UDC 517.432

MATHEMATICS

A. V. KUZHEL’

SPECTRAL ANALYSIS OF UNBOUNDED NON-SELF-ADJOINT OPERATORS IN A SPACE WITH INDEFINITE METRIC

(Presented by Academician L. S. Pontryagin on 10 III 1967)

In this communication, by the method of characteristic matrix functions, a broad class of unbounded non-self-adjoint operators acting in a space with indefinite metric is investigated. In the case of bounded non-self-adjoint or nonunitary operators, analogous results were obtained in the communications \((^{1-3})\).

1. Let \(H\) be a Hilbert space; \((\cdot,\cdot)\) the scalar product in \(H\); \(P\) a bounded self-adjoint continuously invertible operator acting in the space \(H\). We introduce a metric in \(H\) by putting \([f,g]=(Pf,g)\). The space \(H\) with scalar product \([\cdot,\cdot]\) is, generally speaking, a space with indefinite metric, which we shall denote by \(\Pi\), and we shall speak respectively of the \(H\)-metric (Hilbert metric) and the \(\Pi\)-metric (indefinite metric).

A subspace of the space \(\Pi\) is called, as usual, the linear envelope, closed in the \(H\)-metric, of some system of vectors from \(\Pi\). In what follows, an important role is played by projectively complete subspaces, which were introduced independently in the works of E. Shaybe \((^4)\) and I. S. Iokhvidov \((^5)\) and studied in detail by Yu. P. Ginzburg and I. S. Iokhvidov \((^6)\). For us it is important that if \(\Pi_1\) is a projectively complete subspace of the space \(\Pi\), then the subspace* \(\Pi_2=\Pi[-]\Pi_1\) is also projectively complete, and moreover \(\Pi=\Pi_1[+]\Pi_2\).

1. By analogy with the definite case \((^7)\), a linear closed operator \(A\) with dense domain \(D_A\) in \(\Pi\) will be called an \(N\)-operator. Denote by \(\mathscr G_A\) the set of vectors \(f\in D_A\) for which \([Af,g]=[f,Ag]\) for every \(g\in D_A\). The manifold \(\mathscr G_A\) is called the domain of Hermiticity of the operator \(A\) and is the broadest manifold on which \(A=A^0\) (\(A^0\) is the operator adjoint to \(A\) in the \(\Pi\)-metric).

Definition. An \(N\)-operator \(A\) is called a quasi-Hermitian operator of rank \(r\) (or a \(K^r\)-operator) if: 1) the set \(\rho(A,A^0)\) of common regular points of the operators \(A\) and \(A^0\) contains at least one nonreal point; 2) \(\dim D_A=r \pmod{\mathscr G_A}\).

In this case the \(K^r\)-operator \(A\) is called: 1) a \(K_I^r\)-operator if \(D_A=D_{A^0}\); 2) a \(K_{II}^r\)-operator if \(\mathscr G_A=\Pi\); 3) a \(K_{III}^r\)-operator if \(D_A\ne D_{A^0}\), \(\mathscr G_A\ne\Pi\). Thus the set of \(K^r\)-operators is divided into three disjoint classes of operators. We note that the bounded operators studied in \((^3)\) belong to the class of \(K_I^r\)-operators.

1. Let \(i\in\rho(A,A^0)\). Consider the auxiliary operator

\[ B=iR_{-i}-iR_{-i}^{0}-2R_{-i}^{0}R_{-i}\qquad (R_{-i}=(A+iI)^{-1}). \]

* Here and below, \([-]\) and \([+]\) denote, respectively, the signs of orthogonal complement and orthogonal sum in the \(\Pi\)-metric.

This operator maps the space \(\Pi\) into the defect subspace \(\mathfrak M_{-i}\) of the operator \(A\) (which is defined by the equality \(\mathfrak M_{-i}=\Pi[-]\mathfrak M_{-i}\), where \(\mathfrak M_{-i}=(A+iI)D_A\)) and can be represented in the form

\[ B=\sum_{\alpha,\beta=1}^{s}[\cdot,h_\alpha]J_{\alpha\beta}h_\beta \quad (s\ge r), \]

where \(J=\|J_{\alpha\beta}\|\) is a certain Hermitian and unitary matrix. In what follows, we shall call the collection of vectors \(\{h_k\}_{k=1}^{s}\) an \(\alpha\)-basis of the operator \(A\), and the matrix \(J\) the corresponding coefficient matrix.

Let \(\{h_k\}_{k=1}^{s}\) be an \(\alpha\)-basis of the operator \(A\), and let \(J\) be the corresponding coefficient matrix. Then the matrix

\[ \tau=I-2JG \quad (G=\|[h_k,h_i]\|) \tag{1} \]

is a nonsingular matrix and is called the distortion matrix of the operator \(A\). Moreover, the matrix \((J\tau)^{-1}\) can be represented in the form

\[ (J\tau)^{-1}=C^*J^{(0)}C, \tag{2} \]

where \(J^{(0)}\) is a certain unitary and Hermitian matrix, and \(C\) is a nonsingular matrix depending on the choice of the matrix \(J^{(0)}\).

We shall now agree to call the matrix \(\chi_A(\lambda)\), defined by the relation

\[ \chi_A(\lambda)J^{(0)}\chi_A^*(i) = J+i(\lambda+i)\|[(A^0-iI)(A^0-\lambda I)^{-1}h_h,h_i]\|, \tag{3} \]

the characteristic matrix-function (c.m.-f.) of the \(K^r\)-operator \(A\). Setting \(\lambda=i\) in (3) and using relations (1) and (2), we see that \(\chi_A(i)\) is a certain nonsingular solution of the matrix equation \(XJ^{(0)}X^*=J\tau\), and, consequently, the definition of the c.m.-f. is not contradictory.

1. We shall call a subspace \(\Pi_1\subset\Pi\) invariant with respect to the operator \(A\) if \(A(D_A\cap\Pi_1)\subset\Pi_1\) and \(\overline{D_A\cap\Pi_1}=\Pi_1\). It is easy to verify that \(A^0(D_{A^0}\cap\Pi_2)\subset\Pi_2\), where \(\Pi_2=\Pi[-]\Pi_1\). However, it may happen that \(\overline{D_{A^0}\cap\Pi_2}\ne\Pi_2\). In that case, on the basis of the preceding, the subspace \(\Pi_2\) is not invariant with respect to the operator \(A^0\).

Definition. An operator \(A\), acting in the space \(\Pi\), is called a coupling of the operators \(A_1\) and \(A_2\), acting respectively in the subspaces \(\Pi_1\) and \(\Pi_2=\Pi[-]\Pi_1\), if \(\Pi_1\) and \(\Pi_2\) are projection-complete invariant subspaces respectively with respect to the operators \(A\) and \(A^0\), and, moreover,

\[ A_1=A|_{D_A\cap\Pi_1}, \qquad A_2=\bigl[A^0|_{D_{A^0}\cap\Pi_2}\bigr]^0, \]

Theorem 1 (multiplication theorem). Let the \(K^r\)-operator \(A\) be a coupling of a \(K^{r_1}\)-operator \(A_1\) and a \(K^{r_2}\)-operator \(A_2\) \((r_1,r_2\le r)\), and let \(\bar\lambda\) be a common regular point of these operators. Then, in certain \(\alpha\)-bases, the c.m.-f. of the operators \(A,A_1\), and \(A_2\) are connected with one another by the relation

\[ \chi_A(\lambda)=\chi_{A_1}(\lambda)\chi_{A_2}(\lambda). \]

1. We shall agree to call the largest invariant subspace \(\Pi_A\) of the operator \(A\) the additional subspace of this operator, if \(D_A\cap\Pi_A\subset\mathscr G_A\) and in \(\Pi_A\) the operator \(A\) induces a self-adjoint operator. The operator \(A_p\), which coincides with \(A\) on the manifold \(D_A\cap(\Pi[-]\Pi_A)\), is called the simple part of the operator \(A\). In the case when \(A=A_p\), the operator \(A\) is called simple.

Theorem 2 (isomorphism criterion). Let the c.m.-f. of the \(K^r\)-operators \(A\) and \(A'\) coincide. Then for \(J=J'\) the simple parts of these operators are isomorphic, and for \(J=-J'\) they are co-isomorphic \((^3)\).

The preceding results make it possible to establish the following assertions:

Theorem 3. Let \(\lambda_0\) be a nonreal \((+)\)-proper \((^3)\) or \((-)\)-proper value of the \(K^r\)-operator \(A\). If, moreover, \(\overline{\lambda}_0\) is a regular point of the operator \(A\), then \(\det \chi_A(\lambda_0)=0\).

Theorem 4. If \(\det \chi_A(\lambda_0)=0\), then \(\lambda_0\) is a point of the spectrum of the operator \(A\).

Theorem 5. If the characteristic matrix function of a \(K^r\)-operator \(A\), which acts in a nondegenerate space, is a constant matrix, then the operator \(A\) has a nontrivial invariant subspace that coincides with its isotropic part.

1. An operator \(A\) acting in the space \(\Pi\) will be called dissipative if for every \(f\in D_A\), \(\operatorname{Im}[Af,f]\geq 0\). It is easy to verify that the nonreal \((+)\)-proper \(((-)\)-proper) values of a dissipative operator \(A\) lie in the half-plane \(\operatorname{Im}\lambda>0\) \((\operatorname{Im}\lambda<0)\). The following assertions also hold:

I. The eigenvectors of a dissipative \(K^r\)-operator \(A\) corresponding to real or \((0)\)-proper values belong to the domain of Hermiticity \(\mathfrak G_A\) of the operator \(A\).

II. If a simple dissipative \(K^r\)-operator \(A\) acts in a nondegenerate space, then it has no real or \((0)\)-proper values.

1. Let now the \(K^r\)-operator \(A\) act in the space \(\Pi_\chi\) (a Pontryagin space \((^{8,9})\)). Relying on the theorem of M. G. Krein \((^{10})\) on expanding operators, we establish the following proposition:

Theorem 6. A simple dissipative \(K^r\)-operator \(A\), acting in the space \(\Pi_\chi\), has a \(\chi\)-dimensional negative invariant subspace.

We note that this assertion makes it possible to carry out the factorization of the characteristic matrix function and to construct a triangular model of a simple dissipative \(K^r\)-operator (by analogy with the way this was done in \((^2)\) in the case of nonunitary operators).

In conclusion we note the following assertion (which can also be obtained without constructing the triangular model, on the basis of the preceding results and the results of \((^2)\)):

Theorem 7. Let \(A\) be a simple dissipative \(K^r\)-operator acting in the space \(\Pi_\chi\). Then

\[ \det \chi_A(i)\leq \prod_{k=1}^{\chi}\left|\frac{\mu_k-i}{\mu_k+i}\right| \prod_{k=1}^{N}\left|\frac{\lambda_k-i}{\lambda_k+i}\right| \quad (N\leq \infty), \tag{4} \]

where \(\{\mu_k\}_{k=1}^{\chi}\) and \(\{\lambda_k\}_{k=1}^{N}\) are the totalities, respectively, of the \((-)\)-proper and \((+)\)-proper values of the operator \(A\). Moreover, the system of finite-dimensional invariant subspaces of the operator \(A\) corresponding to the eigenvalues of this operator is complete in the space \(\Pi_\chi\) if and only if equality holds in relation (4).

Uman Pedagogical Institute

Received
18 I 1966

CITED LITERATURE

\(^1\) A. V. Kuzhel, Dokl. AN USSR, No. 9, 1135 (1962).
\(^2\) A. V. Kuzhel, Dokl. AN USSR, No. 4, 430 (1963).
\(^3\) A. V. Kuzhel, DAN, 151, No. 4 (1963).
\(^4\) E. Scheibe, Ann. Acad. Sci. Fenn., A I, 294 (1960).
\(^5\) I. S. Iokhvidov, DAN, 139, No. 4 (1961).
\(^6\) Yu. P. Ginzburg, I. S. Iokhvidov, UMN, 17, issue 4 (106) (1962).
\(^7\) A. M. Naimark, Izv. AN SSSR, ser. matem., 4, No. 1 (1940).
\(^8\) L. S. Pontryagin, Izv. AN SSSR, ser. matem., 8, 243 (1944).
\(^9\) I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956).
\(^ {10}\) M. G. Krein, UMN, 5, issue 2 (36) (1950).