UDC 517.948.35

MATHEMATICS

Yu. M. BEREZANSKII

ONE GENERALIZATION OF THE POWER MOMENT PROBLEM

(Presented by Academician A. Yu. Ishlinskii on 26 III 1966)

This note investigates a certain operator moment problem arising from the desire to satisfy the axioms of field theory proposed by Wightman ((1); see also Borchers (2)). In the scheme considered here, so far only a small number of the requirements of these axioms are reflected.

\(1^\circ\). Let \(\mathfrak H\) be a Hilbert space, \(\mathfrak H \otimes \cdots \otimes \mathfrak H = \otimes \mathfrak H^j\) the tensor product of \(j\) copies of the space \(\mathfrak H\) \((j=0,1,\ldots;\) for \(j=0\), \(\otimes \mathfrak H^0=C_1\), where \(C_1\) is the field of complex numbers). Vectors of the space \(\otimes \mathfrak H^j\) will be denoted by \(f_j, g_j,\ldots\); \(\|f_j\|\), \((f_j,g_j)\) are the norm and scalar product in \(\otimes \mathfrak H^j\). In what follows all spaces are assumed separable.

Consider an operator \(K_{jk}\), acting continuously from \(\otimes \mathfrak H^k\) into \(\otimes \mathfrak H^j\), and form the matrix \(K=(K_{jk})_{j,k=0}^{\infty}\). Such a matrix will be called positive definite (p.d.) if, for any finite sequence \(f=(f_0,f_1,\ldots)\), \(f_j\in \otimes \mathfrak H^j\) (\(\Phi(\mathfrak H)\) denotes their aggregate),

\[ \sum_{j,k=0}^{\infty} (K_{jk}f_k,f_j)\geq 0. \tag{1} \]

This definition is preserved if in each \(\otimes \mathfrak H^k\) some subspace is distinguished and \(K_{jk}\) acts continuously from this subspace into the corresponding subspace of \(\otimes \mathfrak H^j\) (then, clearly, the \(f_j\) in (1) must belong to the distinguished subspaces).

Introduce on sequences \(f=(f_0,f_1,\ldots)\) the right shift

\[ Tf=T(f_0,f_1,\ldots)=(0,f_0\otimes e,f_1\otimes e,\ldots), \tag{2} \]

where \(e\in\mathfrak H\) is a fixed vector; in what follows, without loss of generality, one may assume that \(\|e\|=1\). A p.d. matrix will be called moment, if

\[ \sum_{j,k=0}^{\infty} (K_{j+1,k}f_k,g_j\otimes e) = \sum_{j,k=0}^{\infty} (K_{j,k+1}f_k\otimes e,g_j) \quad (f,g\in\Phi(\mathfrak H)), \tag{3} \]

i.e. the shift (2) on finite sequences is Hermitian in the scalar product

\[ \langle f,g\rangle=\sum_{j,k=0}^{\infty}(K_{jk}f_k,g_j). \]

In the case of a one-dimensional space \(\mathfrak H\), for any \(j=0,1,\ldots\), \(\otimes\mathfrak H^j=C_1\); \(f=(f_0,f_1,\ldots)\) is a sequence of numbers; \((K_{jk})_{j,k=0}^{\infty}\) is an ordinary matrix. If it is moment for \(e=1\), then from (3) it follows that necessarily \(K_{jk}=s_{j+k}\), where \(s_j\) is an ordinary moment sequence \((j,k=0,1,\ldots)\). As is known,

\[ s_j=\int_{-\infty}^{\infty}\lambda^j\,d\tau(\lambda) \quad (j=0,1,\ldots), \]

where \(d\tau(\lambda)\) is a certain nonnegative finite measure. We shall now show that an analogous representation also holds for a general moment matrix.

Denote by $\dot{\mathfrak H}$ the one-dimensional subspace of $\mathfrak H$ spanned by the vector $e$, and by $\overset{\infty}{\mathfrak H}$ its orthogonal complement. Let $A_{\alpha\beta}$ be an operator acting continuously from $\otimes \mathfrak H^{\beta-1}\otimes \overset{\infty}{\mathfrak H}$ into $\otimes \mathfrak H^{\alpha-1}\otimes \overset{\infty}{\mathfrak H}$ ($\alpha,\beta=0,1,\ldots$; $\otimes \mathfrak H^{-1}\otimes \overset{\infty}{\mathfrak H}=\dot{\mathfrak H}$; $\otimes \mathfrak H^{0}\otimes \overset{\infty}{\mathfrak H}=\overset{\infty}{\mathfrak H}$; these latter conventions must be kept in mind throughout what follows). Extend this operator to an operator from $\otimes \mathfrak H^{k}$ into $\otimes \mathfrak H^{j}$ ($j\ge \alpha$, $k\ge \beta$) as follows. Embed the space $\otimes \mathfrak H^{\alpha}$ in $\otimes \mathfrak H^{j}$ ($j\ge \alpha$), identifying it with $\otimes \mathfrak H^{\alpha}\otimes e\otimes\cdots\otimes e$. Consider the operator of orthogonal projection of $\otimes \mathfrak H^{k}$ onto $\otimes \mathfrak H^{\beta-1}\otimes \overset{\infty}{\mathfrak H}$; it is equal to $1\otimes\cdots\otimes 1\otimes \overset{\infty}{P}\otimes \dot P\otimes\cdots\otimes \dot P$, where $\dot P$ and $\overset{\infty}{P}$ are the projectors in $\mathfrak H$ onto $\dot{\mathfrak H}$ and $\overset{\infty}{\mathfrak H}$. The required extension is defined by the formula $A_{\alpha\lambda}(1\otimes\cdots\otimes 1\otimes \overset{\infty}{P}\otimes \dot P\otimes\cdots\otimes \dot P)$.

Theorem. In order that the matrix $K=(K_{jk})_{j,k=0}^{\infty}$ be a moment matrix, it is necessary and sufficient that it admit a representation in the form of the following operator integral, convergent in norm:
\[ K_{jk}=\sum_{\alpha=0}^{j}\sum_{\beta=0}^{k}\int_{-\infty}^{\infty}\lambda^{j+k}\,dT_{\alpha\beta}(\lambda) \bigl(1\otimes\cdots\otimes 1\otimes \overset{\infty}{P}\otimes \dot P\otimes\cdots\otimes \dot P\bigr) \qquad (j,k=0,1,\ldots), \tag{4} \]
where $T_{\alpha\beta}(\Delta)$ is an arbitrary bounded operator measure acting from $\otimes\mathfrak H^{\beta-1}\otimes\overset{\infty}{\mathfrak H}$ into $\otimes\mathfrak H^{\alpha-1}\otimes\overset{\infty}{\mathfrak H}$, for which the matrix $(T_{\alpha\beta}(\Delta))_{\alpha,\beta=0}^{\infty}$ is p.o.

In (4) one may, of course, bring the sum under the integral sign and include it in the differential; however, the resulting differential is less convenient to describe than $dT_{\alpha\beta}(\lambda)$.

We obtain the proof of the representation in the following way. The matrix $K$ generates a generalized kernel $\mathcal K$ associated with a chain of positive and negative spaces $H_-=H_0\supset H_+$, where
\[ H_0=\bigoplus_{j=0}^{\infty}\otimes\mathfrak H^{j}, \]
and $H_+$ is chosen in the proper way. Conditions (1) and (3) ensure that the general theorem on integral representation of p.o. kernels (([3], Chapter VIII, § 1)) is applicable to such a kernel, leading to the equality
\[ \mathcal K=\int_{-\infty}^{\infty}\Omega_\lambda\,d\rho(\lambda) \]
(cf. (4)). The elementary kernels $\Omega_\lambda=(\Omega_{\lambda;jk})_{j,k=0}^{\infty}$ will, for each of the indices $j,k$, be eigenfunctions for certain equations of difference type with left shift, adjoint to the shift (2) (see (5)). Finding the general solution of these equations and substituting it into the expression for $\mathcal K$, one can arrive at (4). The finding of the general solution is based on the following lemma:

Lemma. Let the sequence $f=(f_0,f_1,\ldots)$, $f_j\in\otimes\mathfrak H^{j}$, where $\mathfrak H$ is some Hilbert space, satisfy the equation
\[ \bigl(1\otimes\cdots\otimes 1\otimes \dot P\bigr)f_{j+1}=\lambda f_j \qquad (j=0,1,\ldots). \tag{5} \]
Here $\dot P f_1=(f_1,e)\in C_1$, $e\in\mathfrak H$ is fixed, $\|e\|=1$. The general solution of equation (5) has the form
\[ f_j=\sum_{\alpha=0}^{j}\lambda^{j-\alpha} f_{\alpha}^{\infty}\otimes e\otimes\cdots\otimes e \qquad (j=0,1,\ldots), \]

where \(f_\alpha \in \otimes \mathfrak H^{\alpha-1}\otimes \mathfrak H^\infty\) \((\alpha=0,1,\ldots)\) are arbitrarily prescribed “initial data” (equal to \((\underbrace{1\otimes\cdots\otimes 1}_{\alpha-1}\otimes \overset{\infty}{P})f_\alpha\)).

\(2^0\). Suppose that the representation (4) holds. We apply (4) to \(g_k\), multiply scalarly by \(f_j\), and sum over \(j,k\). As a result we obtain Parseval’s equality

\[ \langle f,g\rangle = \sum_{\alpha,\beta=0}^{\infty}\int_{-\infty}^{\infty} \bigl((dT_{\alpha\beta}(\lambda))\widetilde f_\beta(\lambda), \widetilde g_\alpha(\lambda)\bigr), \tag{6} \]

where \(f,g\in \Phi(\mathfrak H)\), and the “Fourier transform” \(\widetilde f(\lambda)=(\widetilde f_0(\lambda),\widetilde f_1(\lambda),\ldots)\) of the vector \(f\) is defined by the equality

\[ \widetilde f_\alpha(\lambda) = \sum_{j=\alpha}^{\infty}\lambda^j (\underbrace{1\otimes\cdots\otimes 1}_{\alpha-1}\otimes \overset{\infty}{P}\otimes \underbrace{\dot P\otimes\cdots\otimes \dot P}_{j-\alpha})f_j \in \otimes \mathfrak H^{\alpha-1}\otimes \mathfrak H^\infty \]

\[ (\alpha=0,1,\ldots). \tag{7} \]

The following inversion formula is valid for the transform (7):

\[ f_n=\frac{1}{n!}\left(\frac{d^n}{d\lambda^n} (\widetilde f_0(\lambda)+\widetilde f_1(\lambda)+\cdots)\right)(0) \quad (n=0,1,\ldots). \]

It follows easily from (7) that \((\widetilde{Tf})_\alpha(\lambda)=\lambda f_\alpha(\lambda)\) \((\alpha=0,1,\ldots)\); therefore, using (6), we conclude that \(\langle Tf,g\rangle=\langle f,Tg\rangle\) \((f,g\in\Phi(\mathfrak H))\), i.e., (4) implies (3).

It is not difficult to show that the totality \(\widetilde\Phi(\mathfrak H)\) of Fourier transforms consists of all finite sequences \((P_0(\lambda),P_1(\lambda),\ldots)\), where \(P_\alpha(\lambda)\) is a vector polynomial of the form \(\lambda^\alpha(a_{\alpha,1}+\lambda a_{\alpha,2}+\cdots+\lambda^n a_{\alpha,n})\), \(a_{\alpha,j}\in \otimes\mathfrak H^{\alpha-1}\otimes\mathfrak H^\infty\). Completion of \(\widetilde\Phi(\mathfrak H)\) with respect to the scalar product (6) leads to a Hilbert space \(\widetilde H_k\), isometric to \(H_k\), the completion of \(\Phi(\mathfrak H)\) with respect to \(\langle\cdot,\cdot\rangle\). The shift (2) on \(\Phi(\mathfrak H)\) generates in \(H_k\) a Hermitian operator \(A\), isometric to the operator of multiplication by \(\lambda\), defined on \(\widetilde\Phi(\mathfrak H)\) in \(\widetilde H_k\). The operator measure \(dT_{\alpha\beta}(\lambda)\) in (4) is determined uniquely by \(K\) if and only if the closure of \(A\) is self-adjoint in \(H_k\). The following sufficient criterion for self-adjointness holds: if \(\|K_{jk}\|\le p_jp_k\) \((j,k=0,1,\ldots)\), and the class \(C(p_n)\) is quasi-analytic, then the closure of \(A\) is self-adjoint in \(H_k\). The proof of this assertion is carried out similarly to the proof of Theorem 5.8, Ch. VIII of \((^3)\) and is essentially set forth in \((^4)\); the literature on the question is presented in the same work. We shall not dwell on the transfer of further results of the power moment problem to our problem.

\(3^0\). We now give some constructions more special than in item \(1^0\). Suppose there is a chain of positive and negative spaces
\(\otimes\mathfrak H_-^j \supseteq \otimes\mathfrak H_0^j \supseteq \otimes\mathfrak H_+^j\) \((j=0,1,\ldots)\); in each such chain its own involution \(\overline{\phantom{u}}=\overline{\phantom{u}}_j\) is introduced (for the terminology and notation see \((^3)\), Ch. I, §§ 1–2 or \((^5)\); we furnish scalar products and norms in \(\otimes\mathfrak H_-^j\), \(\otimes\mathfrak H_0^j\), and \(\otimes\mathfrak H_+^j\) with the indices \(-,0,+\), respectively). Consider the matrix \(K=(K_{jk})_{j,k=0}^{\infty}\), where \(K_{jk}\in \otimes\mathfrak H_-^{j+k}\) and the analogous positive-definiteness condition (1) is satisfied:

\[ \sum_{j,k=0}^{\infty}(K_{jk},u_j\otimes \overline{u}_k)_0\ge 0 \quad (u=(u_0,u_1,\ldots)\in\Phi(\mathfrak H_+)). \tag{8} \]

Let \(e\in\mathfrak H_+\), \(\|e\|_+=1\), and let \(K\) be such that, similarly to (3),

\[ \sum_{j,k=0}^{\infty}(K_{j+1,k},v_j\otimes e\otimes \overline{u}_k)_0 = \sum_{j,k=0}^{\infty}(K_{j,k+1},v_j\otimes \overline{u}_k\otimes e)_0 \quad (u,v\in\Phi(\mathfrak H_+)). \tag{9} \]

We associate with \(K\) the matrix of operators \(K=(K_{jk})_{j,k=0}^{\infty}\), where \(K_{jk}\) acts continuously from \(\otimes \mathfrak H_{+}^{k}\) into \(\otimes \mathfrak H_{+}^{j}\) according to the equality
\[ (K_{jk}u_k,v_j)_{+}=(K_{jk},v_j\otimes \overline{u}_k)_0 \]
(the role of \(\mathfrak H\) is played by \(\mathfrak H_{+}\)). By theorem, item \(1^0\), \(K_{jk}\), and hence also \(K_{jk}\), admits the representation (4). This representation can be written in such a form that, instead of the operators \(T_{\alpha\beta}(\Delta)\), elements of tensor products occur. For this purpose consider a Hilbert space \(\mathfrak H_{++}\subseteq\mathfrak H_{+}\) such that the embedding \(\mathfrak H_{++}\to\mathfrak H_{+}\) is quasi-nuclear, and construct chains
\[ \otimes\mathfrak H_{--}^{j}\supseteq \otimes\mathfrak H_{-}^{j}\supseteq \otimes\mathfrak H_{0}^{j}\supseteq \otimes\mathfrak H_{+}^{j}\supseteq \otimes\mathfrak H_{++}^{j}\qquad (j=0,1,\ldots). \]
In addition, we shall suppose that the involution \(-\) = \(-j\) is continued to the last chain and has the property that
\[ \overline{\varphi_j\otimes\psi_k^{\,j+k}}=\overline{\psi}_k\otimes\overline{\varphi}_j \quad (\varphi_j\in\otimes\mathfrak H_{--}^{j},\ \psi_k\in\otimes\mathfrak H_{--}^{k};\ j,k=0,1,\ldots). \tag{10} \]

By virtue of the theorem on the kernel, for arbitrary \(u_\beta\in\otimes\mathfrak H_{++}^{\beta}\), \(v_\alpha\in\otimes\mathfrak H_{++}^{\alpha}\),
\[ \bigl(T_{\alpha 3}(\Delta)(\underbrace{1\otimes\cdots\otimes 1}_{\beta-1}\otimes \stackrel{\infty}{P})u_\beta,\ v_\alpha\bigr)_{+} = (T_{\alpha 3}(\Delta),\, v_\alpha\otimes\overline{u}_\beta)_0, \]
where
\[ T_{\alpha 3}(\Delta)\in \otimes\mathfrak H_{--}^{\alpha+\beta}. \]
We shall embed \(\otimes\mathfrak H_{--}^{\alpha+\beta}\) in \(\otimes\mathfrak H_{--}^{j+k}\) \((j\geq\alpha,\ k\geq\beta)\), identifying this space with
\[ \otimes\mathfrak H_{--}^{\alpha}\otimes \underbrace{\varepsilon\otimes\cdots\otimes\varepsilon}_{j-\alpha} \otimes \underbrace{\overline{\varepsilon}\otimes\cdots\otimes\overline{\varepsilon}}_{k-\beta} \otimes\mathfrak H_{--}^{\beta} \subseteq \otimes\mathfrak H_{--}^{j+k}; \]
the element corresponding to \(\xi_{\alpha\beta}\) will be denoted by
\[ \xi_\alpha\langle \otimes\underbrace{\varepsilon\otimes\cdots\otimes\varepsilon}_{j-\alpha} \otimes \underbrace{\overline{\varepsilon}\otimes\cdots\otimes\overline{\varepsilon}}_{k-\beta} \rangle_\beta . \]
Here \(\varepsilon\in\mathfrak H_{-}\subseteq\mathfrak H_{--}\) is determined from the equality
\[ (\varepsilon,u)_0=(e,u)_{+}\qquad (u\in\mathfrak H_{+}). \]
It is not hard to see that now representation (4) can be rewritten in the form
\[ K_{jk}=\sum_{\alpha=0}^{j}\sum_{\beta=0}^{k} \int_{-\infty}^{\infty} \lambda^{j+k}\,dT_\alpha \langle\otimes\varepsilon\otimes\cdots\otimes\varepsilon\otimes \overline{\varepsilon}\otimes\cdots\otimes\overline{\varepsilon}\rangle_\beta(\lambda) \]
\[ (j,k=0,1,\ldots). \tag{11} \]

\(4^0\). A functional of Wightman type will be called a sequence
\[ W=(W_0,W_1,\ldots), \]
where \(W_j\in\otimes\mathfrak H_{-}^{j}\), for which the matrix
\[ (K_{jk})_{j,k=0}^{\infty}=(W_{j+k})_{j,k=0}^{\infty} \]
is positive definite, i.e., (8) is fulfilled. Since from (8) it always follows that \(K_{jk}=\overline{K}_{kj}\), \(W\) is real:
\[ W_j=\overline{W}_j\qquad (j=0,1,\ldots). \]
Now condition (9), thanks to (10), is fulfilled automatically for any \(e=\overline e\). Conversely, it is easy to see that if (9) holds for any \(e=\overline e\in\mathfrak H_{+}\), \(\|e\|_{+}=1\), then \(K_{jk}\) depends on the sum:
\[ K_{jk}=W_{j+k}\qquad (j,k=0,1,\ldots). \]
Thus, representation (11) holds for any \(e=\overline e\in\mathfrak H_{+}\), \(\|e\|_{+}=1\), if and only if
\[ (K_{jk})_{j,k=0}^{\infty}=(W_{j+k})_{j,k=0}^{\infty}, \]
where \(W\) is a functional of Wightman type. From what was said in item \(2^0\) it is easy to see that, if the class
\[ C(\sqrt{\|W_{2n}\|_{-}}) \]
is quasi-analytic, then the measure in each representation (11) for \(W_{j+k}\) is determined uniquely.

In conclusion let us note that in items \(3^0,4^0\) one can pass in the usual way from elements of \(\otimes\mathfrak H_{--}^{j}\) to considerations of functionals over nuclear linear topological spaces. In particular, such spaces may be spaces of functions of \(j\) variables \(x_1,\ldots,x_j\in(-\infty,\infty)\); as the involution \(-j\), satisfying (10), one may take the passage
\[ f(x_1,x_2,\ldots,x_{j-1},x_j)\to \overline{f(x_j,x_{j-1},\ldots,x_2,x_1)}, \]
where the bar denotes complex conjugation.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
14 III 1966

CITED LITERATURE

1. A. S. Wightman, Colloques internat. an Lille, Juin, 1957, Paris, 1959, p. 1; Sborn. per. Matematika, 6, No. 4, 96 (1962).
2. H. J. Borchers, Nuovo Cim., 24, No. 2, 214 (1962).
3. Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.
4. Yu. M. Berezanskii, Ukr. Mat. Zhurn., 18, No. 3 (1966).
5. Yu. M. Berezanskii, UMN, 18, No. 1, 63 (1963).