with earlier works of E. Hellinger, are due to Professor J. von Neumann.”

The space $\mathfrak{M}(\varOmega,\mathfrak{B},m)$ is not a Banach space, but each element has norm 1 in the space of measures and the embedding Hilbert space. In our case:

$$\displaystyle{\mathcal{H}_{\otimes }^{2}\left (\mu \right ) \supset \mathcal{M}_{ \otimes }^{\gamma }\left (\mu \right ) \supset \mathfrak{M}_{ \otimes }^{\gamma }\left (\mu \right ).}$$

If $\mathfrak{M}_{\otimes }^{\gamma }\left (\mu \right )$ contains an orthonormal basis for $\mathcal{H}_{\otimes }^{2}\left (\mu \right )$, for every μ, we would have a positive answer to (I).

If the answer to (I) is true, this would explain the appearance of this phenomenon in general and would provide insight into the causes for the failure of certain expected/desired properties of (probability) measures on infinite dimensional spaces. These failures could then be directly linked to the breaking up of the infinite tensor product spaces into orthogonal subspaces as described by Theorem 6.20.