MATHEMATICS

M. E. CHUMAKIN

ON GENERALIZED RESOLVENTS OF AN ISOMETRIC OPERATOR

(Presented by Academician P. S. Novikov on 30 IX 1963)

1. The development of the spectral theory of symmetric* non-self-adjoint operators in a Hilbert space \(H\) was initiated by Carleman \((^{1})\) and Stone \((^{10})\). M. A. Naimark \((^{6})\) was the first to give a general definition of the spectral function of an arbitrary symmetric operator and established an important theorem according to which a family of operators \(E_t(-\infty<t<+\infty)\) in \(H\) is the spectral function of a symmetric operator \(A\) in \(H\) if and only if it admits the representation

\[ E_t h = P\widetilde E_t h \quad (h\in H), \tag{1} \]

where \(\widetilde E_t(-\infty<t<+\infty)\) is the spectral function of some self-adjoint extension \(\widetilde A\) of the operator \(A\) in a Hilbert space \(\widetilde H \supset H\), and \(P\) is the operator of orthogonal projection in \(\widetilde H\) onto \(H\). The definition of M. A. Naimark and formula (1) admit a generalization to any closed Hermitian operator \((^{12})\). Relation (1) may be taken as the basis for the definition of the spectral function of a closed Hermitian operator.

A formula for the generalized resolvents of an arbitrary symmetric operator, convenient for the actual construction of spectral functions, was obtained by A. V. Shtraus \((^{11,12})\). In another form, generalized resolvents of symmetric operators with defect index \((1,1)\) and \((m,m)\) \((m<\infty)\) had earlier been described by M. A. Naimark \((^{7})\) and M. G. Krein \((^{3,4})\). A. V. Shtraus \((^{12})\) also studied the properties characterizing generalized resolvents of Hermitian operators and proved an important theorem on the characteristic properties of generalized resolvents of a closed Hermitian operator.

In the present note we consider properties characterizing generalized resolvents of isometric operators, and establish that, by means of a formula analogous to the formula of A. V. Shtraus, all generalized resolvents of a closed isometric operator can be specified. In doing so, we largely follow the method of \((^{12})\). We note that the spectral functions of isometric operators in the special case when the operator is defined on all of \(H\) were considered by A. I. Plesner \((^{9})\).

2. Let \(U\) be a closed isometric operator acting in a Hilbert space \(H\), and let \(\widetilde U\) be an arbitrary unitary extension of the operator \(U\) (such an extension always exists, possibly with passage to an enlarged space \(\widetilde H \supset H\)). In what follows we shall denote by \(\widetilde E_t\) \((0\leq t\leq 2\pi)\) the spectral function of the operator \(\widetilde U\), by \(P\) the operator of orthogonal projection in \(\widetilde H\) onto \(H\), and by \(D_B\) and \(\Delta_B\), respectively, the domain of definition and the range of values of any operator \(B\).

Definition 1. A family of operators \(E_t\) \((0\leq t\leq 2\pi)\), acting in the Hilbert space \(H\), is called a spectral function of the operator \(U\) if, for every \(h\in H\),

\[ E_t h = P\widetilde E_t h. \]

* Recall that a linear operator \(A\), acting in a Hilbert space \(H\), is called Hermitian if, for any elements \(f,g\) of its domain of definition \(D_A\), \((Af,g)=(f,Ag)\). If, in addition, the manifold \(D_A\) is dense in \(H\), then the operator \(A\) is called symmetric.

Definition 2. A family of operators \(R_\zeta\) \((|\zeta|\ne 1)\) in \(H\) is called a generalized resolvent of the operator \(U\) if

\[ R_\zeta h=P(E-\zeta \widetilde U)^{-1}h\qquad (h\in H). \tag{2} \]

A given isometric operator has, generally speaking, an infinite set of spectral functions and the corresponding generalized resolvents. If \(E_t\), \(R_\zeta\), and \(\widetilde U\) are related as indicated in the definitions, then we shall say that \(E_t\) and \(R_\zeta\) are generated by the extension \(\widetilde U\).

Theorem 1. In order that a family of linear bounded operators \(R_\zeta\), acting in \(H\) \((D_{R_\zeta}=H)\) and depending on the complex parameter \(\zeta\) \((|\zeta|\ne 1)\), be the generalized resolvent of some closed isometric operator, it is necessary and sufficient that the following conditions be satisfied:

1) for some \(\zeta_0\ne 0\) \((|\zeta_0|<1)\) there exists a subspace \(L\subset H\) such that, for every \(\zeta\) \((|\zeta|\ne 1)\) and every \(f\in L\), the equality

\[ (\zeta R_\zeta-\zeta_0R_{\zeta_0})f=(\zeta-\zeta_0)R_\zeta R_{\zeta_0}f; \]

holds;

2) for every \(\varphi\in H\ominus \overline{R_{\zeta_0}L}\), \(R_0\varphi=\varphi\);

3) for every \(\zeta\) \((|\zeta|<1)\) and every \(h\in H\) the inequality

\[ \operatorname{Re}(R_\zeta h,h)\ge \frac12 (h,h); \]

holds;

4) for every \(h\in H\), \(R_\zeta h\) is an analytic vector function of \(\zeta\) in the unit disk \(|\zeta|<1\);

5) for every \(\zeta\) \((0<|\zeta|<1)\)

\[ R_\zeta^{*}=E-R_{1/\bar\zeta}. \]

In Theorem 1 the question concerned intrinsic properties characterizing the generalized resolvents of a closed isometric operator. At the same time, however, the necessary and sufficient conditions were not clarified under which the family of operators \(R_\zeta\) would serve as the generalized resolvent of a given closed isometric operator. The following theorem gives an answer to this question.

Theorem 2. In order that a family of linear bounded operators \(R_\zeta\) in \(H\) \((D_{R_\zeta}=H,\ |\zeta|\ne 1)\) serve as the generalized resolvent of a closed isometric operator \(U\) in \(H\), it is necessary and sufficient that conditions 3)—5) of Theorem 1 be satisfied, as well as the conditions:

\(1')\) for every \(\zeta\) \((|\zeta|\ne 1)\) and every \(g\in D_U\) the equality

\[ R_\zeta(E-\zeta U)g=g; \]

holds;

\(2')\) for every \(\varphi\in H\ominus \overline{D_U}\), \(R_0\varphi=\varphi\).

We now formulate the theorem on the general formula for generalized resolvents of a closed isometric operator, which can be proved on the basis of Theorem 2. In view of the relation

\[ R_{1/\bar\zeta}=E-R_\zeta^{*} \]

it is sufficient to establish this formula for all values of \(\zeta\) with modulus less than one.

Theorem 3. Every generalized resolvent \(R_\zeta\) of a closed isometric operator \(U\) in \(H\) is representable in the form:

\[ R_\zeta=[E-\zeta(U\oplus \Phi(\zeta))]^{-1}\qquad (|\zeta|<1), \tag{3} \]

where \(\Phi(\zeta)\) is some linear operator from \(H\ominus D_U\) into \(H\ominus \Delta_U\), which is an operator-valued function analytic in the unit disk and not exceeding one in norm. Conversely, every operator-valued function \(\Phi(\zeta)\) possessing the listed properties determines by formula (3) some generalized resolvent of the operator \(U\).

Remark 1. From formula (3) it follows that different \(\Phi(\xi)\) correspond to different generalized resolvents \(R_\xi\).

Remark 2. With the aid of formula (3) one easily obtains the result of B. Sz.-Nagy \((^8)\):

If \(T\) is a contraction of a Hilbert space \(H\), then in some space \(\widetilde H \supset H\) there exists a unitary operator \(\widetilde U\) such that

\[ T^n h = P\widetilde U^n h \qquad (h\in H;\ n=0,1,2,\ldots). \tag{4} \]

Indeed, the operator \(T\) can be represented in the form \(T=U\oplus\Phi\), where \(U\) is a closed isometric operator and \(\|\Phi\|\leq 1\) (for example, as \(U\) one may take the operator with \(D_U=\{0\}\)). Therefore, by virtue of formulas (2) and (3), we have

\[ (E-\xi T)^{-1}h = P(E-\xi\widetilde U)^{-1}h \qquad (h\in H,\ |\xi|<1), \]

whence (4) follows.

Theorem 4. Let \(U\) be a closed isometric operator which has no nonzero fixed elements. Then formula (3), where \(\Phi(\xi)\) satisfies the same conditions as in Theorem 3, determines a generalized resolvent of the operator \(U\), generated by some unitary extension \(\widetilde U\) having no nonzero fixed elements, if and only if, for every \(\varphi\in H\ominus \overline{\Delta}_{E-U}\), the condition

\[ \lim_{\xi\to 1}(1-\xi)\bigl([E-\xi(U\oplus\Phi(\xi))]^{-1}\varphi,\varphi\bigr)=0 \]

\[ \left( -\frac{\pi}{2}+\varepsilon<\arg(1-\xi)<\frac{\pi}{2}-\varepsilon;\quad 0<\varepsilon<\frac{\pi}{2} \right). \tag{5} \]

is fulfilled.

Let the operator \(U\) be the same as in Theorem 4. Denote by \(P_1\) and \(P_2\) the operators of orthogonal projection in \(H\) respectively onto \(H\ominus D_U\) and \(H\ominus \Delta_U\). On the set of elements \(P_1(H\ominus\overline{\Delta}_{E-U})\) define an operator \(V\) by the formula

\[ VP_1h=P_2h \qquad (h\in H\ominus\overline{\Delta}_{E-U}). \]

The operator \(V\) is isometric \((^2)\).

Lemma. Let the operator \(U\) be the same as in Theorem 4, and let \(\Phi(\xi)\) be the same as in Theorem 3. If for every \(\varphi\in H\ominus\overline{\Delta}_{E-U}\) relation (5) is fulfilled, then \(\Phi(\xi)\psi=V\psi\) only for \(\psi=0\).

Using Theorem 4 and the lemma, one can prove the following proposition.

Theorem 5. Every generalized resolvent \(R'_\lambda\) of a closed Hermitian operator \(A\) in \(H\) is representable in the form

\[ R'_\lambda=(A_{F(\lambda)}-\lambda E)^{-1} \qquad (\operatorname{Im}\lambda>0), \tag{6} \]

where \(F(\lambda)\) is some linear operator from \(\mathfrak N_i=H\ominus(A-iE)D_A\) into \(\mathfrak N_{-i}=H\ominus(A+iE)D_A\), satisfying the conditions:

1) \(\|F(\lambda)\|\leq 1\);

2) \(F(\lambda)\) is an operator-valued function of the parameter \(\lambda\), analytic in the upper half-plane;

3) for every \(\varphi\in H\ominus\overline{D}_A\)

\[ \lim_{\lambda\to\infty}\lambda\bigl((A_{F(\lambda)}-\lambda E)^{-1}\varphi,\varphi\bigr) = (-\varphi,\varphi) \qquad \left(\varepsilon<\arg\lambda<\pi-\varepsilon;\ 0<\varepsilon<\frac{\pi}{2}\right), \]

and the operator \(A_{F(\lambda)}\) is defined by the formula

\[ A_{F(\lambda)}g = Af+iF(\lambda)\psi_i+i\psi_i \]

on the manifold of elements of the form

\[ g=f+F(\lambda)\psi_i-\psi_i \qquad (f\in D_A,\ \psi_i\in\mathfrak N_i). \]

Conversely, by every operator function possessing properties 1)–3), a certain generalized resolvent of the operator \(A\) is determined by formula (6).

This proposition generalizes Theorem 7 of the work \({}^{12}\).

The result contained in Theorem (5) was previously established by another method by B. I. Loshkarev \({}^{5}\). Let us note that condition 3, which appears in the theorem of the work \({}^{5}\), turns out to be superfluous.

The work was carried out under the supervision of Prof. A. V. Shtraus, to whom the author expresses sincere gratitude.

Ulyanovsk State Pedagogical Institute
named after I. N. Ulyanov

Received
26 IX 1963

CITED LITERATURE

\({}^{1}\) T. Carleman, Sur les équations intégrales singulières a noyau réel et symetrique, Upsala, 1923.
\({}^{2}\) M. A. Krasnosel’skii, Ukr. Math. J., 1, 21 (1949).
\({}^{3}\) M. G. Krein, DAN, 43, No. 8, 339 (1944).
\({}^{4}\) M. G. Krein, DAN, 52, No. 8, 657 (1946).
\({}^{5}\) B. I. Loshkarev, Proceedings of the Second Scientific Conference of the Mathematics Departments of the Pedagogical Institutes of the Volga Region, issue 1, Kuibyshev, 1962.
\({}^{6}\) M. A. Naimark, Izv. Acad. Sci. USSR, Ser. Math., 4, No. 3, 277 (1940).
\({}^{7}\) M. A. Naimark, Izv. Acad. Sci. USSR, Ser. Math., 7, No. 6, 285 (1943).
\({}^{8}\) B. S. Sz.-Nagy, UMN, 11, No. 6 (72), 173 (1956).
\({}^{9}\) A. I. Plesner, DAN, 25, No. 9, 708 (1939).
\({}^{10}\) M. Stone, Linear Transformations in Hilbert Space, N. Y., 1932.
\({}^{11}\) A. V. Shtraus, DAN, 78, No. 2, 217 (1951).
\({}^{12}\) A. V. Shtraus, Izv. Acad. Sci. USSR, Ser. Math., 18, No. 1, 51 (1954).