MATHEMATICS

R. S. ISMAGILOV

SELF-ADJOINT EXTENSIONS OF A SYSTEM OF COMMUTING SYMMETRIC OPERATORS

(Presented by Academician S. L. Sobolev on 29 III 1960)

Consider a linear topological space \(\Phi\), in which commuting linear continuous operators \(A\) and \(B\) act; let \(H\) be the Hilbert space obtained from \(\Phi\) by completion with respect to the continuous nondegenerate scalar product \((x,y)\), and suppose that the operator \(\bar A\) (the closure in \(H\) of the operator \(A\)) is self-adjoint in \(H\), while the operator \(B\) is symmetric. Does there exist a self-adjoint extension of the operator \(B\) that commutes with \(\bar A\)? (Self-adjoint operators are called commuting if their spectral families commute.) Below a solution of this problem is given, with applications to representations of positive-definite functionals.

Definition 1. We shall say that the operator \(\bar A\) is strongly self-adjoint in \(H=\bar\Phi\) if \(\Phi\) contains a subset \(\Phi_1\) such that: a) \(\Phi_1\) is dense in \(\bar\Phi\) in the topology of \(\Phi\); b) if \(x\in\Phi_1\), then in the subspace \(L_x\) spanned by the elements \(\{A^k x,\ k=0,1,2,\ldots\}\), the operator \(A\) has a self-adjoint closure.

A strongly self-adjoint operator is, obviously, self-adjoint in \(H\) (in the usual sense).

I. Denote the set \((B-\lambda E)\Phi\) by \(\Phi^\lambda\), and let \(H^\lambda=\overline{\Phi^\lambda}\). Obviously, \(A\Phi^\lambda\subseteq\Phi^\lambda\) and the operator \(A\) is symmetric in \(H^\lambda\).

Theorem 1. Suppose the following conditions are satisfied: 1) the operator \(A\) is strongly self-adjoint in \(H^\lambda\) for some \(\lambda\) with \(\operatorname{Im}\lambda\ne0\); 2) in \(\Phi\) there is an involution \(x\leftrightarrow x^*\), and \((x,y)=(x^*,y^*)\) for all \(x,y\in\Phi\), and the operators \(A,B\) are real with respect to this involution, i.e. \(Ax^*=(Ax)^*\).

Then the operator \(B\) has a self-adjoint extension commuting with \(\bar A\).

Proof. If the operator \(\bar A\) is strongly self-adjoint in some \(H^{\lambda_0}\) \((\operatorname{Im}\lambda_0\ne0)\), then this is true also for any \(\lambda\), \(\operatorname{Im}\lambda\ne0\); indeed, \(H^{\lambda_0}\) is mapped onto \(H^\lambda\) one-to-one and continuously by the operator

\[ V_{\lambda_0}^{\lambda}=\frac{\bar B-\lambda E}{\bar B-\lambda_0E}, \]

and moreover

\[ V_{\lambda_0}^{\lambda}(A^k x)=A^k\bigl(V_{\lambda_0}^{\lambda}x\bigr)\quad (x\in\Phi^{\lambda_0}). \]

It may therefore be assumed that \(\bar A\) is strongly self-adjoint in \(H^i\). Let \(\Phi^i=(B-iE)\Phi\) and \(\Phi_1\subseteq\Phi^i\) be the set introduced in Definition 1. If \(x\) is any element of \(\Phi\), \(h=Bx+ix\), \(g=Bx-ix\), then from \(\operatorname{Im}(A^nBx,x)=0\) one can obtain that

\[ (A^n h,h)=(A^n g,g) \]

or

\[ \int \lambda^n\,d(E_\lambda h,h)=\int \lambda^n\,d(E_\lambda g,g)=c_n. \]

If \(g\in\Phi_1^i\) (in other words, \(x\in(B-iE)^{-1}\Phi_1^i\)), then from the last equality we obtain

\[ (E_\lambda h,h)=(E_\lambda g,g). \tag{1} \]

Indeed, the condition \(g\in \Phi_1^i\) entails the determinacy of the moment problem (see (1)) \(\int \lambda^n d\sigma(\lambda)=c_n\); hence (1) follows. Substituting, instead of \(h\) and \(g\), \(Bx+ix\) and \(Bx-ix\), we obtain from (1)

\[ \operatorname{Im}(E_\lambda Bx,x)=0. \tag{2} \]

Since \(\Phi_1^i\) is dense in \(\Phi^i\), (2) is true for all \(x\in (B-iE)^{-1}\Phi^i=\Phi\). From (2) it follows that the operator \(E_\lambda B\) is symmetric in \(\Phi\).

Let \(\overline{B}\) be the closure of \(B\) in \(H\), and let

\[ V=\frac{\overline{B}+iE}{\overline{B}-iE},\quad (\overline{B}+iE)D_{\overline{B}}=\Delta_V, \]

\[ (\overline{B}-iE)D_{\overline{B}}=D_V. \]

From what has been said it follows that for all \(x,y\in D_V\) the equality

\[ (E_\lambda x,y)=(E_\lambda Vx,Vy) \tag{3} \]

holds.

Denote \(R^+=H\ominus D_V\) and \(R^-=H\ominus \Delta_V\). All \(E_\lambda\) map \(R^+\) (respectively \(R^-\)) into itself. We must find a unitary extension \(\widetilde V\) of the operator \(V\), mapping \(R^+\) onto \(R^-\) in such a way that for any \(x,y\in R^+\) an equality analogous to (3) holds:

\[ (E_\lambda x,y)=(E_\lambda \widetilde Vx,\widetilde Vy). \tag{4} \]

For this purpose we choose in \(R^+\) such an orthonormal basis \(\{x_i\}\) that all functions \(\varphi_{ij}(\lambda)=(E_\lambda x_i,x_j)\) are real for \(-\infty<\lambda<\infty\). (This can be done as follows: decompose \(R^+\) into mutually orthogonal subspaces, in each of which the operator \(A=\int \lambda\,dE_\lambda\) has simple spectrum; each such subspace \(H_i\) with the operator \(A\) is realized as the space \(L^2(\sigma_i)\), and the operator \(\widetilde E_\Delta^i\) becomes multiplication by the characteristic function of the interval \(\Delta\); now choose in each \(L^2(\sigma_i)\), isomorphic to \(H_i\), an orthonormal basis of real functions.) We now construct the desired extension by setting \(\widetilde Vx_i=x_i^*\) (\(x_i^*\) is the element involutive to \(x_i\)). Since \(E_\lambda\) is a real operator, we have \((E_\lambda x_i^*,x_j^*)=((E_\lambda x_i)^*,x_j^*)=\overline{(E_\lambda x_i,x_j)}\); but \((E_\lambda x_i,x_j)\) is real for all \(\lambda\); therefore \((E_\lambda x_i^*,x_j^*)=(E_\lambda x_i,x_j)\), i.e. (4) is fulfilled, as required.

Usually one has to consider not one scalar product, but an entire family of scalar products in \(\Phi\), in each of which the operators \(A\) and \(B\) are symmetric. It is therefore necessary that the conditions of Theorem 1 be preserved for all these scalar products. Let us consider one case where this occurs.

Definition 2. An element \(x\in \Phi\) will be called Carleman if there exists a numerical sequence \(m_k\) such that

\[ \sum m_{2k}^{-1/2k}=\infty, \]

and the sequence

\[ \left\{\frac{1}{m_k}A^k x\right\} \]

is bounded in the topology of \(\Phi\).

Definition 3. The operator \(A\) in \(\Phi\) will be called Carleman if \(\Phi\) contains a dense set of Carleman elements.

Denote by \(\Phi^K\) the set of Carleman elements in \(\Phi\). Since \(B-\lambda E\) is a continuous operator commuting with \(A\), we have \((B-\lambda E)\Phi^K\subseteq \Phi^K\). Therefore the operator \(A\) in \(\Phi^\lambda\) is also Carleman. From the well-known Carleman criterion for the determinacy of the moment problem (see (1)) we conclude that \(\overline A\) is strongly self-adjoint in each \(H^\lambda\).

Therefore the following is true:

Theorem 1*. If \(A\) is a Carleman operator in \(\Phi\) and the operators \(A\) and \(B\) are real with respect to the involution in \(\Phi\) (where \((x^*,y^*)=\overline{(x,y)}\)), then \(B\) admits a self-adjoint extension in \(H\) commuting with \(\overline A\).

II. Applications.

Let \(\Phi\) be a nuclear algebra or a closed linear subspace of a nuclear algebra with a (unique) characteristic operator \(A\) (defined—

conditions, see \((^2)\), more precisely \((^3)\)); \(T(x)\) is a positive-definite functional on \(\Phi\). As was shown by A. G. Kostyuchenko and B. S. Mityagin, \(T(x)\) admits a (unique) expansion
\[ T(x)=\int \chi_\lambda(x)\,d\sigma(\lambda) \]
in the eigenfunctionals \(\chi_\lambda\) of the operator \(A\), if the latter has in \(H_T\) (this denotes the Hilbert space obtained from \(\Phi\) by completion in the norm
\[ \|x\|=\sqrt{T(x\circ x^*)} \]
) a self-adjoint closure. We shall introduce one class of such spaces.

Definition 4. We shall call a nuclear algebra (or a subspace of a nuclear algebra) \(\Phi\) a Carleman space if \(\Phi\) possesses a unique characteristic operator \(A\), and the latter is Carleman in \(\Phi\). In addition, throughout it is assumed that \(\Phi\) is a space with involution.

Every positive-definite functional on the Carleman space \(\Phi\) admits a unique expansion in the eigenfunctionals of the characteristic operator \(A\); in other words, the closure \(\overline A\) of the operator \(A\) in the space \(H_T\) is always a self-adjoint operator. This follows from the fact that \(A\) is strongly self-adjoint in \(H_T\).

Theorem 2. Let \(\Phi=\Phi_1\otimes\Phi_2\) be the tensor product of nuclear algebras (or linear subspaces of algebras) \((^2,^3)\) \(\Phi_1\) and \(\Phi_2\) with characteristic operators \(A_1\) and \(A_2\), and suppose that \(\Phi_1\) is a Carleman space.

Then every positive-definite functional \(T(x)\) on \(\Phi\) admits an expansion (in general, nonunique) in common eigenfunctionals of the operators \(A_1\) and \(A_2\).

Proof. If the operator \(A_1\) in \(\Phi_1\) is Carleman, then the operator \(A_1\otimes E_2\) (\(E_i\) is the identity operator in \(\Phi_i\)) will be Carleman in \(\Phi\). By Theorem 1, the operator \(E_1\otimes A_2\) has in \(H_T\) a self-adjoint extension commuting with \(\overline{A_1\otimes E_2}\). By the theorem of Kostyuchenko and Mityagin, \(T(x)\) admits the required representation.

In what follows, a space \(\Phi\) with a (unique) characteristic operator \(A\) will be denoted by \((\Phi,A)\).

Theorem 3. Let \(\Phi\) be a space of functions absolutely integrable on \((-\infty,\infty)\), with continuous convolution
\[ \varphi\circ\psi=\int \varphi(x-t)\psi(t)\,dt \]
and with characteristic operator
\[ c\,\frac{d^k}{dx^k}, \]
and suppose that a continuous “dilation” operator
\[ K\varphi(x)=\varphi(nx) \]
is defined in \(\Phi\) for all large \(n\).

Then, if \(\Phi\) contains at least one Carleman element \(\varphi_0(x)\) and
\[ \int \varphi_0(x)\,dx\ne0, \]
then \(\Phi\) is a Carleman space.

Proof. The sequence
\[ \varphi_n(x)=n\varphi_0(nx) \]
is an “identity” in \(\Phi\), i.e.
\[ \varphi_n(x)\circ\varphi(x)\to\varphi(x) \]
for all \(\varphi\in\Phi\) (we assume that
\[ \int\varphi_0(x)\,dx=1 \]
). But all \(\varphi_n(x)\) are, obviously, Carleman elements, as are the elements \(\varphi_n(x)\circ\varphi(x)\). Thus, for any \(\varphi(x)\in\Phi\) we have found a sequence of Carleman elements converging to \(\varphi(x)\), as was required.

Verification that \(\Phi\) is a Carleman space is reduced by this theorem to a certain quasi-analytic problem.

Theorem 4. The spaces
\[ \left(S,\, i\frac{d}{dx}\right),\quad \left(S_\alpha^\beta,\, i\frac{d}{dx}\right)\quad \text{for } \alpha+\beta\ge1, \]
\[ \left(+S_\alpha^\beta,\, \frac{d^2}{dx^2}\right)\quad \text{for } \alpha+\beta\ge1,\ \alpha\ge\frac12, \]
are Carleman spaces*.

Proof. We shall prove, for example, that
\[ \left(+S_\alpha^\beta,\, \frac{d^2}{dx^2}\right) \]
for
\[ \alpha+\beta\ge1,\quad \alpha\ge\frac12 \]
is a Carleman space. Denote by \(G\) and \(G_1\)

* For the notation see \((^1)\), Ch. 4, and \((^3)\), § 4.

respectively, to the regions \((\alpha+\beta \geq 1,\ \alpha \geq 1/2)\) and \((\alpha+\beta \geq 1,\ \beta \leq 1/2)\). If the point \((\alpha,\beta)\in G_1\), then directly from the inequalities defining \(^{+}S_{\alpha}^{\beta}\) one can show that all functions \(\varphi(x)\in {}^{+}S_{\alpha}^{\beta}\) are Carleman. If \((\alpha,\beta)\notin G_1\), i.e. \(\beta>1/2\), then one can find a point \((\alpha',\beta')\in G_1\) such that \(\alpha'\leq \alpha\) and \(\beta'\leq \beta\); then \(^{+}S_{\alpha'}^{\beta'}\subset {}^{+}S_{\alpha}^{\beta}\). But all elements of \(^{+}S_{\alpha'}^{\beta'}\) are Carleman. It follows from Theorem 3 that \(^{+}S_{\alpha}^{\beta}\) is also a Carleman space.

It follows from Theorem 4 that positive-definite functionals on the indicated spaces admit a unique representation.

Theorem 5 (on the extension of a positive-definite function from a strip). A positive-definite function \(f(x_1,\ldots,x_n)\), continuous in the strip \(-\infty<x_i<\infty\) \((i=1,2,\ldots,n-1)\), \(-h\leq x_n\leq h\), of \(n\)-dimensional space, admits an extension to the whole plane by the formula
\[ f(x_1,\ldots,x_n) = \int_{R_n} \exp\left[i\sum x_k\xi_k\right]\,d\sigma(\xi_1,\ldots,\xi_n), \tag{5} \]
where \(\sigma(\xi_1,\ldots,\xi_n)\) is a bounded, nonnegative measure.

Proof. \(f(x_1,\ldots,x_n)\) may be regarded as a functional on the space of functions \(S(\pi_h)\) that vanish outside the strip
\[ \pi_h=\{x:\ -\infty<x_i<\infty\ (i=1,2,\ldots,n-1),\ -h\leq x_n\leq h\} \]
and satisfy the estimates
\[ \left|D^k\varphi(x_1,\ldots,x_n)\right|\leq C_{k,p}|x|^{-p}, \quad \text{where } k=(k_1,\ldots,k_{n-1}), \]
\[ D^k=\partial^{k_1+\cdots+k_{n-1}}/\partial x_1^{k_1}\cdots\partial x_{n-1}^{k_{n-1}}, \quad |x|=(x_1^2+\cdots+x_n^2)^{1/2}. \]

Obviously,
\[ S(\pi_h)= \underbrace{\hat S\hat\otimes\cdots\hat\otimes \hat S}_{n-1\ \text{times}} \hat\otimes K_h, \]
where \(K_h\) is the space of finitely infinitely differentiable functions on \((-h,h)\). The space \(S\) with characteristic operator \(i\,\frac{d}{dx}\) is a Carleman space (Theorem 4). Theorem 1 is applicable (obviously, it remains valid if, instead of one operator \(A\), one takes any number of Carleman operators \(A_i\) \((i=1,2,\ldots,n)\), commuting with one another and commuting with \(B\)): the operator \(i\,\frac{\partial}{\partial x_n}\) admits a self-adjoint extension commuting with the closures of the operators
\[ i\,\frac{\partial}{\partial x_k}\quad (k=1,2,\ldots,n-1). \]
By the Kostyuchenko–Mityagin theorem \((^2)\), \(f(x_1,\ldots,x_n)\) admits the expansion (5) in common eigenfunctions
\[ \chi_{\xi_1\ldots \xi_n} = \exp\left[i\sum x_k\xi_k\right] \]
of the operators
\[ i\,\frac{\partial}{\partial x_k}\quad (k=1,2,\ldots,n). \]
Theorem 5 is proved.

I express my gratitude to B. S. Mityagin for his attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
25 III 1960

REFERENCES

1. M. G. Krein, M. A. Krasnosel’skii, UMN, 2, no. 3 (19), 60 (1947).
2. A. G. Kostyuchenko, B. S. Mityagin, DAN, 131, no. 1 (1960).
3. A. G. Kostyuchenko, B. S. Mityagin, Tr. Moskovsk. matem. obshch., 9, 283 (1960).
4. I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 2, 1958.