Reports of the Academy of Sciences of the USSR

1. Volume 133, No. 3

MATHEMATICS

G. I. ESKIN

A SUFFICIENT CONDITION FOR THE SOLVABILITY OF THE MULTIDIMENSIONAL MOMENT PROBLEM

(Presented by Academician S. L. Sobolev on III 30 1960)

Let \(\Phi_1\) and \(\Phi_2\) be linear spaces with involution, and let \(A\) and \(B\) be linear operators in \(\Phi_1\) and \(\Phi_2\), respectively, real with respect to the involution, i.e. \(A\varphi^*=(A\varphi)^*\) for every \(\varphi\in\Phi_1\) and \(B\psi^*=(B\psi)^*\) for every \(\psi\in\Phi_2\). Let \(\Phi=\Phi_1\otimes\Phi_2\) be the tensor product of \(\Phi_1\) and \(\Phi_2\), i.e. the set of elements \(x\) of the form \(\sum_{k=1}^n \varphi_k\otimes\psi_k\), where \(\varphi_k\in\Phi_1,\ \psi_k\in\Phi_2\). The operators \(A\) and \(B\) and the involution are defined in \(\Phi\) in the natural way. Suppose that a scalar product \((x,y)\), real with respect to the involution, is given in \(\Phi\), i.e. \((x^*,y^*)=\overline{(x,y)}\), and that \(A\) and \(B\) are symmetric operators in this scalar product. As a consequence of the reality of \(A\), and also of \(B\), they have equal deficiency indices. Denote by \(H\) the completion of \(\Phi\) with respect to the scalar product \((x,y)\). For fixed \(\psi_0\in\Phi_2\), obviously, \((\varphi_1\otimes\psi_0,\varphi_2\otimes\psi_0)\) is a scalar product in \(\Phi_1\).

Theorem 1. Let \(\bar A_{\psi_0}\), the closure of \(A\) in the Hilbert space \(H_{\psi_0}\) obtained by completing \(\Phi_1\) with respect to the scalar product \((\varphi_1\otimes\psi_0,\varphi_2\otimes\psi_0)\), be a self-adjoint operator for every fixed \(\psi_0\in\Phi_2\).

Then \(\bar A\), the closure of \(A\) in \(H\), is a self-adjoint operator, and in \(H\) there exists a self-adjoint extension \(\widetilde B\) of the operator \(\bar B\) such that \(\bar A\) and \(\widetilde B\) commute in \(H\), i.e. their spectral families commute.

The case when the closures of \(A\) and \(B\) in \(H\) are self-adjoint operators was considered by A. G. Kostyuchenko and B. S. Mityagin in \((^1)\). The case when only \(A\) is self-adjoint was considered by R. S. Ismagilov under the assumption that \(\Phi\) is a Carleman space \((^2)\). Theorem 1 generalizes R. S. Ismagilov’s result, which makes it possible to obtain certain new results for the multidimensional moment problem and the representation of positive-definite functionals.

Proof of Theorem 1. The operators \(A\) and \(B\), obviously, commute on \(\Phi\). Therefore, for any non-real \(\lambda\) the equality

\[ R_\lambda Bx = BR_\lambda x \quad \text{for } x\in\Phi_\lambda=(A-\lambda E)\Phi_1\otimes\Phi_2, \tag{1} \]

holds, where \(R_\lambda\) is the resolvent of the operator \(\bar A\). Denote by \(B_\lambda\) the operator \(B\) considered on the linear manifold \(\Phi_\lambda\). Equality (1) holds for any \(x\in D_{\bar B_\lambda}\), where \(\bar B_\lambda\) is the closure of \(B_\lambda\) in \(H\), since \(R_\lambda\) is a bounded operator. We shall show that \(\bar B_\lambda\supset B\), i.e. \(D_{\bar B_\lambda}\supset\Phi\) and for \(x\in\Phi\) one has \(\bar B_\lambda x=Bx\).

Indeed, let \(\varphi_0\) be any element of \(\Phi_1\) and \(\psi_0\) any element of \(\Phi_2\). From the condition of the theorem it follows that the set \((A-\lambda E)\Phi_1\) is dense in \(H_{\psi_1}\), where \(\psi_1=(B-iE)\psi_0\in\Phi_2\), so that there exists a sequence \(\varphi_n^{(1)}=(A-\)

\(-\lambda E)\varphi_n\), where \(\varphi_n\in\Phi_1\), converging to \(\varphi_0\) in the norm \(H_{\psi_1}\). But, since \(B\) is symmetric, \(\|(B-iE)x\|^2=\|Bx\|^2+\|x\|^2\), so that convergence in \(H_{\psi_1}\) means that simultaneously
\[ (A-\lambda E)\varphi_n\otimes\varphi_0\to\varphi_0\otimes\psi_0,\qquad (A-\lambda E)\varphi_n\otimes B\psi_0\to\varphi_0\otimes B\psi_0 \]
in the norm \(H\). Consequently, \(\varphi_0\otimes\psi_0\in D_{\overline{B}_\lambda}\) and, hence, \(\Phi\subset D_{\overline{B}_\lambda}\). On the other hand, since \(B_\lambda\subset B\), we have \(\overline{B}_\lambda\subset \overline{B}\). Thus, \(\overline{B}_\lambda=\overline{B}\). Hence, for any \(x\in\Phi\) we have \(\overline{B}R_\lambda x=R_\lambda\overline{B}x\). It follows that for any \(y\in D_{B^*}\),
\[ (\overline{B}R_\lambda x,y)=(R_\lambda x,B^*y)= \int_{-\infty}^{\infty}\frac{1}{\lambda-\xi}\,d(E_\xi x,B^*y). \]

Next,
\[ (R_\lambda\overline{B}x,y)= \int_{-\infty}^{\infty}\frac{1}{\lambda-\xi}\,d(E_\xi \overline{B}x,y). \]

Comparing the last two equalities, from the uniqueness of the Stieltjes transform we have, for any \(\xi\), that \((E_\xi \overline{B}x,y)=(E_\xi x,B^*y)\). The equality
\[ (\overline{B}x,E_\xi y)=(x,E_\xi B^*y) \]
holds for all \(x\in\Phi\). It follows that if \(y\in D_{B^*}\), then also \(E_\xi y\in D_{B^*}\), and
\[ B^*E_\xi y=E_\xi B^*y, \tag{2} \]
i.e., the operator \(B^*\) commutes with \(E_\xi\).

Further, \(D_{B^*}=D_{\overline{B}}\dotplus M_+\dotplus M_-\), where \(B^*M_+=iM_+\) and \(B^*M_-=-iM_-\). From (2) it follows that \(E_\xi M_\pm\subset M_\pm\) and \(E_\xi D_{\overline{B}}\subset D_{\overline{B}}\). In consequence of the reality of \(A\) and \(B\), the operators \(B^*\) and \(E_\xi\) will also be real. Therefore \(M_-=M_+^*\), i.e., if \(x\in M_+\), then \(x^*\in M_-\). Since every self-adjoint extension of the operator \(B\) is a part of \(B^*\), in order that some extension \(\widetilde{B}\) commute with \(E_\xi\), it is necessary that from \(y\in D_{\widetilde{B}}\) it should follow that \(E_\xi y\in D_{\widetilde{B}}\). Let
\[ D_{\widetilde{B}}=D_{\overline{B}}\dotplus M_+\dotplus VM_+, \]
where \(V\) is an isometric operator mapping \(M_+\) onto \(M_-\). Then
\[ E_\xi D_{\widetilde{B}}=E_\xi D_{\overline{B}}\dotplus E_\xi M_+\dotplus E_\xi VM_+, \]
and, in order that \(E_\xi D_{\widetilde{B}}\subset D_{\widetilde{B}}\), it is necessary that \(E_\xi VM_+=VE_\xi M_+\). Let \(x_0\) be an arbitrary element of \(M_+\). Denote by \(M_+^{(1)}\) the closure of the linear span of the vectors \(E_\Delta x_0\), where \(\Delta\) is an arbitrary interval. \(M_+^{(1)}\subset M_+\). Let \(M_-^{(1)}=M_+^{(1)*}\subset M_-\). Define the operator \(V^{(1)}\) from \(M_+^{(1)}\) into \(M_-^{(1)}\) as follows:
\[ V^{(1)}E_\Delta x_0=E_\Delta x_0^*, \]
and on the remaining vectors by linearity and continuity. \(V^{(1)}\) maps \(M_+^{(1)}\) isometrically onto \(M_-^{(1)}\), since the operator \(E_\xi\) and the scalar product \((x,y)\) are real with respect to the involution. It is obvious that
\[ E_\xi V^{(1)}M_+^{(1)}=V^{(1)}E_\xi M_+^{(1)} \]
for any \(\xi\). If the orthogonal complement to \(M_+^{(1)}\) in \(M_+\) is not empty, then choose there an arbitrary element \(y_0\) and do the same, etc., until \(M_+\) is exhausted.

Thus one can construct an isometric operator \(V\) such that \(E_\xi VM_+=VE_\xi M_+\). The corresponding extension \(\widetilde{B}\) will commute with \(E_\xi\): \(\widetilde{B}E_\xi x=E_\xi\widetilde{B}x\) for all \(x\in D_{\widetilde{B}}\). Such an extension \(\widetilde{B}\) is not unique, if \(\overline{B}\) is not a self-adjoint operator. Let \(R_\mu\) be the resolvent of \(\widetilde{B}\). Since
\[ (\widetilde{B}-\mu E)E_\xi x=E_\xi(\widetilde{B}-\mu E)x, \]
then, applying \(R_\mu\) to both sides of the equality, we obtain
\[ E_\xi R_\mu y=R_\mu E_\xi y \]
for all \(y\in H\). Using again the spectral representation of the resolvent and the uniqueness of the Stieltjes transform, we shall have \(E_\xi F_\eta=F_\eta E_\xi\), where \(F_\eta\) is the spectral family of \(\widetilde{B}\). The theorem is proved.

Let now \(\Phi_1\) and \(\Phi_2\) be nuclear algebras, and let the scalar product be given by a continuous positive-definite functional \(T(x)\), i.e. one such that \(T(x\cdot x^*)\ge 0\). Then, assuming the conditions fulfilled

of Theorem 1, from the general scheme in (1) we obtain a representation for \(T(x)\):

\[ T(x)=\int X_\lambda(x)\,d\sigma(\lambda), \tag{3} \]

where \(X_\lambda\) are the common eigenfunctionals of the operators \(A'\) and \(B'\), adjoint to \(A\) and \(B\) in \(\Phi'\). The measure \(\sigma(\lambda)\), generally speaking, is not unique.

Let us give some applications of Theorem 1.

1. Let \(c_{mn}\) be a double sequence of moments, i.e., for any sequence \(\{\xi_{mn}\}\)

\[ \sum_{m,n} c_{m_1+m_2,\,n_1+n_2}\xi_{m_1,n_1}\overline{\xi}_{m_2,n_2}\geqslant 0. \tag{4} \]

The question is posed of when the two-dimensional moment problem is solvable (see \((^3)\)), i.e., when there exists a nonnegative measure \(\sigma(\lambda,\mu)\) such that

\[ c_{mn}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\lambda^m\mu^n\,d\sigma(\lambda,\mu). \]

Theorem 2. The two-dimensional moment problem is solvable if, for any fixed \(n_0\), the one-dimensional moment problem \(a_m=c_{m,2n_0}+c_{m,2(n_0+1)}\) is determinate; moreover, the measure \(\sigma(\lambda,\mu)\), generally speaking, is not unique.

Theorem 2 follows from Theorem 1 if we assume that \(\Phi_1\) and \(\Phi_2\) are spaces of sequences in which only a finite number of terms are nonzero, and \(A\) and \(B\) are shift operators, i.e., \(A\{\xi_m\}=\{\xi_{m-1}\}\) and \(B\{\eta_n\}=\{\eta_{n-1}\}\); the involution is simply passage to the complex-conjugate sequence, and the scalar product is defined by means of (4).

1. Let \(f(x,y)\) be a continuous function positive-definite in the sense of Bochner in the rectangle \(P=[-a,a;\,-b,b]\), i.e.,

\[ \int_P f(x_1-x_2,y_1-y_2)\varphi(x_1,y_1)\overline{\varphi(x_2,y_2)}\,dx_1dx_2dy_1dy_2\geqslant 0 \tag{5} \]

for all finite infinitely differentiable functions from \(P\). In this case \(\Phi_1=K(-a,a)\), \(\Phi_2=K(-b,b)\). The operators \(A\) and \(B\) are respectively \(i\,d/dx\) and \(i\,d/dy\). The involution is given by the equality \([\varphi(x,y)]^*=\overline{\varphi(-x,-y)}\), and the scalar product is determined by (5).

The inequality holds

\[ \int_P f(x_1-x_2,y_1-y_2)\varphi(x_1)\overline{\varphi(x_2)}\psi(y_1)\overline{\psi(y_2)}\,dP\leqslant \]

\[ \leqslant \left(\int_{-b}^{b}|\psi(y)|\,dy\right)^2 \int_{-a}^{a}\int_{-a}^{a} f(x_1-x_2,0)\varphi(x_1)\overline{\varphi(x_2)}\,dx. \tag{6} \]

The function \(f(x,0)\) is obviously a positive-definite function. From (6) it follows that convergence in the scalar product generated by \(f(x,0)\) implies convergence in the scalar product (5) for any fixed \(\psi(y)\), which is stronger than the requirement of Theorem 1. Therefore the following theorem holds:

Theorem 3. If \(f(x,0)\) admits a unique extension to the entire \(X\)-axis as a positive-definite function, then \(f(x,y)\) has the representation

\[ f(x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(\lambda x+\mu y)}\,d\sigma(\lambda,\mu), \]

where \(\sigma(\lambda,\mu)\) is a nonnegative measure, generally speaking not unique, so that \(f(x,y)\) admits an extension to the whole plane as a positive-definite function.

Devinatz in (4) proved an analogous theorem under the assumption that \(\hat f(x,0)\) on the interval \((-a+\varepsilon, a-\varepsilon)\) admits a unique extension to the whole axis as a positive-definite function.

1. Theorem 1 is easily generalized to the case when \(\Phi=\Phi_1\otimes\cdots\otimes\Phi_k\), the operators \(A_i\) \((1\le i\le k)\) act in \(\Phi_i\), and the closure \(\overline{A_i}\) \((1\le i\le k-1)\) is a self-adjoint operator in the Hilbert space obtained by completing \(\Phi_i\) with respect to the scalar product

\[ (\varphi,\varphi)_i = (\psi_{10}\otimes\cdots\otimes\psi_{i-1,0}\otimes\varphi\otimes\psi_{i+1,0}\otimes\cdots\otimes\psi_{k0}, \]

\[ \psi_{10}\otimes\cdots\otimes\psi_{i-1,0}\otimes\varphi\otimes\psi_{i+1,0}\otimes\cdots\otimes\psi_{k0}) \]

for arbitrary \(\psi_{j0}\in\Phi_j\) \((j\ne i)\). Then there exists a self-adjoint extension \(\widetilde A_k\) of the operator \(A_k\) such that all \(\overline{A_i}\) \((1\le i\le k-1)\) commute with \(\widetilde A_k\). The fact that the \(\overline{A_i}\) commute with one another was proved in (1).

Theorem 2 takes the following form in the multidimensional case: if the one-dimensional moment problem

\[ a_{n_i}=c_{2n_{10},\ldots,2n_{i-1},0,n_i,2n_{i+1},0,\ldots,2n_{k0}} + c_{2n_{10},\ldots,2n_{i-1},0,n_i,2n_{i+1},0,\ldots,2(n_{k0}+1)} \]

is uniquely solvable for \(1\le i\le k-1\) and for arbitrary \(n_{j0}\) \((j\ne i)\), then the multidimensional moment problem

\[ c_{n_1,\ldots,n_k} = \int \lambda_1^{n_1}\cdots\lambda_k^{n_k}\,d\sigma(\lambda_1,\ldots,\lambda_k) \]

is solvable and, generally speaking, not uniquely.

Similarly, if \(f(x_1,\ldots,x_k)\) is a positive-definite function in a \(k\)-dimensional parallelepiped, then if \(f(x_1,0,\ldots,0)\), \(f(0,x_2,\ldots,0),\ldots,f(0,\ldots,x_{k-1},0)\) admit unique extensions to the whole axis as positive-definite functions, then \(f(x_1,\ldots,x_k)\) also admits an extension to the whole \(k\)-dimensional space as a positive-definite function.

Moscow State University
named after M. V. Lomonosov

Received
25 III 1960

REFERENCES CITED

1. A. G. Kostenko, B. S. Mityagin, DAN, 131, No. 1 (1960).
2. R. S. Ismagilov, DAN, 133, No. 3 (1960).
3. A. G. Kostenko, B. S. Mityagin, DAN, 131, No. 6 (1960).
4. A. Devinatz, Acta Math., 102, No. 1–2 (1959).