MATHEMATICS

A. G. KOSTYUCHENKO and B. S. MITYAGIN

ON POSITIVE-DEFINITE FUNCTIONALS ON NUCLEAR SPACES

(Presented by Academician S. L. Sobolev on 10 XI 1959)

Recently, in connection with the development of the theory of linear topological spaces and their applications, the question has arisen of representing positive-definite functionals defined on various linear topological spaces*. The simplest theorem of this type is the Bochner—Schwartz theorem (¹). In the works of I. M. Gelfand and S. Doshin (²) and N. Ya. Vilenkin (³), the general form of a positive-definite functional on the even space ( {}^{+}S^{1/2}_{1/2} ) was found.

In the present work a certain general approach is proposed, essentially a synthesis of M. G. Krein’s idea, embodied in his method of directing functionals, and the general theorem (⁶) on expansion in eigenfunctions of arbitrary self-adjoint operators. Such an approach to the question is not new (⁴, ⁵); however, until now it has been connected with the consideration of only one operator and therefore did not make it possible to obtain multidimensional theorems. We have already obtained a number of multidimensional theorems, both known and new.

(1^\circ). We shall consider only nuclear linear topological spaces (\Phi). Let a continuous Hermitian form ((\varphi,\psi)) be given in (\Phi). Complete (\Phi) with respect to this scalar product to a complete Hilbert space (H). Then (⁶) (\Phi \subset H \subset \Phi'), where (\Phi') is the space conjugate to (\Phi). Consider in (\Phi) a system of symmetric operators (A_1, A_2,\ldots,A_n), which have self-adjoint mutually commuting extensions in (H)**.

Theorem 1. If the system ({A_i}) of symmetric operators has in (H) self-adjoint mutually commuting extensions, then this system has a complete system of generalized eigenfunctionals over (\Phi), (\chi_{\lambda_1\ldots\lambda_n}), and the representation holds

[
(\varphi,\psi)=\sum_{\alpha}\int_{-\infty}^{\infty}\cdots\int
\overline{(\chi_{\lambda_1\ldots\lambda_n},\varphi)}
(\chi_{\lambda_1\ldots\lambda_n},\psi)\,
d\sigma_{\alpha}(\lambda_1,\ldots,\lambda_n),
\tag{1}
]

where (\sigma_{\alpha}(\lambda_1,\ldots,\lambda_n)) is a system of finite measures in (n)-dimensional space.

If the system of commuting operators jointly has simple spectrum***, then the representation is possible

[
(\varphi,\psi)=\int_{-\infty}^{\infty}\cdots\int
\overline{(\chi_{\lambda_1\ldots\lambda_n},\varphi)}
(\chi_{\lambda_1\ldots\lambda_n},\psi)\,
d\sigma(\lambda_1,\ldots,\lambda_n).
\tag{1a}
]

* The question of representing a positive-definite functional on the normalized ball is completely solved by the Raikov—Weil theorem.

** Self-adjoint operators are called commuting if their spectral families commute.

*** It is said that a system of commuting self-adjoint operators ({A_i}) has jointly simple spectrum if there exists a vector (g\in H) such that the closed linear hull of vectors of the form (E_{\lambda_1\lambda_2\ldots\lambda_n}g) is the whole space (H). Here
[
E_{\lambda_1\lambda_2\ldots\lambda_n}
=
E_{\lambda_1}^{(1)}E_{\lambda_2}^{(2)}\cdots E_{\lambda_n}^{(n)},
]
and (E_{\lambda_k}^{(k)}) is the spectral family of the operator (A_k).

The proof of Theorem 1 is obtained almost by a verbatim repetition of the proof of the theorem on expansion in eigenfunctionals of a single operator (see also (7)).

2°. Definition 1. A complete linear topological space (\Phi) is called a nuclear algebra (ring) with involution if (\Phi) is a nuclear space and two operations are defined on (\Phi), continuous in its topology: commutative multiplication (x\circ y) and involution (x\to x^*), with the usual properties.

Definition 2. A continuous operator (A) on a nuclear algebra with involution will be called characteristic if, for any two elements (x,y\in\Phi),

[
Ax\circ y^ = x\circ (Ay)^ .
]

Definition 3. We shall say that a continuous functional (T) is positive definite on a nuclear algebra with involution if, for all (x\in\Phi), (T(x\circ x^*)\ge 0).

As a rule, (\Phi) has no unit, but we shall assume that there exists a unit sequence ({\delta_n}), i.e. (\delta_n\in\Phi) and (x\circ\delta_n\to x) in the topology of (\Phi) for every (x). Let us give several simple properties related to the notions introduced.

1. (A(x\circ y)=Ax\circ y=x\circ Ay).
2. A characteristic operator is real, i.e. ((Ax)^=Ax^).
3. The ring (\mathfrak A) of characteristic operators is commutative.
4. The subring (\Phi_) of all Hermitian elements, i.e. such that (x=x^), is dense in (\mathfrak A).
5. Every positive-definite functional is real, i.e. (T(x^*)=\overline{T(x)}).

An arbitrary positive-definite functional (T) defines on (\Phi) a scalar product in a natural way: ((x,y)_T=T(x\circ y^*)). Completing (\Phi) with respect to this scalar product to a complete Hilbert space (H). Every characteristic operator is symmetric on (\Phi) in the scalar product ((x,y)_T). The following assertion is valid: the closure (\overline A) in (H) of any characteristic operator (A) has equal deficiency indices.

3°. Suppose further that there exists a system (A_1,A_2,\ldots,A_n) of characteristic operators such that:

1) for each (\lambda=(\lambda_1,\ldots,\lambda_n)) there is at most one common eigenfunctional (\chi_\lambda), and moreover for each (\chi_\lambda) there exists

[
\lim_{n\to\infty}(\chi_\lambda,\delta_n^*)=b(\lambda)\ne 0;
]

2) either a) the closures (\overline A_i) of the operators (A_i) in (H_T) are self-adjoint pairwise commuting operators in any (H_T); or b) (\overline A_i) admit self-adjoint pairwise commuting extensions in (H_T).

Using formula (1a), we obtain a representation of the positive-definite functional

[
T(x)=\int_{-\infty}^{\infty}\overline{(\chi_\lambda,x)}\,d\sigma(\lambda).
\tag{2}
]

Here (\chi_\lambda=\chi_{\lambda_1\ldots\lambda_n}) are the common eigenfunctionals of the system of operators ({A_i}), normalized in such a way that (\lim_{n\to\infty}(\chi_\lambda,\delta_n^*)=1). It is not difficult to see that the (\chi_\lambda) are multiplicative functionals, i.e. (\chi_\lambda(x\circ y)=\chi_\lambda(x)\chi_\lambda(y)).

An important question in the theory of representation of positive-definite functionals is the question of the uniqueness of the measure in formula (2). Theorem 2 gives a criterion for uniqueness.

Theorem 2. If the closures of the operators ({A_i}) are commuting self-adjoint operators in (H_T), obtained by completing (\Phi) with respect to (T(x\circ y^*)), then the measure (\sigma(\lambda)) in formula (2) is determined uniquely.

In a certain sense the converse assertion is also true: if the representation is unique, then there do not exist two different commuting extensions of the system of operators.

(4^\circ.) We have made several assumptions concerning the operators (A_i). One of them, namely the commutativity condition, is in each concrete case checked with great difficulty. In this section we shall consider a class of nuclear algebras with involution of simple structure, for which the commutativity condition will always be fulfilled.

Let (\Phi_1) be a nuclear algebra with involution and let (A_1) be a characteristic operator such that the equation (A_1\chi_\lambda=\lambda\chi_\lambda) in (\Phi_1) can have only one (up to a constant factor) solution (\chi_\lambda).

Consider two nuclear algebras (\Phi_1) and (\Phi_2) with involutions (_1) and (_2). Their projective tensor product (\Phi_1\widehat{\otimes}\Phi_2=\Phi) is a nuclear algebra with involution (*), defined by the equality

[
(\varphi_1\otimes\varphi_2)^=\varphi_1^{{1}}\otimes\varphi_2^{*,\qquad}
\varphi_1\in\Phi_1,\ \varphi_2\in\Phi_2 .
]

Let (\Phi_1) (and respectively (\Phi_2)) have the property that the closure in any (H_T) of the characteristic operator (A_1) (respectively (A_2)) is already a self-adjoint operator. In (\Phi_1\widehat{\otimes}\Phi_2) one can define in the natural way the operators (A_1\otimes I_2) and (I_1\otimes A_2)*.

Theorem 3. Under the assumptions made, the closures of the operators (A_1\otimes I_2), (I\otimes A_2) are self-adjoint commuting operators in (H_T), obtained by completing (\Phi_1\widehat{\otimes}\Phi_2) with respect to any scalar product generated by a positive-definite functional.

(5^\circ.) Example 1. Consider the space (S_\alpha^\beta=S_{\alpha_1\ldots\alpha_n}^{\beta_1\ldots\beta_n}) (8). The space (S_\alpha^\beta) is nuclear and

[
S_\alpha^\beta=S_{\alpha_1}^{\beta_1}\widehat{\otimes}\ldots\widehat{\otimes}S_{\alpha_n}^{\beta_n}\tag{9}.
]

Multiplication and involution are introduced in the following way:

[
(\varphi\circ\psi)(s)=\int_{R_n}\varphi(\xi)\psi(s-\xi)\,d^n\xi,\qquad
\varphi^*(s)=\overline{\varphi(-s)}.
]

In (S_{\alpha_k}^{\beta_k}), as the characteristic operator (A_k), we take the operator

[
A_k=i\,d/ds_k .
]

Theorem 4. Every positive-definite functional on (S_{\alpha_1\ldots\alpha_n}^{\beta_1\ldots\beta_n}) admits the representation

[
T(\varphi)=\int_{R_n}\left{\int_{R_n}e^{i(s,\lambda)}\varphi(s)\,d^n(s)\right}\,d\sigma(\lambda_1,\ldots,\lambda).
]

The measure (\sigma(\lambda)) is determined uniquely and satisfies the condition

[
\int_{R_n}\exp\left(-\sum_{k=1}^n b_k|\lambda_k|^{1/\beta_k}\right)d\sigma(\lambda)<\infty
\quad\text{for any } b_k>0 .
]

Conversely, every such functional is positive-definite on (S_\alpha^\beta).

Example 2. In (S_\alpha^\beta) one can introduce the involution in another way: (\varphi^*(s)=\overline{\varphi(s)}). The characteristic operators in this case will be (B_k=d/ds_k). Positive-

* (I_1, I_2) are the identity operators respectively in (\Phi_1) and (\Phi_2).

functionals associated with such an involution should naturally be called positive-definite functionals in the sense of S. N. Bernstein.

Theorem 5. A positive-definite functional in the sense of S. N. Bernstein on (S_{\alpha}^{\beta}), (\alpha<1), admits the representation
[
T(\varphi)=\int_{\lambda}\left{\int_{R_n} e^{-(s,\lambda)}\varphi(s)\,d^n s\right}\,d\sigma(\lambda).
]

The measure (\sigma(\lambda)) is uniquely determined and satisfies the condition
[
\int_{R_n}\exp\left(\sum_{k=1}^{n} a_k |\lambda_k|^{\frac{1}{1-\alpha_k}}\right)\,d\sigma(\lambda)<\infty
\quad \text{for any } a_k>0.
]

The converse is also true: every such functional is a positive-definite functional in the sense of S. N. Bernstein.

Example 3. Consider the space ({}^{+}S_{\alpha}^{\beta}) of all even functions from (S_{\alpha}^{\beta}) depending on one variable. The involution and multiplication are the same as before. The principal characteristic operator will be the operator (C=d^2/ds^2).

Theorem 6. Every positive-definite functional on ({}^{+}S_{\alpha}^{\beta}) admits the representation
[
T=\int_{0}^{\infty}\cos\sqrt{\mu}\,s\,d\sigma_1(\mu)
+\int_{0}^{\infty}\operatorname{ch}\sqrt{\lambda}\,s\,d\sigma_2(\lambda);
]
the measure (\sigma_2\equiv 0), if (\alpha\ge 1).

The question of the uniqueness of the measures (\sigma_i) is equivalent to the question of self-adjointness in (H_T) of the operator (C=d^2/ds^2). Using results on the uniqueness of the solution of the Cauchy problem for the heat equation, one can show that (d^2/ds^2) is self-adjoint if (\alpha\ge 1/2). If (\alpha<1/2), then, as shown in ((^2)), uniqueness, generally speaking, does not hold.

Theorem 7. Every continuous positive-definite functional (T) on ({}^{+}S_{\alpha_1\ldots \alpha_n}^{\beta_1\ldots \beta_n}), (\alpha_k\ge 1/2), is representable in the form
[
T=\int_{+} e^{i(s,\lambda)}\,d\sigma(\lambda_1,\ldots,\lambda_n),
]
where (\sigma(\lambda_1,\ldots,\lambda_n)) is an even measure on the direct product of the sets (\Gamma_k={\lambda:\operatorname{Re}\lambda_k\cdot\operatorname{Im}\lambda_k=0}), if (\alpha_k<1), and (\Gamma_k={\lambda:\operatorname{Im}\lambda_k=0}), if (\alpha_k\ge 1). The measure (\sigma) is uniquely determined and satisfies the condition, for any (c_k>0), (d_k>0),
[
\int_{\Gamma_1\times\cdots\times\Gamma_k}
\exp\left(
-\sum_{k=1}^{n} c_k |\operatorname{Re}\lambda_k|^{\frac{1}{\beta_k}}
+\sum_{k=1}^{n} d_k |\operatorname{Im}\lambda_k|^{\frac{1}{1-\alpha_k}}
\right)
\,d\sigma(\lambda_1,\ldots,\lambda_k)<\infty.
]

For the case (S_{1/2}^{1/2}), by another method this result was obtained earlier by I. M. Gel'fand and Xia Dao-xing ((^2)) in the one-dimensional case and by N. Ya. Vilenkin ((^3)) in the multidimensional case. We also note the work of E. Vul ((^{10})), devoted to the question of uniqueness of measures in the representation of a positive-definite functional.

Moscow State University
named after M. V. Lomonosov

Received
4 XI 1959

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