UDC 513.88 + 517.948.35 + 517.948.5

MATHEMATICS

N. K. NIKOLSKII

UNICELLULARITY AND NONUNICELLULARITY OF WEIGHTED SHIFT OPERATORS

(Presented by Academician V. I. Smirnov on 15 III 1966)

Let \(\{H_i\}_{i=0}^{\infty}\) be a sequence of Hilbert spaces; \(l^2\{H_i\}_0^\infty\) the space of sequences \(X\), \(X=\{X_i\}_{i=0}^{\infty}\), \(X_i\in H_i\), \(i\geq 0\), with finite norm
\[ \|X\|=\left(\sum_0^\infty |X_i|^2\right)^{1/2}, \]
where \(|X_i|\) is the norm of the element \(X_i\) in the space \(H_i\), \(i=0,1,\ldots\). If \(H_0=H_1=\cdots=H\), then we write \(l^2(H)=l^2\{H_i\}_0^\infty\). Let \(H_0\subset H_1\subset\cdots\). Define in the space \(l^2\{H_i\}_0^\infty\) the shift operator \(S\) by the equalities \(SX=Y\), \(X\in l^2\{H_i\}_0^\infty\), where \(X=\{X_i\}_0^\infty\), \(Y=\{Y_i\}_0^\infty\) and \(Y_0=0\), \(Y_i=X_{i-1}\), \(i\geq 1\), and the operator \(\Lambda\) of multiplication by a (bounded) sequence of complex numbers \(\{\lambda_i\}_0^\infty\) by the equalities \(\Lambda X=Y\), \(X\in l^2\{H_i\}_0^\infty\), where \(X=\{X_i\}_0^\infty\), \(Y=\{Y_i\}_0^\infty\) and \(Y_i=\lambda_i X_i\), \(i\geq 0\). Finally, let \(C^q\) be the \(q\)-dimensional complex space and \(l^2=l^2(C^1)\).

Consider in the space \(l^2\{H_i\}_0^\infty\) the operator \(T=S\Lambda\) (operators of this kind are naturally called weighted shift operators). Of known interest is a complete description of the invariant subspaces* of the operators \(S\Lambda\) in the spaces \(l^2\{H_i\}_0^\infty\) (see also \((^6)\)). Beginning with the work of A. Beurling \((^2)\) on the operator \(S\) in the space \(l^2\), constant attempts have been made in this direction (see, for example, \((^{1,\,3-5,\,7})\)). All these works (except \((^{1,\,3})\)) adjoin A. Beurling’s investigations. In the present note the study, begun in \((^1)\), of invariant subspaces of the operators \(S\Lambda\) in the spaces \(l^2\), \(l^2(C^q)\) (and also in “two-sided” sequence spaces, see no. 4) is continued. In particular, it is established that there exist nonunicellular (see the definition below) operators \(S\Lambda\), acting in \(l^2\), with a sufficiently complicated structure of invariant subspaces (including ones such that \(\lambda_k\ne 0\), \(k\geq 0\), \(\lim_k\lambda_k=0\)).

1. Unicellularity of the operators \(S\Lambda\) in the space \(l^2\).

Denote by \(P_k\) \((k=0,1,\ldots)\) the operator acting from \(l^2\{H_i\}_0^\infty\) into \(H_k\) according to the rule \(P_kX=X_k\), if \(X\in l^2\{H_i\}_0^\infty\), \(X=\{X_i\}_0^\infty\). Recall that an operator \(T\), acting in some Hilbert space \(\mathcal H\), is called unicellular if, for any two of its invariant subspaces \(H\) and \(H'\), at least one of the inclusions \(H\subset H'\), \(H'\subset H\) holds. For the operators \(T=S\Lambda\) acting in \(l^2\), unicellularity means that every \(T\)-invariant subspace coincides with one of the subspaces
\[ l_n^2=\{X:X\in l^2,\; P_kX=0,\ \text{if } k<n\},\qquad n=0,1,\ldots . \]

Denote, as usual, by \(l^p\), \(1\leq p<\infty\), the space of sequences of complex numbers summable to the power \(p\).

Theorem 1. Consider in the space \(l^2\) an operator \(T=S\Lambda\) such that \(\lambda_i\ne 0\), \(i\geq 0\). Let
\[ r_i^{(k)}=|\lambda_i\cdots \lambda_{i+k}|,\qquad i=0,1,\ldots;\quad k=0,1,\ldots, \]
and suppose that the following conditions are fulfilled:

* Here and below, subspaces are always assumed to be closed.

1) There exists an integer \(k,\ k \geq 0\), such that \(r_i^{(k)} \downarrow 0,\ i \to \infty\) (i.e. \(\lim_i r_i^{(k)}=0,\ r_0^{(k)} \geq r_1^{(k)} \geq \ldots\)).

2) There exists an integer \(n,\ n \geq 0\), such that \(\{r_i^{(n)}\}_{i=0}^{\infty} \in \bigcup_{p<\infty} l^p\).

Then the operator \(T\) is unicellular.

Remark 1. Conditions 1) and 2) of the theorem mean that the sequence \(\{\lambda_i\}_0^\infty\) splits into a finite number of subsequences of the form \(\{\lambda_{i+kq}\}_{k=0}^\infty,\ i=0,1,\ldots,q-1\), in each of which \(|\lambda_{i+kq}|\) is nonincreasing as \(k \to \infty\), and, moreover,

\[ \{\lambda_{kq}\lambda_{1+kq}\ldots \lambda_{q-1+kq}\}_{k=0}^{\infty} \in \bigcup_{p<\infty} l^p . \]

Example. \(\lambda_{2k}=1,\ k=0,1,\ldots;\ \lambda_{2k+1}=1/(2k+1),\ k=0,1,\ldots\).

Remark 2. Theorem 1 remains valid if the space \(l^2\) is replaced by \(l^p,\ 1<p<\infty\). If \(p=1\), then it suffices to restrict oneself only to requirement 1) in the conditions of Theorem 1 in order to obtain unicellularity of the operator \(T\).

2. Invariant subspaces of the operators \(S\Lambda\) in \(l^2(C^q)\). Consider the operator \(T=S\Lambda\) in the space \(l^2(H)\), and let

\[ T_1=\sum_{k=0}^{\infty} \hat{\varphi}_k T^k, \]

where \(\hat{\varphi}_k,\ k=0,1,\ldots\), are operators in \(l^2(H)\) such that \(\hat{\varphi}_k(\{X_i\}_0^\infty)=\{\varphi_k X_i\}_0^\infty\), and \(\varphi_k\) are bounded operators in \(H\). We shall say of operators \(T_1\) with \(\{\|\hat{\varphi}_k\|\cdot |\lambda_0\ldots \lambda_{k-1}|\}_1^\infty \in l^2\) that they are \(l^2\)-functions of the operator \(T\) (and in this case the series \(\sum_0^\infty \hat{\varphi}_k T^k\) converges in the operator norm), and we shall write \(T_1=\varphi(T)\).

Let us introduce one more notation. Let \(Y\) be a subset of the Hilbert space \(\mathcal H\); let \(\Gamma\) be an operator acting from \(Y\) into \(\mathcal H\) such that \(\Gamma Y\) is orthogonal to \(Y\). Put \(Y \dotplus \Gamma Y=\{X+\Gamma X:X\in Y\}\).

Theorem 2. Let \(T=S\Lambda\) be an operator acting in \(l^2(C^q)\), and let the sequence \(\{\lambda_k\}_0^\infty\) satisfy all the conditions of Theorem 1. Let \(M,\ M \subset l^2(C^q)\), be an invariant subspace of the operator \(T\): \(TM \subset M\). Then there exist: an integer \(k,\ k\geq 0\), subspaces \(B_i,\ i\geq 0\),

\[ B_0 \subset B_1 \subset \ldots \subset B_k = B_{k+1}=\ldots,\quad B_{k-1}\neq B_k,\quad B_i \subset C^q,\ i\geq 0, \]

and an \(l^2\)-function \(\varphi(T)\) of the operator \(T\), with \(\hat{\varphi}_0=0\), mapping \(l^2\{B_i\}_0^\infty\) into \(l^2\{C^q \ominus B_i\}_0^\infty\), such that

\[ M=l^2\{B_i\}_0^\infty \dotplus \varphi(T)l^2\{B_i\}_0^\infty . \tag{1} \]

Under the assumptions made, the triple \(k,\ \{B_i\}_0^\infty,\ \varphi(T)\) is uniquely determined by the subspace \(M\). Conversely, every subspace of the form (1) is invariant with respect to the operator \(T\).

Corollary. Every \(T\)-invariant subspace \(M,\ M\subset l^2(C^q)\), is generated by no more than \(q\) of its elements (i.e. there are \(X^i,\ 1\leq i\leq q,\ X^i\in M\), such that the closed linear span of the elements \(T^nX^i,\ n\geq 0,\ 1\leq i\leq q\), coincides with \(M\)).

3. Non-unicellularity of the operators \(S\Lambda\) in \(l^2\). Consider again the operators \(T=S\Lambda\) in the space \(l^2\). We shall now show that conditions of the “regularity” type for the decrease of \(|\lambda_k|,\ k\to\infty\) (see Theorem 1) are essential for the unicellularity of such operators. In this connection one may note that the very fact of the existence of non-unicellular operators \(S\Lambda\) in the space \(l^2\) such that \(\lim_k \lambda_k=0,\ \lambda_k\neq 0,\ k\geq 0\), is important. In this sense the question is settled by the following example:

Example. The operator \(T=S\Lambda\) in the space \(l^2\) such that \(\lambda_n=2^{-n}\).

if \(n \ne 2^p+1,\ p \ge 1\), and \(\lambda_n=2^{-2(p+1)^2}\), if \(n=2^p+1,\ p \ge 1\), is not unicellular.

However, a more general assertion also holds.

Theorem 3. Consider in the space \(l^2\) the operator \(T=S\Lambda\), \(\lambda_k \ne 0;\ |\lambda_k|\ge |\lambda_{k+1}|\), \(k=0,1,\ldots\). Let \(P=\{n_k\}_{k=1}^{\infty}\) be a sequence of natural numbers \(n_k,\ k=1,2,\ldots\), and \(n_{k+1}/n_k>\varkappa>1,\ k\ge 1\). Then there exists a sequence \(\{\mu_k\}_1^\infty\), \(\mu_k>0\), \(\mu_k\downarrow 0,\ k\to\infty\), such that, if \(\widetilde T=S\widetilde\Lambda\), where: \(\widetilde\lambda_n=\lambda_n\), if \(n\notin P\), and \(\widetilde\lambda_n=\mu_k\), if \(n=n_k,\ k=1,2,\ldots\), with \(|\widetilde\lambda_k|\le \mu_k,\ k\ge 1\), then the operator \(\widetilde T\) is not unicellular.

4. Operators \(S\Lambda\) in the space \(l_Z^2\). Let \(Z\) be the set of all integers and
\[ l_Z^p=\{X=\{X_k\}_{k\in Z}:\ X_k\in C^1,\ k\in Z;\ \|X\|=(\sum_k |X_k|^p)^{1/p}<\infty\}. \]
Define in the space \(l_Z^p\) the shift operator \(S\) by the equalities \(SX=Y\), \(X\in l_Z^p\), where \(X=\{X_k\}_{k\in Z}\), \(Y=\{Y_k\}_{k\in Z}\) and \(Y_k=X_{k-1}\), \(k\in Z\), and the operator \(\Lambda\) of multiplication by the (bounded) sequence \(\{\lambda_k\}_{k\in Z}\) by the equalities \(\Lambda X=Y\), \(X\in l_Z^p\), where \(X=\{X_k\}_{k\in Z}\), \(Y=\{Y_k\}_{k\in Z}\) and \(Y_k=\lambda_k X_k,\ k\in Z\). If \(\lambda\) is a complex number (\(\lambda\ne 0\)), put \(\operatorname{sign}\lambda=\lambda/|\lambda|\), and let \(m_k=\operatorname{sign}(\lambda_0\ldots\lambda_{k-1})\), \(k\ge 1\), \(m_k=\operatorname{sign}(\lambda_{-1}\ldots \lambda_{k-1})^{-1}\), \(k\le 0\) (we assume, of course, that \(\lambda_k\ne 0,\ k\in Z\)). Finally, let \(M\) be the operator of multiplication by the sequence \(\{m_k\}_{k\in Z}\), acting in the space \(l_Z^2\), and \(P_WX=\{Y_k\}_{k\in Z}\), \(Y_k=X_k\), \(k\in W\), and \(Y_k=0\), \(k\notin W\), where \(W\) is a subset of \(Z\).

Theorem 4*. Consider the operator \(T=S\Lambda\) in the space \(l_Z^2\) and let \(\lambda_0=1,\ \lambda_k\ne 0,\ k\in Z\). Then we have:

1) The following assertions are equivalent:

a) \(T\) has nontrivial invariant subspaces;

b) \(Z\) is decomposed into a finite number of arithmetic progressions \(Z_1,\ldots,Z_n\), on each of which the modulus \(|\lambda_p|\), \(p\in Z_i\), \(i=1,\ldots,n\), is constant;

c) there exists a natural number \(n\) such that \(|T|^n=rS^n\), where \(r>0\) and \(|T|^n=S^n|\Lambda|\), \(|\Lambda|\) is the operator of multiplication by the sequence \(\{|\lambda_k|\}_{k\in Z}\).

2) Let \(Z_1,\ldots,Z_n\) be disjoint arithmetic progressions with difference \(n\) and \(\bigcup_1^n Z_i=Z\). If \(|\lambda_k|=r_i,\ k\in Z_i\), where \(r_i>0,\ i=1,\ldots,n\), then the following assertions are equivalent:

a) the subspace \(H,\ H\subset l_Z^2\), reduces the operator \(T\);

b) \(H=M^{-1}H_0\), where
\[ H_0=\sum_{i=1}^{n}\oplus P_{Z_i}H_0 \equiv \sum_{i=1}^{n}H_i \]
and \(SH_i=H_{i+1}\), \(1\le i\le n-1,\ SH_n=H_1,\ H_1=\{\{a_{nk}\}_{k\in Z}:\ \{a_p\}_{p\in Z}\in \widehat H\}\), where \(\widehat H\) is a subspace of \(l_Z^2\), \(S\widehat H=\widehat H\);

c) \(H=M^{-1}(\chi_E L^2)\), \(E\subset\{\lambda:\ |\lambda|=1\}\), \(e^{2\pi i/n}E=E\). (Here \(L^2\) is the space of measurable functions on the circle \(\{\lambda:\ |\lambda|=1\}\), square-summable, and \(\chi_E L^2=\{\chi_E f:\ f\in L^2\}\), \(\chi_E\) is the characteristic function of the set \(E\). The correspondence of a function to the sequence of its Fourier coefficients is the natural isomorphism of the spaces \(L^2\) and \(l_Z^2\); it is precisely in this sense that the equality in item c) is to be understood.)

The proof of Theorem 4 is based on the following lemma, useful also in some other questions.

Lemma. Let \(T\) be a continuous operator in a Hilbert space \(\mathcal H\) such that \(T=VR\), where \(V\) is isometric and \(R\) is self-adjoint. Suppose, moreover, that the operator \(R\) is one-to-one (or, equivalently, \(R\mathcal H\) is dense in \(\mathcal H\)). In order that a subspace

* An analogous assertion can also be formulated for the operators \(S\Lambda\) in the spaces \(l^2\{H_i\}_0^\infty\), but the scope of the present note does not permit doing this (see (5) for the case \(\lambda_k=1\)).

\(L, L \subset \mathcal H\), reduces the operator \(T\), it is necessary and sufficient that it reduce the operators \(V\) and \(R\).

We now turn to the invariant (but not necessarily reducing) subspaces of the operators \(S\Lambda\) in \(l_z^2\). It is easy to see that the subspaces

\[ l_k^2=\{X:\ X\in l_z^2,\ P_iX=0,\ i<k\},\quad k\in Z, \]

are invariant with respect to any operator of the form \(T=S\Lambda\). The unicellularity of such an operator means that the subspaces \(l_k^2\), \(k\in Z\), exhaust all \(T\)-invariant subspaces of \(l_z^2\). It is easy to give examples of non-unicellular operators \(S\Lambda\) in \(l_z^2\) (operators having reducing subspaces or having at least one eigenvalue).

From the results of the preceding section it also follows easily that the operator \(T=S\Lambda\) is non-unicellular in the space \(l_z^p\), if at least one of the operators \(T_{\pm}=S\Lambda_{\pm}\) is non-unicellular in the space \(l^p\), where \(\Lambda_+(\Lambda_-)\) is the operator of multiplication by the sequence \(\{\lambda_k\}_{k=0}^{\infty}\) (\(\{\lambda_{-k}\}_{k=0}^{\infty}\), respectively).

However, the general question of the existence of unicellular operators of the form \(T=S\Lambda\) in the space \(l_z^p\) remains open.

Leningrad State University
named after A. A. Zhdanov

Received
24 II 1966

CITED LITERATURE

\(^{1}\) N. K. Nikol’skii, Vestn. LGU, ser. matem., No. 7, issue 2, 68 (1965).
\(^{2}\) A. Beurling, Acta Math., 81, 1, 2, 239 (1949).
\(^{3}\) W. F. Donoghue, Pacif. J. Math., 7, No. 2, 1031 (1957).
\(^{4}\) P. L. Duren, Trans. Am. Math. Soc., 99, No. 2, 320 (1961).
\(^{5}\) P. R. Halmos, J. reine u. angew. Math., 208, 102 (1961).
\(^{6}\) P. R. Halmos, In: Lectures on Modern Mathematics, 1, N. Y., 1963, pp. 1—32.
\(^{7}\) D. Sarason, Mem. Am. Math. Soc., No. 56 (1965).