UDC 517.948.35:513.88

MATHEMATICS

A. V. STRAUS

ON EXTENSIONS, CHARACTERISTIC FUNCTIONS, AND GENERALIZED RESOLVENTS OF SYMMETRIC OPERATORS

(Presented by Academician I. M. Vinogradov, 7 IV 1967)

1. Let \(A\) be a closed symmetric operator in a Hilbert space \(H\). The domain \(D_A\) of the operator \(A\) is not assumed to be dense in \(H\). For an arbitrary nonreal \(z\) set \(\mathfrak N_z = H \ominus (A - zE)D_A\). As is known \((^1)\), \(D_A\) and \(\mathfrak N_z\) are linearly independent, and \(D_A\), \(\mathfrak N_z\), and \(\mathfrak N_{\bar z}\) are linearly independent if and only if \(\overline{D}_A = H\). Put \(\mathfrak P_z = \mathfrak N_z \cap (D_A + \mathfrak N_{\bar z})\). \(\mathfrak P_z = \{0\}\) if and only if \(\overline{D}_A = H\). In the Cartesian product \(\mathfrak N_z \times \mathfrak N_{\bar z}\) we distinguish the linear manifold

\[ \mathfrak G_z = \{[\psi,\varphi] : \varphi - \psi \in D_A\}. \]

\(\mathfrak G_z\) is the graph of an isometric operator \(X_z\), with domain \(\mathfrak P_z\) and range \(\mathfrak P_{\bar z}\) \(( (^1),\) Theorem 9). It is clear that \(X_z^{-1} = X_{\bar z}\). In a somewhat different way the operator \(X_z\) was defined in \((^2)\), where it plays an essential role in the construction of symmetric extensions of the operator \(A\).

In the present paper the operator \(X_z\) is used in describing dissipative and other extensions of the operator \(A\). A formula is established connecting \(X_z\) with the characteristic function of the operator \(A\). Using these terms, we describe the totality of all generalized resolvents of the operator \(A\).

2. A linear operator \(B\) in \(H\) is called dissipative if, for every \(f \in D_B\), \(\operatorname{Im}(Bf,f) \geq 0\) (cf. \((^3)\)). Such an operator \(B\) is called maximal dissipative if it has no proper dissipative extensions in \(H\). We shall call a linear operator \(B\) in \(H\) accumulative if \(\operatorname{Im}(Bf,f) \leq 0\) \((f \in D_B)\); analogously to the preceding, we define the notion of a maximal accumulative operator. The domain \(D_B\) of any closed maximal dissipative or accumulative operator \(B\) is dense in \(H\) \((^3)\).

Denote by \(\mathfrak K_z\) \((\operatorname{Im} z \ne 0)\) the class of all linear nonexpanding operators \(F\) from \(\mathfrak N_z\) into \(\mathfrak N_{\bar z}\) \((D_F \subset \mathfrak N_z)\), and by \(\mathfrak F_z\) the totality of all \(F \in \mathfrak K_z\) for which \(D_F = \mathfrak N_z\). We shall call a linear operator \(F\) from \(\mathfrak N_z\) into \(\mathfrak N_{\bar z}\) admissible if \(F\psi = X_z\psi\) only for \(\psi = 0\). If the operator \(F \in \mathfrak F_z\) is admissible, then the adjoint operator \(F^* \in \mathfrak F_{\bar z}\) is also admissible.

Theorem 1. For any nonreal \(z\) in the lower (upper) half-plane, the formulas

\[ D_B = D_A + (F - E)D_F, \tag{1} \]

\[ B(f + F\psi - \psi) = Af + zF\psi - \bar z \psi \quad (f \in D_A,\ \psi \in D_F) \tag{2} \]

establish a one-to-one correspondence between the totality of all admissible operators \(F \in \mathfrak K_z\) and the totality of all dissipative (accumulative) extensions \(B\) of the operator \(A\). Moreover,

\[ F = (B - \bar z E)(B - zE)^{-1}\big|_{\mathfrak N_z \cap (B - zE)D_B}. \]

The operator \(B\) is closed and maximal if and only if it corre-

corresponds to the admissible operator \(F\in\mathfrak F_z\). In this case the operator \(B^*\), adjoint to \(B\), corresponds (with \(z\) replaced by \(\bar z\)) to the operator \(F^*\in\mathfrak F_{\bar z}\).

We shall say that the linear operators \(S\) and \(T\) in \(H\) are formally adjoint if, for all \(f\in D_S\) and \(g\in D_T\),
\[ (Sf,g)=(f,Tg). \]

Theorem 2. For every nonreal \(z\) there exists a one-to-one correspondence between the totality of all extensions \(B\) of the operator \(A\), formally adjoint to \(A\), and the totality of all linear manifolds \(\mathfrak L\subset \mathfrak N_z\times\mathfrak N_{\bar z}\) satisfying the condition \([\psi,X_z\psi]\in\mathfrak L\) only when \(\psi=0\). This correspondence is defined by the formulas
\[ D_B=D_A\dotplus\{\varphi-\psi:\ [\psi,\varphi]\in\mathfrak L\}, \]
\[ B(f+\varphi-\psi)=Af+z\varphi-\bar z\psi \qquad (f\in D_A,\ [\psi,\varphi]\in\mathfrak L), \]
which is equivalent to the equality
\[ \mathfrak L=\{[\psi,\varphi]\in\mathfrak N_z\times\mathfrak N_{\bar z}:\ \varphi-\psi\in D_B,\ B(\varphi-\psi)=z\varphi-\bar z\psi\}. \]

The operator \(B\) is closed if and only if the corresponding linear manifold \(\mathfrak L\) is closed.**

1. Denote by \(A_\lambda\) \((\operatorname{Im}\lambda\ne0)\) the extension of the operator \(A\), defined on the linear manifold \(D_{A_\lambda}=D_A\dotplus\mathfrak N_z\) by the formula
   \[ A_\lambda(f+\varphi)=Af+\lambda\varphi \qquad (f\in D_A,\ \varphi\in\mathfrak N_\lambda). \]

According to Theorem 1, \(A_\lambda\) for \(\operatorname{Im}\lambda>0\) \((\operatorname{Im}\lambda<0)\) is a closed maximal dissipative (accumulative) extension of the operator \(A\), and \(A_\lambda^*=A_{\bar\lambda}\). Fix an arbitrary nonreal \(\lambda_0\) and denote by \(\Pi\) the open upper or lower half-plane containing \(\lambda_0\). By the same Theorem 1, for every \(\lambda\in\Pi\) the operator \(A_\lambda\) corresponds to the admissible operator \(C(\lambda)\in\mathfrak F_{\bar\lambda}\), defined by the formula
\[ C(\lambda)=(A_\lambda-\lambda_0E)(A_\lambda-\bar\lambda_0E)^{-1}\bigm|_{\mathfrak N_{\bar\lambda}}, \]
The operator function \(C(\lambda)\) \((\lambda\in\Pi)\) is called the characteristic function of the operator \(A\).*** It depends analytically on \(\lambda\), and
\[ \|C(\lambda)\|\le \left|\frac{\lambda-\lambda_0}{\lambda-\bar\lambda_0}\right| \qquad (\lambda\in\Pi). \]

Put
\[ \Pi_\varepsilon=\{\lambda\in\Pi:\ \varepsilon<|\arg\lambda|<\pi-\varepsilon\}\quad (0<\varepsilon<\pi/2). \]
Let \(\Phi(\lambda)\) \((\lambda\in\Pi)\) be an arbitrary analytic operator function whose values are linear nonexpanding operators mapping one Hilbert space \(\mathfrak N\) into another \(\mathfrak N'\). Put, then,
\[ \Omega_\Phi=\left\{h\in\mathfrak N:\ \lim_{\lambda\to\infty,\ \lambda\in\Pi_\varepsilon}|\lambda|(\|h\|-\|\Phi(\lambda)h\|)<\infty\right\}, \]
and denote by \(\Phi_0(\infty)\) the operator with domain of definition \(\Omega_\Phi\), given by the formula
\[ \Phi_0(\infty)h=\lim_{\lambda\to\infty,\ \lambda\in\Pi_\varepsilon}\Phi(\lambda)h \qquad (h\in\Omega_\Phi). \tag{3} \]
According to the results of the work \((^9)\), \(\Omega_\Phi\) is a linear manifold and for every \(h\in\Omega_\Phi\) the strong limit (3) exists.

* This theorem intersects with some results of the papers \((^{2-5})\) and, in particular, with Theorem 1.1.1 of \((^3)\).

** Theorem 2 is related to the results of the paper \((^6)\).

*** The concept of the characteristic function of a linear operator was first introduced by M. S. Livšic \((^7)\) for isometric and densely defined symmetric operators with defect index \((1,1)\) and their extensions. Subsequently this concept was generalized and modified in various ways. In the sense of the definition proposed in \((^8)\), the operator function \(C(\lambda)\) \((\lambda\in\Pi)\) considered here is the characteristic function of the operator \(A_{\lambda_0}\).

Theorem 3. The formulas
\[ \Phi_{\lambda_0}=\Omega_C,\qquad X_{\lambda_0}=C_0(\infty) \]
hold.

From this, in particular, it follows that

Theorem 4*. The domain of definition of the operator \(A\) is dense in \(H\) if and only if, for every nonzero \(\varphi\in\mathfrak N\),
\[ \lim_{\lambda\to\infty,\ \lambda\in\Pi_\varepsilon} [|\lambda|(\|\varphi\|-\|C(\lambda)\varphi\|)]=\infty. \]

1. We shall agree to denote by \(A_F\) the operator \(B\supset A\) corresponding to the admissible operator \(F'\in\mathfrak F_{\lambda_0}\) according to formulas (1), (2) for \(z=\lambda_0\). In an analogous sense the notation \(A_{F^*}\) will be used, so that \((A_F)^*=A_{F^*}\).

Recall that the operator-valued function \(R_\lambda\) (\(\operatorname{Im}\lambda\ne0\)) is a generalized resolvent of the operator \(A\) if and only if it can be represented in the form \(R_\lambda=P(\widetilde A-\lambda E)^{-1}|H\), where \(\widetilde A\) is some self-adjoint extension of the operator \(A\) with exit into a Hilbert space \(\widetilde H\supset H\), and \(P\) is the orthoprojector in \(\widetilde H\) onto \(H\).**

Theorem 5. The formula
\[ R_\lambda= \begin{cases} (A_{F(\lambda)}-\lambda E)^{-1} & (\lambda\in\Pi),\\ (A_{F^*(\bar\lambda)}-\lambda E)^{-1} & (\bar\lambda\in\Pi) \end{cases} \tag{4} \]
establishes a one-to-one correspondence between the set of all generalized resolvents \(R_\lambda\) (\(\operatorname{Im}\lambda\ne0\)) of the operator \(A\) and the set of all analytic operator-valued functions \(F(\lambda)\) (\(\lambda\in\Pi\)) with values in \(\mathfrak F_{\lambda_0}\), satisfying the condition
\[ F_0(\infty)\psi=X_{\lambda_0}\psi \quad \text{only for } \psi=0. \tag{5} \]
This condition is equivalent to the following:
\[ \lim_{\lambda\to\infty,\ \lambda\in\Pi_\varepsilon} \left|\frac1\lambda\bigl([E-C(\lambda)F(\lambda)]^{-1}\psi,\psi\bigr)\right|=0 \quad \text{for every } \psi\in\mathfrak N_{\lambda_0}.*** \tag{6} \]

Theorem 6. If \(F(\lambda)\) (\(\lambda\in\Pi\)) is an analytic operator-valued function with values in \(\mathfrak F_{\lambda_0}\) satisfying condition (5) or (6), then
\[ \lim_{\lambda\to\infty,\ \lambda\in\Pi_\varepsilon} \left\{\frac1\lambda [E-C(\lambda)F(\lambda)]^{-1}\right\}=0 \]
in the sense of strong convergence.

Ulyanovsk State Pedagogical Institute
named after I. N. Ulyanov

Received
5 IV 1967

CITED LITERATURE

\(^{1}\) M. A. Naimark, Izv. AN SSSR, Ser. Mat., 4, No. 1, 53 (1940).
\(^{2}\) M. A. Krasnosel’skii, DAN, 59, No. 1, 13 (1948).
\(^{3}\) R. S. Phillips, Sborn. per. Matematika, 6, 4, 11 (1962).
\(^{4}\) A. V. Shtraus, Izv. AN SSSR, Ser. Mat., 18, No. 1, 51 (1954).
\(^{5}\) B. I. Lomkarev, Tr. II nauch. konf. matem. kafedr pedagog. inst. Povolzh’ya, vol. 1, Kuibyshev, 1962, p. 66.
\(^{6}\) A. I. Plesner, DAN, 66, No. 4, 557 (1949).
\(^{7}\) M. S. Livshits, Matem. sborn., 19 (61), 2, 239 (1946).
\(^{8}\) A. V. Shtraus, Izv. AN SSSR, Ser. Mat., 24, No. 1, 43 (1960).
\(^{9}\) A. V. Shtraus, Izv. AN SSSR, Ser. Mat., 30, No. 6, 1325 (1966).
\(^{10}\) A. V. Shtraus, DAN, 67, No. 4, 611 (1949).
\(^{11}\) M. A. Naimark, Izv. AN SSSR, Ser. Mat., 4, No. 3, 277 (1940).
\(^{12}\) M. A. Naimark, Izv. AN SSSR, Ser. Mat., 7, No. 6, 285 (1943).

* Cf. \((^{10})\), Theorem 5. In the case of an operator with defect index \((1,1)\), the theorem 4 obtained here is close to one result of M. S. Livshits \((^{7})\), Theorem 15).

** The notions of the spectral function and generalized resolvent of a densely defined symmetric operator were first defined by M. A. Naimark \((^{11,12})\). In the case of a non-densely defined closed symmetric operator these notions are extended in \((^{4})\).

*** If \(\overline{D_A}=H\), then conditions (5), (6) are trivial. For this case formula (4) was established in \((^{4})\). There the theorem on the properties characterizing generalized resolvents of a non-densely defined operator was also proved. On its basis, B. I. Lomkarev \((^{5})\) extended formula (4) to the case of an operator \(A\) with non-dense domain of definition; however, the additional condition imposed on \(F(\lambda)\) had a rather complicated form.