UDC 513.88 : 517.948

MATHEMATICS

N. K. NIKOL’SKII, B. S. PAVLOV

BASES OF EIGENVECTORS OF COMPLETELY NONUNITARY CONTRACTIONS

(Presented by Academician V. I. Smirnov on 16 V 1968)

In the present note we study systems of eigenvectors of completely nonunitary contractions (see also \((^1)\)).

0.1. Let \(E\) be a Hilbert space, \(R\) the ring of bounded operators in \(E\); \(H^2(E)\) the space of all functions, regular in the unit disk
\(D=\{z:\ |z|<1\}\), with values in \(E\) and with norm

\[ \|f\|_2^2 \stackrel{\mathrm{def}}{=} \sup_{0<r<1} \frac{1}{2\pi}\int_0^{2\pi} \|f(re^{it})\|_E^2\,dt < \infty . \]

Further, let \(S\) be the shift operator in \(H^2(E)\)

\[ (Sf)(z)=zf(z), \qquad |z|<1,\qquad f\in H^2(E); \]

\(S^*\) the adjoint operator, and \(T\) the restriction of \(S^*\) to some \(S^*\)-invariant subspace of the form
\(K=H^2(E)\ominus \Theta H^2(E)\). Here \(\Theta\) is a regular bounded function in \(D\) with values in \(R\), unitary almost everywhere on the unit circle (an inner \((^{2,3})\) operator-function).

It was shown in \((^2)\) that every linear operator \(A\) acting in a Hilbert space and satisfying the conditions \(\|A\|\leq 1\),
\(\operatorname{s-lim}_{n\to\infty} A^n = \operatorname{s-lim}_{n\to\infty} A^{*n}=0\), is unitarily equivalent to some operator of the form \(T\). The corresponding function \(\Theta\), up to inessential factors, coincides with the characteristic function of the operator \(A^*\) (see \((^{2,4})\)).

0.2. We shall investigate the system of eigenfunctions (e.f.) of the operator \(T\) under the same assumptions as in the note \((^1)\): the spectrum of the operator \(T\) in the disk \(D\) consists of isolated points \(\bar z_k\), \(k=1,2,\ldots\), which are simple poles of the resolvent, and the system of e.f. of \(T\) is complete in \(K\).

As in \((^1)\), let \(\pi_k\) (respectively \(\Delta_k\)) be the projector onto the kernel of the operator \(\Theta(z_k)\) (respectively \(\Theta^*(z_k)\)), \(|z_k|<1\). Under the assumptions made (see \((^{5,6})\)) \(\Theta\) factors in the form \(\Theta=\Theta_k B_k\), where both factors are inner and

\[ B_k(z)=\frac{z_k-z}{1-\bar z_k z}\,\frac{\bar z_k}{|z_k|}\,\pi_k+(1-\pi_k),\qquad |z|<1, \]

\(\bar z_k\) is an eigenvalue of \(T\), and \(\Theta_k(z_k)\) is an isomorphism in \(E\). Hence it follows, in particular, that
\(\dim \operatorname{Ker}\Theta(z_k)=\dim \operatorname{Ker}\Theta^*(z_k)\).

Let \(e_k^i\) be any orthonormal basis in \(\pi_k E\). The system of (all) e.f. of the operator \(T\) (respectively \(T^*\)) corresponding to the eigenvalue \(\bar z_k\) (respectively \(z_k\)) is formed by the functions

\[ \varphi_k^i=(1-\bar z_k z)^{-1}\Theta_k^{*-1}(z_k)e_k^i, \]

\[ \psi_k^i=(1-\bar z_k z)^{-1}\Theta_k(z)e_k^i. \tag{1} \]

The correspondingly normalized systems \(\{\varphi_k^i\}\), \(\{\psi_k^i\}\) are biorthogonal in \(H^2(E)\) (see \((^1)\)). In what follows we shall assume that the system

\(\{\varphi_k^i\}\) is uniformly minimal, i.e.

\[ \sup_{k,i}\left\|\Theta_k^{*-1}(z_k)e_k^i\right\|<\infty . \tag{2} \]

Let us note that condition (2) entails the \(\omega\)-linear independence \({}^{(6)}\) of the system \(\{\varphi_k^i\}\).

1.1. Theorem. Under assumptions 0.2 the following conditions are equivalent:

1. The system (1) of root vectors of the operator \(T\) forms a Bari basis.

2. The infinite product converges (cf. \({}^{(8)}\))

\[ \prod_{k,i}\left\|\Theta_k^{*-1}(z_k)e_k^i\right\|. \]

3.

\[ \sum_k \left\|D_{\Theta_k^*}\Delta_k\right\|_{\gamma_2}^{\,2}<\infty, \]

where

\[ D_{\Theta_k^*}=\left[I-\Theta_k(z_k)\Theta_k^*(z_k)\right]^{1/2}. \]

1. The Gram matrix

\[ \left\{\delta_{ki,lj}-\left(\varphi_k^i,\varphi_l^j\right)\|\varphi_k^i\|^{-1}\|\varphi_l^j\|^{-1}\right\} \]

generates in \(l^2\) a Hilbert–Schmidt operator.

1.2. Theorem. Suppose that assumptions 0.2 are satisfied and that the system of eigenspaces (e.s.) of the operator \(T\) is uniformly minimal, i.e. \((1)\)

\[ \sup_k\left\|\Delta_k\Theta_k^{*-1}(z_k)\pi_k\right\|<\infty . \]

The system of e.s. forms a Bari basis \({}^{(6)}\), if the condition

\[ \sum_{\substack{i,k\\ i\ne k}} (1-|z_i|^2)(1-|z_k|^2)\left|1-\bar z_i z_k\right|^{-2} \left\|\Delta_i\Delta_k\right\|_{R}^{2}<\infty \]

is satisfied.

From Theorems 1.1 and 1.2 one easily obtains the known spectral characterizations of Bari bases of root vectors of contraction operators \({}^{(7,8)}\).

2.1. Let the eigenvalues of the operator \(T\) have only one limit point \(z=1\), and suppose that for some \(a\), \(a<\pi\), the condition

\[ \left|\arg((I-T)f,f)\right|<a,\qquad f\in K. \tag{3} \]

is fulfilled. Further, let

\[ \delta_k=\inf_{i,\ i\ne k}\left\{|z_k-z_i|\left|1-\bar z_k z_i\right|^{-1}\right\}^{m_i}, \qquad m_k=\dim\operatorname{Ker}\Theta(z_k). \]

2.2. Theorem. Suppose that assumptions 0.2 and (3) are satisfied. If

\[ 0<\prod_k \delta_k^{m_k}<\infty, \tag{4} \]

then the system of root vectors of the operator \(T\) is a Bari basis. In the case \(\dim E=1\), condition (4) is also necessary.

The proof of this assertion is obtained from Theorem 1.1 and certain estimates \({}^{(9)}\) for Blaschke products with zeros lying in an angle.

3.1. We now consider an operator \(T\) whose sequence of eigenvalues decomposes into a certain number of Carleson subsequences. Recall \({}^{(6,8,10)}\) that a Carleson sequence \(\{\zeta_k\}_{k=1}^{\infty}\), \(|\zeta_k|<1\), is characterized by the property

\[ \delta=\inf_k\prod_{i\ne k}\left|\frac{\zeta_i-\zeta_k}{1-\bar\zeta_i\zeta_k}\right|>0. \tag{C} \]

In works \({}^{(1,8)}\) it was shown that the system of root vectors of the operator \(T\) forms a Riesz basis \({}^{(7)}\), if the sequence of its eigenvalues \(\{\bar z_k\}\) is Carleson.

Assume that the sequence \(\{\bar z_k\}_{k=1}^{\infty}\) of eigenvalues of the operator \(T\) is the union of a finite number of Carleson sequences \(\{z_n^{(i)}\}_{n=1}^{\infty}\), \(i=1,2,\ldots,m\). Then (see \((^1,^6,^{10})\)) the matrices

\[ \left\{ \frac{(1-|z_l^{(i)}|^2)^{1/2}(1-|z_n^{(i)}|^2)^{1/2}} {1-\bar z_l^{(i)}z_n^{(i)}} \right\}_{l,n=1}^{\infty}, \qquad i=1,2,\ldots,m, \]

generate positive definite operators in \(l^2\), whose lower and upper bounds (\(\gamma_i,\Gamma_i\), respectively) depend only on the constant \(\delta^{(i)}\) in condition (C).

Suppose, moreover, that there exist orthoprojectors \(P^1,P^2,\ldots,P^m\) in \(E\) such that

\[ P_{\operatorname{Ker}\Theta^*(z_k^{(i)})}\equiv \Delta_k^{(i)}\to P^i \]

in norm as \(k\to\infty\), \(i=1,2,\ldots,m\).

Then we shall say that the system of eigenfunctions of the operator \(T\) decomposes into \(m\) Carleson series \(\{K(P^i,\gamma^i,\Gamma^i)\}\), \(i=1,2,\ldots,m\).

3.2. Theorem. The system of eigenfunctions of the operator \(T\) forms a Riesz basis if conditions 0.2, 3.1 are satisfied and

\[ \min_i \gamma_i>\max_i \sum_{j,\;j\ne i} a^{ij}\Gamma_i^{1/2}\Gamma_j^{1/2}. \tag{5} \]

Here \(a^{ij}=\cos(P^iE\wedge P^jE)\), \(i,j=1,2,\ldots,m\).

3.3. Remark. If \(a^{ij}=0\), \(i\ne j\), then condition (5) is also necessary.

The following assertion, in a certain sense, is the converse of Theorem 3.2.

3.4. Theorem. Suppose that conditions 0.2 are satisfied, the system of eigenfunctions of the operator \(T\) forms a Riesz basis, \(\dim \Delta_kE=1\), \(k=1,2,\ldots\), and the projectors \(\Delta_k\), \(k\ge 1\), form a compact family. Then the set of eigenvalues \(\{\bar z_k\}\) is the union of a finite number of Carleson sequences.

3.5. Corollary. If \(\dim E<\infty\), and the system of eigenfunctions of the operator \(T\) forms a Riesz basis, then the sequence \(\{\bar z_k\}\) is the union of a finite number of Carleson sequences.

4.1. Applying Theorems 1.1 and 1.2 and some results of Remark \((^1)\) in the case \(\dim E=1\), we arrive at criteria for the basis property of certain known systems of functions. Consider, for example, the system of powers \(\varphi_k(x)=x^{\lambda_k}\), \(x\in(0,1)\), \(\operatorname{Re}\lambda_k>-1/2\), \(k=1,2,\ldots\), in the space \(L_2(0,1)\). In order that the system \(\varphi_k\) form a Riesz basis in the closure of its linear span, it is necessary and sufficient that, for the numbers \(\mu_k=i(\bar\lambda_k+1/2)\), the Carleson condition for the half-plane

\[ \inf_k \prod_{l,\;l\ne k} \left|\frac{\mu_k-\mu_l}{\mu_k-\bar\mu_l}\right|>0. \tag{6} \]

be satisfied. If \(\operatorname{Im}\lambda_k=0\), then (6) becomes the lacunarity condition obtained in \((^{11})\). If the \(\lambda_k\) lie in the angle \(|\arg(\lambda_k+1/2)|\le \alpha<\pi\), then (6) is equivalent (see \((^9)\)) to the condition

\[ \inf_{k,l,\;l\ne k} \left|\frac{\mu_k-\mu_l}{\mu_k-\bar\mu_l}\right|>0. \]

Necessary and sufficient conditions for the quadratic closeness of the basis \(\{\varphi_k\}\) to an orthonormal one follow at once from Theorems 1.1 (or 1.2) and 2.2. Everything said applies equally to the system \(\psi_k(x)=\exp i\mu_k x\), \(x\in(0,\infty)\), \(k=1,2,\ldots\), in the space \(L_2(0,\infty)\), and to the system \(\chi_k(z)=(z-\mu_k)^{-1}\), \(\operatorname{Im}z>0\), \(k=1,2,\ldots\), in the Hardy space \(H^2\) in the half-plane \((^{10})\).

* In other words, the defect operators \(D_T,D_{T^*}\) are finite-dimensional.

In conclusion, we note that analogues of the results of this note can also be obtained for dissipative operators, if the disk \(D\) is replaced by the upper half-plane.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Leningrad State University
named after A. A. Zhdanov

Received
27 IV 1968

REFERENCES

\(^{1}\) N. K. Nikol’skii, B. S. Pavlov, DAN, 184, No. 3 (1969).
\(^{2}\) B. Sz.-Nagy, C. Foias, Analyse Harmonique des Opérateurs de l’espace de Hilbert, Budapest, 1967.
\(^{3}\) H. Helson, Lectures on Invariant Subspaces, N. Y.—London, 1964.
\(^{4}\) M. S. Brodskii, M. S. Livshits, UMN, 13, No. 1, 3 (79) (1958).
\(^{5}\) Yu. P. Ginzburg, Izv. vyssh. uchebn. zaved., Matematika, No. 1 (1963).
\(^{6}\) N. K. Nikol’skii, B. S. Pavlov, Notes of seminars of the Leningrad Branch of the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR, 6 (1968).
\(^{7}\) I. Ts. Gokhberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, “Nauka,” 1965.
\(^{8}\) E. E. Kandel’son, On Conditions for the Basis Property of Systems of Root Vectors of Certain Classes of Operators, Dissertation, Kharkov State University, 1968.
\(^{9}\) S. A. Vinogradov, Interpolation Problems for Analytic Functions Continuous in the Closed Disk, and for Functions with a Sequence of Coefficients from \(l^p\), Dissertation, Leningrad State University, 1968.
\(^{10}\) K. Hoffman, Banach Spaces of Analytic Functions, IL, 1963.
\(^{11}\) V. I. Gurarii, V. I. Matsaev, Izv. AN SSSR, ser. matem., 30, No. 1 (1966).