V. I. PLYUSHCHEVA

ON THE INTEGRAL REPRESENTATION OF CONTINUOUS HERMITIAN-INDEFINITE KERNELS

(Presented by Academician N. N. Bogolyubov, 5 III 1962)

Let (\chi) be some nonnegative integer ((\chi < \infty)). A continuous Hermitian kernel (K(x,y)), (x,y \in (a,b)) ((-\infty \leq a,b \leq +\infty)), will be called Hermitian-indefinite with (\chi) negative squares (h.i.) if, for any (x_1,\ldots,x_n \in (a,b)), the form

[
\sum_{j,k=1}^{n} K(x_j,x_k)\xi_k\overline{\xi_j}
\tag{1}
]

has no more than (\chi) negative squares, and at least one such form has exactly (\chi) negative squares.

In the case of kernels of the form (K(x,y)=k(y-x)) ((-\infty < x < \infty)), M. G. Krein ((^1)) obtained an integral representation of h.i. functions (k(t)) of Bochner-theorem type. Below this result is generalized to the case of kernels that are expanded in eigenfunctions of a differential operator. In the case of a difference operator, a similar representation was obtained by the author in ((^2)^*) by developing the method of M. G. Krein and I. S. Iokhvidov ((^3)) for h.i. sequences (c_j). The method of proof in this note is a continual variant of the reasoning in ((^2)); it also uses the theory of representation of positive definite kernels developed by M. G. Krein ((^4)) and Yu. M. Berezanskii ((^5)). We note that in this note, in particular, the problem of extending h.i. functions from a finite interval to the whole axis is solved without additional restrictions.

1. Let (K(x,y)) be h.i. We associate with it a space of type (\Pi_\chi) ((^6)). To this end, consider the manifold (C_0^\infty(a,b)) of finite infinitely differentiable functions on ((a,b)) with scalar product

[
\langle \varphi,\psi\rangle
=
\int_a^b \int_a^b K(x,y)\varphi(y)\overline{\psi(x)}\,dx\,dy .
\tag{2}
]

It follows from the definition of an h.i. function that there exist (\chi) linearly independent functions of the form
[
\xi_m(x)=\sum_j \xi_j^{(m)}\delta(x-x_j)\quad (m=1,\ldots,\chi),
]
for which

[
\int_a^b \int_a^b K(x,y)\xi_m(y)\overline{\xi_n(x)}\,dx\,dy
=
\sum_{j,k} K(x_j,x_k)\xi_k^{(m)}\overline{\xi_j^{(n)}}
=
\begin{cases}
-1 & \text{when } m=n,\
0 & \text{when } m\ne n.
\end{cases}
\tag{3}
]

Since every generalized function can be approximated by finite infinitely differentiable functions, in (C_0^\infty(a,b)) there will be found—

* The requirement imposed in that article that, among (\det |F_{jk}|{j,k=1}^{n}), for large (n), there occur values different from zero can be dropped. The results of that work are also valid for the case of a differential expression with variable coefficients, owing to the possibility of extending a Hermitian operator to a self-adjoint one with passage to a broader space of type (\Pi\chi) with the same (\chi). This fact was communicated to the author by M. G. Krein.

there are (\varkappa) linearly independent negative functions (e_1,\ldots,e_\varkappa): (\langle e_m,e_m\rangle<0). Applying to them the usual orthogonalization process, we obtain a basis of a negative (\varkappa)-dimensional subspace in (C_0^\infty(a,b)). It also follows from the definition of an h.i. function that in (C_0^\infty(a,b)) there do not exist negative subspaces of dimension greater than (\varkappa). Identifying in (C_0^\infty(a,b)) functions (\varphi) and (\psi) such that (\varphi-\psi) is an element of an isotropic subspace, and using Theorem 1.4 ((^6)), we obtain that (C_0^\infty(a,b)) can be completed to a space of type (\Pi_\varkappa).

1. Consider on ((a,b)) an ordinary differential expression

[
\mathcal L[u]=\sum_{0\le k\le r} a_k(x)\frac{d^k u}{dx^k}
\tag{4}
]

with complex-valued coefficients, each of which, for simplicity of exposition, we shall assume to be infinitely differentiable. Suppose that there exists an h.i. kernel (K(x,y)) for which

[
\mathcal L_x[K(x,y)]=\overline{\mathcal L}_y[K(x,y)]
\tag{5}
]

(the bar denotes passage to the complex-conjugate coefficients; the equality is understood in the sense of Schwartz distributions).

Theorem 1. To every h.i. kernel for which (5) holds there corresponds at least one polynomial (P(\lambda)) of degree (\varkappa) such that the kernel, generalized in the sense of Schwartz,
[
\Phi(x,y)=\overline{P}(\overline{\mathcal L})_y\,P(\overline{\mathcal L})_x[K(x,y)]
]
is positive definite.

We outline the proof. The operator (L_0), defined on (C_0^\infty(a,b)) by means of the differential expression (\mathcal L^+) ((+) denotes passage to the expression formally adjoint to (\mathcal L)), is Hermitian in (\Pi_\varkappa), constructed from (K(x,y)). It is not difficult to see that this definition of (L_0) is correct. The operator (L_0) can be extended to a self-adjoint operator (L), generally speaking, with range in a wider space of type (\Pi_\varkappa) with the same (\varkappa).* By a theorem of L. S. Pontryagin ((^8)), for (L) there exist an invariant nonnegative (\varkappa)-dimensional subspace (\mathcal T), in which all eigenvalues of the operator (L) have nonnegative imaginary part, and (\mathcal T'), an invariant nonnegative subspace of dimension (\varkappa), in which all eigenvalues of (L) have nonpositive imaginary part. Let (P(\lambda)) and (\overline{P}(\lambda)) be the characteristic polynomials of (L) in (\mathcal T) and (\mathcal T'), respectively. The operator (P(L')), where (L') is the restriction of (L) to (\mathcal T+\mathcal T'+C_0^\infty(a,b)), annihilates (\mathcal T), and (\overline{P}(L')) annihilates (\mathcal T'). Therefore (M={P(\mathcal L^+)\varphi},\ \varphi\in C_0^\infty(a,b)), is orthogonal to (\mathcal T'). By Theorem 1.2 ((^6)), (M), as orthogonal to a (\varkappa)-dimensional nonnegative subspace in (\Pi_\varkappa), is itself nonnegative, i.e.
[
\langle P(\mathcal L^+)\varphi,\;(\mathcal L^+)\varphi\rangle\ge 0
\quad (\varphi\in C_0^\infty(a,b)),
]
and this means precisely that
[
\Phi(x,y)=\overline{P}(\overline{\mathcal L})_y\,P(\overline{\mathcal L})_x[K(x,y)]
]
is positive definite.

1. Since
   [
   \Phi(x,y)=\overline{P}(\overline{\mathcal L})_y\,P(\overline{\mathcal L})_x[K(x,y)]
   ]
   is a positive definite generalized kernel, using the works ((^5,^9)) (or the method of directing functionals ((^{11}))), we obtain the representation, in the weak sense,

[
\overline{P}(\overline{\mathcal L})y\,P(\overline{\mathcal L})_x[K(x,y)]
=
\int}^{\infty
\sum_{j,k=0}^{r-1}
\chi_j(x,\lambda)\,\overline{\chi_k(y,\lambda)}\,d\rho_{j,k}(\lambda),
\tag{6}
]

where (\chi_j(x,\lambda)) ((j=0,1,\ldots,r-1)) is a fundamental system of solutions of the equation (\mathcal L[v]=\lambda v), satisfying initial data of the form

[
\frac{d^k}{dx^k}\chi_j(x,\lambda)\bigg|{x=a}
=
\delta
\quad (k=0,1,\ldots,r-1)
]
((a) is an arbitrary point),

* The possibility of such an extension was communicated to the author by M. G. Krein and I. S. Iokhvidov.

(|d\rho_{j,k}(\lambda)|_{j,k=0}^{r-1}) is a matrix distribution function. It remains to solve the differential equation (6) with respect to (K(x,y)).

Theorem 2. Let (\mathscr L) be an ordinary differential expression of the form (4), and let (K(x,y)) be an H.-i. kernel for which (5) holds. Then (K(x,y)) admits the integral representation

[
K(x,y)=T_\rho(x,y)+\int_{-\infty}^{\infty}
\frac{\displaystyle \sum_{j,k=0}^{r-1}\chi_j(x,\lambda)\overline{\chi_k(y,\lambda)}-S_\rho^{(j,k)}(x,y,\lambda)}
{Q_0^2(\lambda)}\,d\sigma_{j,k}(\lambda);
\tag{7}
]

here (|d\sigma_{j,k}(\lambda)|_{j,k=0}^{r-1}) is a matrix distribution function,

[
Q_0(\lambda)=\prod_{n=1}^{s}(\lambda-a_n)^{\rho_n},
]

where (a_1,\ldots,a_n) are all the real distinct zeros of the polynomial (P(\lambda)), and (\rho_n) are their multiplicities; (S_\rho^{(j,k)}(x,y,\lambda)) are any corrections regularizing the integrals; (T_\rho(x,y)=\overline{T_\rho(y,x)}) is a suitably chosen Hermitian solution of the equation

[
\overline{P(\mathscr L)_y}\,P(\mathscr L)_x\,u(x,y)=0.
]

As in (1), the function (S_\rho^{(j,k)}(x,y,\lambda)) ((\rho>0)) will be called a regularizing correction in the integrals (7) if (S_\rho^{(j,k)}(x,y,\lambda)=0) for (|\lambda|>\rho), while for (|\lambda|\leq \rho) the function (S_\rho^{(j,k)}(x,y,\lambda)) is equal to the product (Q_0^2(\lambda)) by the sum of the principal parts of the function
(\chi_j(x,\lambda)\overline{\chi_k(y,\lambda)}/Q_0^2(\lambda)) with respect to all its poles.

1. Let us consider two examples.

Example 1. Let (k(x)) ((-\infty<x<+\infty)) be a continuous function such that the kernel (K(x,y)=k(x+y)) is H.-i. For simplicity we shall take (\chi=1). The expression (\mathscr L[u]=du/dx) satisfies (5). It is not difficult to write a representation for (K(x,y)=k(x+y)). Then, replacing (x+y) by (t), for (k(t)) we obtain: either

[
k(t)=(At+B)e^{a_1t}+
\int_{-\infty}^{\infty}
\frac{e^{\lambda t}-e^{a_1t}[t(a_1+\lambda)+1]}
{(\lambda-a_1)^2}\,d\sigma_{0,0}(\lambda),
]

or

[
k(t)=Ce^{\alpha_1 t}+D e^{\bar{\lambda}1 t}+
\int(\lambda),}^{\infty} e^{\lambda t}\,d\sigma_{0,0
]

where (A,B,C,D) are certain constants, and (\lambda_1) is a certain non-real number.

Example 2. Let (k(x)), (-\infty<x<+\infty), be a continuous function such that the kernel (K(x,y)=k(y-x)) is H.-i. Then (\mathscr L[u]=i\,du/dx) satisfies (5), and for (k(x)) we obtain the representation, given by M. G. Krein in ((^1)):

[
k(x)=h_\rho(x)+\int_{-\infty}^{\infty}
\frac{e^{i\lambda x}-S_\rho(x,\lambda)}
{Q_0^2(\lambda)}\,d\sigma(\lambda),
\tag{8}
]

where (h_\rho(x)) is a suitably chosen Hermitian solution of the equation

[
P\left(-i\frac{d}{dx}\right)\overline{P}\left(-i\frac{d}{dx}\right)h=0.
]

In the case of an H.-i. kernel on a one-dimensional interval ((-a,a)), as is clear from the preceding, we obtain the same representation (8). The existence of such a representation shows that every H.-i. function on a finite interval can be extended to an H.-i. function on the whole axis.

1. The result stated above can be generalized to the case of kernels (K(x,y)) defined for (x,y\in G), where (G) is a domain of (n)-dimensional space—

properties; in this case (\mathscr L) is a partial differential expression. Analogously to the preceding, one can prove that the kernel (\Phi(x,y)=\overline{P(\mathscr L)_y}\,P(\mathscr L)_x[K(x,y)]) is positive definite and therefore admits an integral representation. In obtaining a representation of the kernel (K(x,y)), difficulties arise on which we shall not dwell here.

In conclusion, the author expresses his gratitude to Yu. M. Berezanskii for valuable comments and for guidance in this work.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
28 II 1962

REFERENCES

1. M. G. Krein, DAN, 125, No. 1 (1959).
2. V. I. Plyushcheva, Ukr. matem. zhurn., No. 1 (1962).
3. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 8, 413 (1958).
4. M. G. Krein, DAN, 53, No. 1 (1946).
5. Yu. M. Berezanskii, Matem. sborn., 47 (89), 2 (1959).
6. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956).
7. M. G. Krein, Ukr. matem. zhurn., 1, No. 4 (1949).
8. L. S. Pontryagin, Izv. AN SSSR, ser. matem., 8, 243 (1944).
9. K. Maurin, Bull. Acad. Polon. Sci., Ser. math., 6, No. 3, 149 (1958).
10. A. Devinatz, Trans. Am. Math. Soc., 77, 455 (1954).
11. M. G. Krein, Sborn. tr. Inst. matem. AN USSR, 10, 83 (1948).