UDC 517.948:513.88

MATHEMATICS

I. S. IOKHVIDOV

UNITARY EXTENSIONS OF ISOMETRIC OPERATORS IN THE PONTRYAGIN SPACE \(\Pi_1\) AND EXTENSIONS IN THE CLASS \(\mathfrak P_1\) OF FINITE SEQUENCES OF THE CLASS \(\mathfrak P_{1;n}\)

(Presented by Academician P. S. Novikov on 30 V 1966)

1. A finite sequence of complex numbers \(\{c_p\}_0^{\,n-1}\) \((\bar c_0=c_0)\) belongs, by definition, to the class \(\mathfrak P_{\chi;n}\), if the Toeplitz form

\[ \sum_{p,q=0}^{n-1} c_{p-q}\xi_p\bar \xi_q \quad (c_{-p}=\bar c_p;\ p=0,1,\ldots,n-1) \tag{1} \]

has exactly \(\chi\) \((\ge 0)\) positive squares. If
\[ \Delta_{n-1}=\det\|c_{p-q}\|_{p,q=0}^{\,n-1}\ne 0 \]
(a nondegenerate sequence), the sequence \(\{c_p\}_0^{\,n-1}\) can be extended in infinitely many ways to a sequence \(\{c_p\}_0^\infty\) of the class \(\mathfrak P_\chi\), i.e. such that all forms (1), for arbitrarily large \(n\), preserve exactly \(\chi\) positive squares \(((^1),\) Ch. V).

In the case where \(\chi=1\), the sequences \(\{c_p\}_0^\infty\in\mathfrak P_1\) may be of three types, according as \(\{c_p\}_0^\infty\) is bounded (elliptic type), and, in the case of unboundedness, according as the finite or infinite (necessarily existing) limit
\[ \lim_{p\to\infty} (|c_p|/p^2) \]
exists (parabolic and hyperbolic types, respectively) \((^1,^2)\). In this connection there naturally arises the question of the existence, for a finite sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak P_{1;n}\) \((\Delta_{n-1}\ne0)\), of infinite extensions of one or another type in the class \(\mathfrak P_1\). Some of the very first results in this direction for real extensions of real sequences \(\{c_p\}_0^{\,n-1}\in\mathfrak P_{1;n}\) were contained in \((^{1-3})\), and for complex sequences \(\{c_p\}_0^{\,1}\in\mathfrak P_{1;2}\)—in \((^2)\).

In the present note the problem under consideration is connected with the more general problem of extending isometric operators in the Pontryagin space \(\Pi_\chi\) (and, in particular, \(\Pi_1\)) to unitary operators in the same or in a wider space of the same type*.

2. We begin by formulating (with a certain refinement) a theorem whose second part, in essence, was already contained in the reasoning of § 9 (Ch. III) of \((^1)\).

Theorem 1. In order that an isometric operator \(V\) in the space \(\Pi_\chi\) admit unitary extensions in this same space \(\Pi_\chi\) or in some wider space \(\widetilde{\Pi}_\chi\supset \Pi_\chi\), it is necessary and sufficient that \(V\) be bounded and continuously invertible. Unitar—

\[ \rule{0.25\linewidth}{0.4pt} \]

* For definitions of all notions occurring here and below, connected with the geometry of the spaces \(\Pi_\chi\) and operators acting in these spaces, see, for example, \((^1)\).

** The embedding \(\Pi_\chi\to\widetilde{\Pi}_\chi\) is isometric. The necessity of the requirement of boundedness and continuous invertibility of the operator \(V\) for the existence of its unitary extensions was first clarified in \((^4)\).

extensions without exit from \(\Pi_\varkappa\) are possible if and only if the defect numbers of the operator \(V\) coincide.

Let now \(\varkappa=1\). According to a well-known theorem \((^{1})\), a unitary operator \(U\) in the space \(\Pi_1\) always has a nonnegative eigenvector: \(U f_0=\lambda_0 f_0\), \(f_0\ne0\), \((f_0,f_0)\ge0\), where \((f,g)\) is the indefinite inner product specified in \(\Pi_1\). If the operator \(U\) has a positive eigenvector \(f_0\), then necessarily \(|\lambda_0|=1\) for the corresponding eigenvalue \(\lambda_0\). Such an operator \(U\) will be called elliptic. If \(U\) is not elliptic, but there exists an eigenvector \(f_0: U f_0=\lambda_0 f_0\) with \((f_0,f_0)=0\) and \(|\lambda_0|=1\), then \(U\) is called parabolic. Finally, if \(U f_0=\lambda_0 f_0\) and \(|\lambda_0|\ne1\), then necessarily \((f_0,f_0)=0\), and there exists a neutral eigenvector \(f_0'\): \((f_0',f_0')=0\), \((f_0,f_0')=1\), \(U f_0'=\bar{\lambda}_0^{-1}f_0'\), while the operator \(U\) is called hyperbolic. The three cases listed exclude one another and exhaust all possibilities \((^{1})\).

The uniquely determined eigenvalues \(\lambda_0\) appearing in the definitions given above will be called critical (cf. \((^{5})\)), and the pair \(\{\lambda_0,\bar{\lambda}_0^{-1}\}\) (in the hyperbolic case) will be called the critical pair for the operator \(U\).

1. For an arbitrary complex \(\lambda\) and any linear operator \(T\) acting in the space \(\Pi_1\) with domain of definition \(\mathfrak D_T\), define the subspace
   \[ \mathfrak M_\lambda(T)=\{Tf-\lambda f\}_{f\in\mathfrak D_T}. \]

In Theorems 2 and 3 below, where unitary extensions of an isometric operator \(V(\mathfrak D_V\subset\Pi_1)\) are discussed, extensions without exit from \(\Pi_1\) are meant if the defect numbers of the operator \(V\) are equal, and extensions with exit into \(\widetilde\Pi_1(\supset\Pi_1)\) otherwise.

Theorem 2. In order that a bounded and continuously invertible isometric operator \(V(\mathfrak D_V\subset\Pi_1)\) admit an elliptic unitary extension with critical number \(\varepsilon\) \((|\varepsilon|=1)\), it is necessary and sufficient that the subspace \(\mathfrak M_\varepsilon(V)\) be negative or equal to \(\{0\}\).

Theorem 3. In order that a bounded and continuously invertible operator \(V(\mathfrak D_V\subset\Pi_1)\) admit a hyperbolic unitary extension \(U\) with critical pair \(\{\lambda_0,\bar{\lambda}_0^{-1}\}\) \((|\lambda_0|\ne1)\), it is necessary and sufficient that there exist a pair of neutral vectors \(f_0,f_0'\in\Pi_1\): \((f_0,f_0)=(f_1,f_1)=0\), \((f_0,f_0')=1\), such that
\[ f_0\perp \mathfrak M_{\bar{\lambda}_0^{-1}}(V),\qquad f_0'\perp \mathfrak M_{\lambda_0}(V). \]
If, however, the operator \(V\) has no eigenvectors, then a sufficient condition is the degeneracy of the metric on the subspace \(\mathfrak M_{\lambda_0}(V)\).

The proofs of Theorems 1–3 (in their “sufficiency” part) contain a description of the procedure for obtaining all extensions of the corresponding type. We do not give here (because of its bulkiness) the criterion we also obtained for the existence of a parabolic extension with a prescribed critical number \(\varepsilon\) \((|\varepsilon|=1)\).

1. The criteria formulated in Theorems 1–3 (essentially purely geometric ones) make it possible to obtain answers to the questions posed in item 1. Indeed, as was shown in Ch. V \((^{4})\), with every infinite sequence \(\{c_p\}_0^\infty\in\mathfrak P_\varkappa\) there are associated a certain space \(\Pi_\varkappa\) and a unitary operator \(U\) \((U\Pi_\varkappa=\Pi_\varkappa)\) such that, for some vector \(e_0\in\Pi_\varkappa\), we have \((U^p e_0,e_0)=c_p\) \((p=0,1,2,\ldots)\). In a similar way, with every nondegenerate finite sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak P_{\varkappa;n}\) there is associated a finite-dimensional space \(\Pi_\varkappa^{(n)}\) \((\dim\Pi_\varkappa^{(n)}=n)\) and in it an isometric “shift” operator \(V\) with defect index \((1,1)\), possessing the property that, for some \(e_0\in\Pi_\varkappa^{(n)}\), we have \(c_p=(V^p e_0,e_0)\) \((p=0,1,\ldots,n-2)\). Thus, the problems of existence and classification of extensions of nondegenerate sequences \(\{c_p\}_0^{\,n-1}\in\mathfrak P_{1;n-1}\) reduce to the corresponding problems of extension of isometric-

…operators to unitary ones (without leaving or with leaving \(\Pi_1^{(n)}\)). In the case of extensions without leaving \(\Pi_1^{(n)}\) we shall obtain sequences \(\{c_p\}_0^\infty \in \mathfrak{P}_1\) of finite rank \(n\) \((^1)\), i.e., for them

\[ \det\|c_{p-q}\|_{p,q=0}^{m}=0\quad (m\geqslant n). \]

The critical numbers obtained with the aid of the above-mentioned procedure of unitary extensions then occur in the integral representations of the extended sequences \(\{c_p\}_0^\infty\) (see \((^1)\), Chap. V); in view of this, these numbers may also be called the critical numbers of the sequences \(\{c_p\}_0^\infty\).

Theorem 4. A nondegenerate sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_{1;n}\) admits extensions of elliptic type \(\{c_p\}_0^\infty\in\mathfrak{P}_1\) with critical number \(\varepsilon\) \((|\varepsilon|=1)\) if and only if the sequence \(d_p=L_\varepsilon[c_p]\) \((p=0,1,\ldots,n-2)\) is (strictly) positive \((^6)\); moreover exactly one of these extensions will have rank \(n\).

Let us explain that here by \(L_\lambda\), where \(\lambda(\ne0)\) is an arbitrary complex number, we denote a finite-difference (so-called defining) operator (cf. \((^1)\)), acting by the formula

\[ L_\lambda[c_p]=\bar{\lambda}c_{p+1}-(1+|\lambda|^2)c_p+\lambda c_{p-1} \quad (p=0,1,\ldots,n-2). \]

As examples show, there exist finite nondegenerate sequences of the class \(\mathfrak{P}_{1;n}\) which admit no extensions of elliptic type at all; this is easily checked by means of Theorem 4. For example, for \(n=3\) * such a sequence is \(\{0,1,4\}\in\mathfrak{P}_{1;3}\).

Theorem 5. In order that a nondegenerate sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_{1;n}\) admit an extension \(\{c_p\}_0^\infty\in\mathfrak{P}_1\) of parabolic type with critical number \(\varepsilon\) \((|\varepsilon|=1)\), it is necessary that the sequence \(\{L_\varepsilon[c_p]\}_0^{\,n-2}\) be nonnegative \((^6)\). The degeneracy of this sequence is necessary and sufficient for the existence of the corresponding parabolic extension of rank \(n\), which (for fixed \(\varepsilon\)) is unique.

Application of the last criterion leads to the following result:

Theorem 6. A nondegenerate sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_{1;n}\) admits no more than \(2(n-1)\) distinct parabolic extensions \(\{c_p\}_0^\infty\in\mathfrak{P}_1\) of rank \(n\); the critical numbers \(\varepsilon\) of these extensions are found from the equation

\[ \det\|L_\varepsilon[c_{p-q}]\|_{p,q=0}^{\,n-2}=0. \]

For \(n=2\) there always exist exactly two distinct parabolic extensions of rank 2. Examples show that for \(n>2\) a sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_1\) may have no parabolic (and at the same time no elliptic—cf. below with Theorem 8) extensions at all. Such, for example, is the sequence \(\{0,1,5\}\in\mathfrak{P}_{1;3}\). There exist examples of sequences \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_{1;n}\) admitting parabolic extensions of rank \(n\) in number \(k\), where \(k\) is any integer, \(0\leqslant k\leqslant 2(n-1)\).

Theorem 7. Every nondegenerate sequence \(\{c_p\}_0^{\,n-1}\in\mathfrak{P}_{1;n}\) admits hyperbolic extensions \(\{c_p\}_0^\infty\in\mathfrak{P}_1\), including a continual family (depending on one real parameter) of extensions of rank \(n\), whose critical numbers (i.e., critical pairs \(\{\lambda,\bar{\lambda}^{-1}\}\), \(0<|\lambda|\ne1\)) are determined from the equation

\[ \det\|L_\lambda[c_{p-q}]\|_{p,q=0}^{\,n-2}=0. \]

Our proof of Theorem 7 is based, in particular, on a recently discovered rule for computing the signature of an arbitrary Hermitian Toeplitz form \((^7)\). Closely adjacent to Theorem 7 is the simpler

* We note that for \(n=2\) a nondegenerate sequence \(\{c_p\}_0^1\) of class \(\mathfrak{P}_{1;2}\) always admits extensions \(\{c_p\}_0^\infty\in\mathfrak{P}_1\) of rank 2 of all three types (\((^2)\), Theorem 4). This fact is now obtained as a very special consequence of the theorems of the present work.

Theorem 7. In order that a nondegenerate sequence \(\{c_p\}_{0}^{\,n-1}\in \mathfrak{P}_{1;n}\) admit extensions of hyperbolic type with critical pair \(\{\lambda,\bar{\lambda}^{-1}\}\) \((0<|\lambda|\ne 1)\), it is necessary that the sequence \(\{L_\lambda[c_p]\}_{0}^{\,n-2}\) be nonnegative. The degeneracy of this sequence is necessary and sufficient for the existence of the corresponding hyperbolic extension of rank \(n\), which (for fixed \(\lambda\)) is determined uniquely.

Theorem 8. If a nondegenerate sequence \(\{c_p\}_{0}^{\,n-1}\in \mathfrak{P}_{1;n}\) has an “elliptic extension,” then there are infinitely many extensions of this type, among which the extensions of rank \(n\) form a nonempty family depending on one parameter ranging over a certain open set on the real axis. In this case the sequence admits extensions (including those of rank \(n\)) of all three types.

In conclusion we note that Theorems 4–8 are essentially equivalent to the corresponding facts in the theory of extensions of isometric “shift” operators in finite-dimensional spaces \(\Pi_1\). We shall not dwell here on possible generalizations of the theory developed above to the case \(\chi>1\).

Odessa Civil Engineering Institute

Received
26 V 1966

CITED LITERATURE

\(^{1}\) I. S. Iokhvidov, M. G. Krein, Tr. Mosk. Matem. Obshch., 5, 169 (Chapters I–III) (1956); 8, 413 (Chapters IV–VI) (1959).
\(^{2}\) I. S. Iokhvidov, Dokl. AN ArmSSR, 42, No. 3 (1966).
\(^{3}\) I. S. Iokhvidov, M. G. Krein, Tr. Mosk. Matem. Obshch., 15 (1966).
\(^{4}\) I. S. Iokhvidov, UMN, 16, 4 (100), 175 (1961).
\(^{5}\) M. G. Krein, G. K. Langer, DAN, 152, 1, 39 (1963).
\(^{6}\) N. I. Akhiezer, M. G. Krein, On Some Questions in the Theory of Moments, Kharkov, 1938.
\(^{7}\) I. S. Iokhvidov, DAN, 169, No. 6, 1258 (1966).