Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 4

MATHEMATICS

A. V. KUZHEL

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

(Presented by Academician I. G. Petrovskii on February 14, 1963)

In this note a characteristic matrix-function is constructed and the spectrum of a broad class of bounded non-self-adjoint operators acting in a space with an indefinite metric \(\Pi\) is studied. In doing so we do not additionally assume that the space \(\Pi\) contains only finitely many negative (or positive) lineals. We note that in a Hilbert space results analogous to those given here were obtained in the works of M. S. Livshits \((^{1,2})\) and M. S. Brodskii \((^{2-4})\).

The method that we shall use (the method of characteristic matrix-functions) was first applied in a space with an indefinite metric in the author’s works \((^{5-7})\) in the study of nonunitary operators. In those works, however, it was assumed that the space \(\Pi\) satisfies the axioms of I. S. Iokhvidov and M. G. Krein \((^8)\) (i.e., is a space with an indefinite metric “of finite rank”). Here this is not assumed.

1. Let \(H\) be a Hilbert space, \((f,g)\) the scalar product in \(H\), and \(P\) a bounded invertible Hermitian operator acting in \(H\). We introduce a metric in the space \(H\) by setting \([f,g]=(Pf,g)\). The scalar product \([f,g]\) has the same properties as \((f,g)\), with one exception: the scalar square \([f,f]\) may be negative or equal to zero for \(f\ne 0\). In what follows, the space \(H\) with scalar product \([f,g]\) will be denoted by \(\Pi\), and we shall speak correspondingly of the \(H\)-metric and the \(\Pi\)-metric. Further, the notation \(\Pi_2=\Pi[-]\Pi_1\) means that the manifold \(\Pi_2\) is orthogonal in the \(\Pi\)-metric to the manifold \(\Pi_1\) \((\Pi_k\subset \Pi)\). The notation \(\Pi_1[+]\Pi_2\) means that the manifolds \(\Pi_1\) and \(\Pi_2\) are orthogonal in the \(\Pi\)-metric and linearly independent. In what follows we use the following assertion: if \(\Pi_1\) is a nondegenerate subspace of the space \(\Pi\) (i.e., from \(\varphi\in\Pi_1\) and \([\varphi,\Pi_1]=0\) it follows that \(\varphi=0\)), then \(\Pi_2=\Pi[-]\Pi_1\) is also a nondegenerate subspace, and moreover \(\Pi=\Pi_1[+]\Pi_2\).

2. Let \(A\) be a linear bounded operator acting in \(\Pi\) (boundedness, continuity, and closedness of an operator are understood in the \(H\)-metric). The operator adjoint to \(A\) in the \(\Pi\)-metric (in the \(H\)-metric) will be denoted by \(A^0\) \((A^*)\). Then \(A^0=P^{-1}A^*P\). In this case the properties of the adjunction signs \(0\) and \(*\) are analogous.

Let \(E_A=\Pi[-]G_A\), where \(G_A\) is the totality of vectors \(f\) in \(\Pi\) such that \(Af=A^0f\). The operator \(A\) is called a \(K^r\)-operator if \(\dim E_A=r\). In this case the operator

\[ \frac{A-A^0}{i} \]

can be represented in the form

\[ \frac{A-A^0}{i}\sum_{k,i=1}^{s}[\cdot,g_k]J_{ki}g_i, \]

where \(s \ge r\), and \(J=\|J_{ki}\|\) is a certain Hermitian and unitary matrix (the coefficient matrix). The collection of vectors \(\{g_k\}_{k=1}^s\) is called an \(\alpha\)-basis of the operator \(A\). The \(\alpha\)-basis of the operator \(A\) may be chosen so that its linear span coincides with \(E_A\) (as will also be assumed in what follows).

Definition. The matrix-function \(\chi_A(\lambda)\), defined by the relation

\[ \chi_A(\lambda)=I+i\left\|\left[(A^0-\lambda I)^{-1}g_k,g_i\right]\right\|J, \]

where \(\{g_k\}_1^s\) is an \(\alpha\)-basis of the operator \(A\), and \(J\) is the corresponding coefficient matrix, is called the characteristic matrix-function of the operator \(A\).

The characteristic matrix-function of the operator \(A\) is analytic in the set of regular points of the operator \(A^0\) and satisfies the relation \(\chi_A(\lambda)J\chi_A^*(\bar\lambda)=J\). Moreover \(\chi_{A^0}(\lambda)=\chi_A^{-1}(\lambda)\).

Let us agree to call an eigenvalue \(\lambda\) of the operator \(A\) a \((+)\)-eigenvalue if the corresponding eigenvector \(f\) is positive (i.e. \([f,f]>0\)). The notions of \((-)\)-eigenvalues and \((0)\)-eigenvalues are introduced analogously.

Theorem 1. Suppose that the nonreal spectrum \(K^r\) of the operator \(A\) consists only of eigenvalues of this operator.

Then every nonreal zero of the function \(\det \chi_A(\lambda)\) is a \((+)\)-eigenvalue or a \((-)\)-eigenvalue of the operator \(A\). Conversely, if \(\lambda_0\) is a nonreal \((+)\)-eigenvalue or \((-)\)-eigenvalue of the operator \(A\) and, moreover, \(\bar\lambda_0\) is a regular point of the operator \(A^0\), then \(\det \chi_A(\lambda_0)=0\).

We note that in the case when \(r=1\), the condition imposed on \(\bar\lambda_0\) may be omitted.

1. Let \(\Pi_p=\Pi[-]\Pi_A\), where \(\Pi_A\) is the largest invariant subspace of \(G_A\) in which the operator \(A\) induces a self-adjoint (in the \(\Pi\)-metric) operator. The subspace \(\Pi_p\) coincides with the closed linear span of \(\{A^n g_k\}\), where \(k=1,2,\ldots,s;\ n=0,1,2,\ldots\). The operator \(A_p=A|_{\Pi_p}\) is called the simple part of the operator \(A\). Further, let the operator \(V\) map \(\Pi_1\) onto \(\Pi_2\) and, for any \(f\) and \(g\) from \(\Pi_1\), satisfy the condition

\[ [Vf,Vg]_2=\theta [f,g]_1 \qquad (\theta=\pm 1). \]

Then for \(\theta=1\) the operator \(V\) is called isometric, and for \(\theta=-1\), coisometric. Operators \(A_1\) and \(A_2\), acting respectively in the spaces \(\Pi_1\) and \(\Pi_2\), are called isomorphic if there exists an isometric operator \(V\) mapping \(\Pi_1\) onto \(\Pi_2\) (or \(\Pi_2\) onto \(\Pi_1\)) such that \(VA_1=A_2V\) (or \(A_1V=VA_2\)). Coisomorphism of operators is defined analogously.

Theorem 2. Suppose \(\chi_{A_1}(\lambda)\equiv \chi_{A_2}(\lambda)\). Then, if \(J_2=J_1\), the simple parts of the operators \(A_1\) and \(A_2\) are isomorphic, while if \(J_2=-J_1\), they are coisomorphic.

We note that in a space with an indefinite metric, for the characteristic matrix-function of a non-self-adjoint operator the identity \(\chi_A(\lambda)\equiv I\) is possible.* It turns out that in this case the subspace \(\Pi_p\) (on which the simple part of the operator \(A\) is defined) coincides with its isotropic part and, consequently, is a nontrivial invariant subspace of the operator \(A\).

1. Consider in the subspace \(\Pi_1\) of the space \(\Pi\) the operator \(A_1=P_1A\), where \(P_1\) is the projection operator in \(\Pi\) onto \(\Pi_1\). The \(\alpha\)-basis of the operator \(A_1\) consists of the vectors \(g_{1k}=P_1g_k\) \((k=1,2,\ldots,s)\), where \(\{g_k\}_1^s\) is the \(\alpha\)-basis of the operator \(A\). In this case the coefficient matrices \(J_1\) and \(J\), corresponding to the indicated \(\alpha\)-bases, coincide. By analogy with how this was done in \((4)\), the characte-

* I. S. Iokhvidov drew my attention to this possibility.

the characteristic matrix-function
\(\chi_{A_1}(\lambda)=I+i\left\|\left[(A^0-\lambda I)^{-1}g_{1k},g_{1i}\right]\right\|J_1\)
of the operator \(A_1\) will be called the projection of the matrix-function \(\chi_A(\lambda)\) onto the subspace \(\Pi_1\).

Theorem 3 (multiplication theorem). Let \(\Pi_1\) be a nondegenerate subspace of the operator \(A\) and \(\Pi_2=\Pi\ominus\Pi_1\). Then an arbitrary characteristic matrix-function of the operator \(A\) is the product of its projections onto \(\Pi_1\) and \(\Pi_2\).

Using this theorem and the preceding results, we arrive at the relation

\[ \sum_{k=1}^{m}|\operatorname{Im}\lambda_k|\leqslant \|P\|\sum_{k=1}^{s}\|g_k\|^2, \tag{1} \]

where \(\{\lambda_k\}_{k=1}^{m}\) is the set of \((+)\)-eigenvalues and \((-)\)-eigenvalues of the operator \(A\), and \(\|P\|\) and \(\|g_k\|\) are the norms of the operator \(P\) and the vector \(g_k\) in the \(H\)-metric.

It follows from relation (1) that the limit points of the set of \((+)\)-eigenvalues and \((-)\)-eigenvalues of the operator \(A\) can lie only on the real axis.

1. Let us now consider the case when the coefficient matrix \(J\) is Hermitian positive (or Hermitian negative). Without loss of generality, we may assume that \(J=\theta I\), where \(\theta=\pm1\). In this case:

I. The \((+)\)-eigenvalues of the \(K^r\)-operator \(A\) lie in the domain \(\theta\operatorname{Im}\lambda\geqslant0\), and the \((-)\)-eigenvalues lie in the domain \(\theta\operatorname{Im}\lambda\leqslant0\).

II. The eigenvectors of the \(K^r\)-operator \(A\) corresponding to \((0)\)-eigenvalues or to real eigenvalues belong to the subspace \(G_A\).

III. A simple \(K^r\)-operator \(A\) has no \((0)\)-eigenvalues, nor any real eigenvalues.

1. The following necessary condition for completeness holds:

Theorem 4. Let the system of nonzero root vectors of the \(K^r\)-operator \(A\) be complete in the space \(\Pi\). Then

\[ 2\sum_{k=1}^{N}\operatorname{Im}\lambda_k = \sum_{k=1}^{s}[g_k,g_k]J_k \quad (N\leqslant\infty), \]

where \(\{\lambda_k\}_{k=1}^{N}\) is the set of all \((+)\)-eigenvalues and \((-)\)-eigenvalues of the operator \(A\), and \(\{g_k\}_{k=1}^{s}\) is an \(\alpha\)-basis of this operator.

1. In conclusion, we point out one problem important for the questions under consideration. There is reason to suppose that an arbitrary characteristic matrix-function of a \(K^r\)-operator \(A\) can be represented as the product of a \(J\)-nonexpanding and a \(J\)-noncompressing matrix-function. In other words, there is reason to suppose that, for the matrix-functions under consideration, there holds a matrix analogue of the well-known theorem of R. Nevanlinna stating that an arbitrary function of bounded type can be represented as the quotient of two functions bounded in some domain. The solution of this problem will make it possible to obtain a spectral decomposition (triangular model) of \(K^r\)-operators in a space with indefinite metric, and also to clarify the question of the behavior of the spectrum of an arbitrary bounded \(K^r\)-operator.

Uman Pedagogical Institute

Received
12 II 1963

CITED LITERATURE

1. M. S. Livshits, Matem. sborn., 34 (76), 1 (1954).
2. M. S. Brodskii, M. S. Livshits, UMN, 13, issue 1 (79) (1958).
3. M. S. Brodskii, DAN, 97, No. 5 (1954).
4. M. S. Brodskii, Matem. sborn., 39 (81), 2 (1956).
5. A. V. Kuzhel, Dokl. AN USSR, No. 8 (1961).
6. A. V. Kuzhel, Dokl. AN USSR, No. 5 (1962).
7. A. V. Kuzhel, Dokl. AN USSR, No. 9 (1962).
8. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956).