A. V. STRAUS

ON SPECTRAL EXPANSIONS OF SYMMETRIC OPERATORS

(Presented by Academician I. M. Vinogradov, 12 IV 1963)

1. Consider a closed symmetric operator (A), acting in a Hilbert space (H) and having a domain of definition (D_A) dense in (H). The operator (A) is not assumed to be self-adjoint, so that (A) is a part of the operator (A^) adjoint to it, and in general the domain of definition (D_{A^}) of the operator (A^) is wider than (D_A). As is known, there exists a resolution of the identity (E_\lambda) ((-\infty<\lambda<+\infty)) such that, for any finite interval (\Delta=[\mu_1,\mu_2)) and any (f\in H), the relations hold:

[
E(\Delta)f\in D_{A^}, \qquad
A^E(\Delta)f=\int_{\mu_1}^{\mu_2}\lambda\,dE_\lambda f,
\tag{1}
]

where (E(\Delta)=E_{\mu_2}-E_{\mu_1}). This resolution of the identity is, in general, not orthogonal and not unique. Each such resolution of the identity is called a spectral function of the operator (A).** According to a well-known theorem of M. A. Naimark ((^{4}); see also ((^{1}), pp. 369—372)), for every spectral function (E_\lambda) ((-\infty<\lambda<+\infty)) of the operator (A) there exists, in some Hilbert space (\widehat H\supset H), a self-adjoint extension (\widetilde A) of the operator (A) such that the orthogonal spectral function (\widetilde E_\lambda) ((-\infty<\lambda<+\infty)) of the operator (\widetilde A) is related to (E_\lambda) by the formula

[
E_\lambda f=P\widetilde E_\lambda f \qquad (f\in H),
\tag{2}
]

where (P) is the projection operator in (\widehat H) onto (H).

Taking (1) into account, one may regard the equality

[
f=\int_{-\infty}^{+\infty} dE_\lambda f
\tag{3}
]

as a certain formula for expansion in generalized eigen-elements of the operator (A^*).

Let, for example, (A) be a symmetric ordinary differential operator with minimal domain of definition in (\mathcal L^2(a,b)), generated by a differential expression (l[y]) of order (2n); the endpoints of the interval ((a,b)) are not assumed to be regular. The spectral functions of such an operator are described in ((^{5})). In this case formula (3) is realized in the form of an expansion in solutions of the equation

[
l[y]-\lambda y=0,
\tag{4}
]

which also play the role of generalized eigen-elements of the operator (A^*). However, if at least one of the endpoints of the interval ((a,b)) is regular, then, as has been shown in a number of papers ((^{6-11})), there is a broad class of spectral expansions such that in each of them not all solutions of equation (4) take part,

* The operator function (E_\lambda) ((-\infty<\lambda<+\infty)) is assumed to be continuous from the left. On the definition of a resolution of the identity see, for example, ((^{1})), pp. 359, 360.

** Such spectral functions were considered for the first time by Carleman ((^{2})) for symmetric integral operators and by Stone ((^{3})) for operators in an abstract Hilbert space. The definition of a spectral function given here is due to M. A. Naimark ((^{4})).

only those which satisfy certain boundary conditions depending on (\lambda).

In the present paper we consider the spectral functions (E_\lambda) ((-\infty<\lambda<+\infty)) of a symmetric operator (A) acting in an abstract Hilbert space. It turns out that in this general case as well there are “boundary conditions,” depending on the parameter (\lambda), which are satisfied, in a certain sense, by the generalized eigenvectors of the operator (A^*) that participate in the expansion (3).

2. Let (E_\lambda) ((-\infty<\lambda<+\infty)) be some resolution of the identity in (\tilde H), and let (\alpha) and (\beta) be fixed real numbers. Denote by (K_0(E_\lambda;\alpha,\beta)) the linear manifold of vector functions (g(\lambda)) ((\alpha\leq \lambda\leq \beta)) with values in (H), admitting a representation
[
g(\lambda)=\varphi_1(\lambda)g_1+\varphi_2(\lambda)g_2+\cdots+\varphi_m(\lambda)g_m,
]
where (\varphi_k(\lambda)) ((k=1,2,\ldots,m)) are arbitrary continuous complex-valued functions of the parameter (\lambda), and (g_k) ((k=1,2,\ldots,m)) are arbitrary elements of (H); for different vector functions these elements and their number may be different.*

For any vector functions (g(\lambda), h(\lambda)\in K_0(E_\lambda;\alpha,\beta)) there exists the integral
[
\int_\alpha^\beta ((dE_\lambda)g(\lambda),h(\lambda))
=\lim_{\omega\to 0}\sum_{k=0}^{n-1}
\bigl((E_{\lambda_{k+1}}-E_{\lambda_k})g(\lambda'k),h(\lambda'_k)\bigr),
\tag{5}
]
where
[
\alpha=\lambda_0<\lambda_1<\cdots<\lambda_n=\beta,\qquad
\lambda'_k\in[\lambda_k,\lambda]\ (k=0,1,\ldots,n-1);
]
[
\omega=\max(\lambda_{k+1}-\lambda_k).
]

Let us introduce the aggregate (K_1(E_\lambda;\alpha,\beta)) of vector functions (g(\lambda)) ((\alpha\leq \lambda\leq \beta)) with values in (H) satisfying the following condition: for every (\varepsilon>0) there exists, for the function (g(\lambda)), a function (g_\varepsilon(\lambda)\in K_0(E_\lambda;\alpha,\beta)) such that
[
\overline{\lim_{\omega\to 0}}\sum_{k=0}^{n-1}
\bigl((E_{\lambda_{k+1}}-E_{\lambda_k})(g(\lambda'k)-g\varepsilon(\lambda'k)),
g(\lambda'_k)-g\varepsilon(\lambda'_k)\bigr)<\varepsilon,
]
where (\lambda_k,\lambda'_k), and (\omega) have the same meaning as before.

(K_1(E_\lambda;\alpha,\beta)) is, obviously, a linear manifold, and together with a vector function (g(\lambda)) it also contains (\varphi(\lambda)g(\lambda)), whatever continuous complex-valued function (\varphi(\lambda)) may be. It is easy to see that for any (g(\lambda)) and (h(\lambda)) from (K_1(E_\lambda;\alpha,\beta)) the integral (5) exists.**

Lemma. If (E_\mu) ((-\infty<\lambda<+\infty)) is a spectral function of a symmetric operator (A), then for any vector functions (g(\lambda)) and (h(\lambda)) from (K_1(E_\lambda;\alpha,\beta)) there exists the integral
[
\int_\alpha^\beta (A^(dE_\lambda)g(\lambda),h(\lambda))
=
\lim_{\omega\to 0}\sum_{k=0}^{n-1}
\bigl(A^(E_{\lambda_{k+1}}-E_{\lambda_k})g(\lambda'_k),h(\lambda'_k)\bigr)
]

* In an analogous way one can define the linear manifold (K_0(E_\lambda;-\infty,+\infty)), requiring additionally that the functions (\varphi_k(\lambda)) be finite.

** If the value of the integral (5) is taken as the scalar product ([g,h]) of the vector functions (g(\lambda)) and (h(\lambda)), then (K_1(E_\lambda;\alpha,\beta)) becomes a Hilbert space, in general incomplete. The completion of the space (K_1(E_\lambda;\alpha,\beta)) is, obviously, also a completion for (K_0(E_\lambda;\alpha,\beta)) in this same metric. We note that the completion of the space (K_0(E_\lambda;-\infty,+\infty)) essentially coincides with the space (\tilde H) discussed in M. A. Naimark’s theorem (see (1), pp. 363–366).

and the formula holds
[
\int_{\alpha}^{\beta}(A^*(dE_\lambda)g(\lambda),\,h(\lambda))
=\int_{\alpha}^{\beta}((dE_\lambda)\lambda g(\lambda),\,h(\lambda))
=\int_{\alpha}^{\beta}((dE_\lambda)g(\lambda),\,\lambda h(\lambda)).
]

1. Let (A) be a closed symmetric operator in (H), let (\widetilde A) be a self-adjoint operator in (\widetilde H \supset H) which is an extension of the operator (A), and let (E_\lambda) ((-\infty<\lambda<+\infty)) be the spectral function of the operator (A) associated with the spectral function (\widetilde E_\lambda) of the operator (\widetilde A) by formula (2). For any complex (\lambda) denote by (\widetilde{\mathcal L}\lambda) the manifold of all (\widetilde f\in D}) such that (\widetilde A\widetilde f-\lambda\widetilde f\in H). Put (\mathcal L_\lambda=P\widetilde{\mathcal L\lambda), where (P) is the projection operator in (\widetilde H) onto (H). It is easy to see that (D_A\subset\mathcal L\lambda\subset D_{A^*}). We note that the manifold (\mathcal L_\lambda), for any complex (\lambda), is determined uniquely by specifying the spectral function (E_\lambda) ((-\infty<\lambda<+\infty)) of the operator (A); here it is not necessary to assume that the operator (\widetilde A) is given.

Theorem 1. Let (g(\lambda)) ((\alpha\leqslant\lambda\leqslant\beta)) be a vector function with values in (H), satisfying the following conditions: 1) for every (\lambda\in[\alpha,\beta]), (g(\lambda)\in\mathcal L_\lambda); 2) the vector functions (g(\lambda)) and (A^g(\lambda)) belong to the manifold (K_1(E_\lambda;\alpha,\beta)); 3) for the vector function (h(\lambda)=A^g(\lambda)-\lambda g(\lambda)), for every (\mu\in[\alpha,\beta]) the equality
[
\lim_{\eta\to0}\left[\frac{1}{\eta^2}\int_{\mu}^{\mu+\eta}
((dE_\lambda)(h(\lambda)-h(\mu)),\,h(\lambda)-h(\mu))\right]=0
]
holds.

Then, for any (f\in H) and any (\mu_1,\mu_2\in[\alpha,\beta]), the equality
[
\int_{\mu_1}^{\mu_2}\left[(A^(dE_\lambda)f,\,g(\lambda))-((dE_\lambda)f,\,A^g(\lambda))\right]=0.
\tag{6}
]

Without dwelling on the proof of the theorem, we shall only note that we carry it out approximately in the same way as M. G. Krein justifies his method of directing functionals (\bigl((^{12});) see also ((^1)), pp. 448—454(\bigr)).

Corollary. Let the vector function (g(\lambda)) ((\alpha\leqslant\lambda\leqslant\beta)) satisfy condition 1) of Theorem 1 and have the form
[
g(\lambda)=\varphi_1(\lambda)g_1+\varphi_2(\lambda)g_2+\cdots+\varphi_m(\lambda)g_m,
]
where (g_k\in D_{A^*}), and (\varphi_k(\lambda)) are complex-valued functions satisfying the Lipschitz condition ((k=1,2,\ldots,m)). Then equality (6) holds.

Denote by (\mathfrak N_z), for any nonreal (z), the defect subspace of the operator (A), consisting of all (u\in D_{A^}) satisfying the equation (A^u=\overline z\,u). Fix some nonreal value (z_0). As shown in ((^8,{}^{13})), for any nonreal (z) from the half-plane ((\operatorname{Im} z\cdot\operatorname{Im} z_0>0)) the manifold (\mathcal L_z) defined above is representable in the form
[
\mathcal L_z=D_A+[F(z)-E]\mathfrak N_{z_0},
]
where (F(z)) is some linear operator from (\mathfrak N_{z_0}) into (\mathfrak N_{\overline z_0}), satisfying the conditions: a) (|F(z)|\leqslant1); b) (F(z)) is an analytic operator function of (z) in the half-plane ((\operatorname{Im} z\cdot\operatorname{Im} z_0>0)). The correspondence between the collection of all such operator functions (F(z)) and the set of all spectral functions (E_\lambda) ((-\infty<\lambda<+\infty)) is one-to-one.

Theorem 2. Let (g(\lambda)) ((\alpha\leqslant\lambda\leqslant\beta)) be a vector function with values in (H) satisfying the following conditions: 1) (g(\lambda)) is the continuous continuation of the vector function (g(z)=f_0+F(z)u-u) from the half-plane ((\operatorname{Im} z\cdot\operatorname{Im} z_0>0)), where (f_0\in D_A), (u\in\mathfrak N_{z_0}); (\operatorname{Im}(A^*g(\lambda),\,g(\lambda))=0) for every (\lambda\in[\alpha,\beta]). Then equality (6) holds.

1. Let us explain the meaning of the theorem by the example of a symmetric differential operator (A) with minimal domain of definition in (\mathcal L^2(0,+\infty)), which is generated by an ordinary differential expression (l[y]) of arbitrary even order (2n). For any functions (y=y(x)) and (v=v(x)) from (D_{A^}) the equality
   [
   (A^y,v)-(y,A^v)=[y,v]0^{+\infty},
   \tag{7}
   ]
   holds, where ([y,v]) is a known bilinear form in the functions (y(x)), (v(x)) and their quasi-derivatives up to order ((2n-1)), inclusive. If (E\lambda(-\infty<\lambda<+\infty)) is the spectral function of the operator (A), then for any real (\mu) and any function (f(x)\in \mathcal L^2(0,+\infty)) the formula
   [
   (E_\mu-E_0)f=\int_0^\mu y(x,\lambda,f)\,d\rho(\lambda),
   \tag{8}
   ]
   holds, where (\rho(\lambda)) is a certain scalar distribution function, and (y(x,\lambda,f)) is a certain solution of equation (4), depending linearly on (f). Let (g(\lambda)) be a vector function of (\lambda) ((\alpha\leq \lambda\leq \beta)) with values in (H=\mathcal L^2(0,+\infty)), such that (g(\lambda)=g(x,\lambda)), where (g(x,\lambda)) is a scalar function of (x) and (\lambda) ((0\leq x<+\infty;\ \alpha\leq \lambda\leq \beta)). Suppose that (g(\lambda)) satisfies all the conditions of Theorem 1 or 2 and that for every (\lambda\in[\alpha,\beta]), (g(x,\lambda)) is a finite function of (x). Taking (6), (7), and (8) into account, we conclude that for any (\mu_1,\mu_2\in[\alpha,\beta]) the equality
   [
   \int_{\mu_1}^{\mu_2} [\,y(x,\lambda,f),g(x,\lambda)\,]{x=0}\,d\rho(\lambda)=0
   ]
   holds. Hence it follows that almost everywhere on the interval ([\alpha,\beta]) with respect to the measure (\rho(\lambda)),
   [
   [\,y(x,\lambda,f),g(x,\lambda)\,]=0,
   ]
   i.e., (y(x,\lambda,f)) satisfies a certain boundary condition depending on (\lambda).

Received
4 IV 1963

References

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* One may set (\rho(\lambda)=\operatorname{Sp}T(\lambda-0)), where (T(\lambda)) is the matrix spectral distribution function (see (5), formulas (5.2) and (5.11)). We note that, without introducing any new means, the results established in (5) can be extended to symmetric ordinary differential operators of arbitrary order with complex coefficients (cf. (14)).