UDC 513.88 + 517.535.4

MATHEMATICS

N. K. NIKOLSKII

SPECTRAL SYNTHESIS FOR THE SHIFT OPERATOR AND ZEROS IN CERTAIN CLASSES OF ANALYTIC FUNCTIONS, SMOOTH UP TO THE BOUNDARY

(Presented by Academician V. I. Smirnov on March 7, 1969)

This note considers the problem of spectral synthesis* for invariant subspaces of the shift operator in weighted classes of sequences (or of the corresponding functions). This problem is related to a certain uniqueness (or quasianalyticity) theorem for functions regular in the unit disk. The proof of the latter theorem makes it possible to formulate, under known restrictions, a criterion for the possibility of spectral synthesis.

§ 1. Notation.

1.1. \(l^p(w)\equiv l^p(w_n^p)\) is the space of all sequences \(x=\{x_n\}_{n=0}^{\infty}\) of complex numbers summable with power \(p\), \(1\leq p\leq\infty\), with weight \(\{w_n^p\}_{n=0}^{\infty}\) (bounded, if \(p=\infty\)), equipped with the usual norm,

\[ l^p(w)=l^p,\quad \text{if } w_n\equiv 1,\ n\geq 0. \]

\[ l_0^p(w)=\{x\in l^p(w):x=\{x_n\}_{n=0}^{\infty},\ \lim_n x_n w_n=0\},\quad 1\leq p\leq\infty. \]

1.2. If \(x\in l^p(w)\), then

\[ \hat{x}(Z)=\sum_{n=0}^{\infty} x_n Z^n \]

is a formal power series convergent for complex \(z\),

\[ |z|<\underline{\lim}_n\, w_n^{1/n}. \qquad l_A^p(w)=\{\hat{x}:x\in l^p(w)\}. \]

1.3. \(S\) is the shift operator in the space of (all) sequences of complex numbers:

\[ S(x_0,x_1,\ldots)=(0,x_0,x_1,\ldots), \]

or

\[ (S\hat{x})(Z)=Z\hat{x}(Z) \]

in the sense of multiplication of formal power series.

1.4. If \(\lambda_n=w_{n+1}/w_n,\ n\geq 0\), then \(T=S\Lambda\) is the weighted shift operator in the space of (all) sequences, and

\[ \Lambda\{x_n\}_{n=0}^{\infty}=\{\lambda_n x_n\}_{n=0}^{\infty}. \]

1.5.

\[ r(T)\equiv r_w(S)\overset{\mathrm{def}}{=}\overline{\lim}_n\, w_n^{1/n}. \]

1.6. If \(f\) is a function from \(l_A^p(w)\), then \(N(f)\) is the family of all its zeros (with multiplicities taken into account) lying in

\[ D_r=\{z:|z|<r_w(S)\} \]

or in

\[ \overline{D}_r=\{z:|z|\leq r_w(S)\}, \]

if \(l_A^p(w)\subset C(\overline{D}_r)\).

* One says that a linear operator \(T\) admits spectral synthesis if any of its (closed) invariant subspaces is generated by the root vectors of the operator \(T\) contained in it. Together with a simple description of all invariant subspaces of the operator \(T\), the possibility of spectral synthesis leads to effective criteria for completeness of systems of the form \(\{T^n x\}_{n=0}^{\infty}\).

§ 2. Additional assumptions.

α. \(\{\log w_n\}_{0}^{n=\infty}\) is a concave sequence, \(r_w(S)=1,\ w_n \uparrow\).

β. \(n^{-3/2}\log w_n \downarrow 0,\ r_w(S)=1,\ w_n \uparrow\).

γ. \(l_A^p(w)\) is a Banach algebra (with multiplication of formal power series), \(r_w(S)=1,\ w_n \uparrow\).

In addition, throughout this note it is assumed that \(w_n>0\), and
\[ \sup_n w_{n+1}/w_n<\infty \]
(the boundedness of the operator \(S\) in \(l^p(w)\)).

§ 3. Assertions. The aim of the note is to clarify the connections between the following assertions:

A. \(l_A^p(w)\) is contained in \(C^\infty(\overline D_1)\) and is a quasianalytic class of functions in \(\overline D_1\).

B. \(f\in l_A^p(w)\) and \(N(f)\) is infinite \(\Rightarrow f\equiv 0\) (uniqueness theorem).

C. \(S\)-invariant subspaces in \(l_A^p(w)\) are determined by their zeros* (i.e., for \(S^*\) in \(l_0^{p'}(w_n^{-p'})\) spectral synthesis is possible).

D.
\[ \sum_{n=0}^{\infty}\frac{\log w_n}{n^{3/2}+1}=+\infty . \]

§ 4. Theorems.

Theorem 1**. \(\mathrm{B}\xRightarrow{\beta}\Gamma \xLeftrightarrow{\alpha}\mathrm{A}\xLeftrightarrow{\beta'}\mathrm{B}\).

Theorem 2. Let the sequence \(\{w_n\}_0^\infty\) satisfy condition β, while condition D is not fulfilled for it. Then:

1) For any sequence \(\{\lambda_n\}_{n=1}^\infty,\ |\lambda_n|<1,\ \sum_n(1-|\lambda_n|)<\infty\), lying in the angle
\[ \{\xi:\ |\arg(1-\xi)|\le \theta<\pi/2\}, \]
there exists \(f,\ f\in l_A^p(w)\), such that \(f(\lambda_n)=0,\ n\ge 1,\ f\not\equiv 0\).

2) There exists an outer \({}^{(1)}\) function \(m\), continuous in \(\overline D_1\), \(m\in l_A^p(w)\), such that \(\psi_\alpha m\in l_A^p(w)\) for every \(\alpha,\ \alpha>0\),
\[ \psi_\alpha(z)=\exp\left(\alpha\,\frac{z+1}{z-1}\right),\quad |z|<1, \]
and \(\psi_\alpha m,\ \alpha>0\), generate distinct \(S\)-invariant subspaces in \(l^p(w)\).

Theorem 3. Let the sequence \(\{w_n\}_0^\infty\) be such that either \(r_w(S)=0\), or condition β is fulfilled. Then each of the following assertions follows from the preceding one:

1) \(T=S\Lambda\) is a basis operator in \(l_0^p\) in the sense of the definition given in (2).

2) \(l_A^p\bigl((w_{n+j})_{n\ge0}\bigr)\) is an algebra for every \(j,\ j\ge0\), admitting spectral synthesis of ideals.

3) \(l_A^p\bigl((w_{n+j})_{n\ge0}\bigr)\) is an algebra for every \(j,\ j\ge0\), and either \(r(T)=0\), or \(r(T)>0\) and condition D is fulfilled.

Moreover***, if \(r(T)=0\), then \(3)\Rightarrow 1)\).

Lemma. In order that the space \(l_A^p(w)\) be a Banach algebra, it is sufficient, and for \(p=1,\infty\) also necessary, that
\[ \sup_{n\ge0}\left(\sum_{k=0}^{n}\frac{w_n^{p'}}{w_k^{p'}w_{n-k}^{p'}}\right)^{1/p'}<\infty,\qquad \frac1p+\frac1{p'}=1. \]

* That is, if \(SM\subset M\), \(M\) is a closed subspace in \(l_A^p(w)\), then
\[ M\overset{\mathrm{def}}{=}M_E=\{f\in l_A^p(w): N(f)\supset E\}, \]
where \(E\) is some family of complex numbers.

** The proposition \(\mathrm{A}\Rightarrow \mathrm{B}\) means that A implies B under condition β.

*** After the manuscript of the note had been submitted for publication, R. Gellar informed the author in a letter that for a basis operator \(T=S\Lambda\) one always has \(r(T)=0\).

Corollary. The operator \(T=S\Lambda\) with \(r(T)=0\) is basic in \(l_0^p\) if

\[ \sup_{n\geqslant 0}\left(\sum_{k=0}^{n} \frac{w_{n+j}^{p'}}{w_{k+j}^{p'} w_{n-k+j}^{p'}} \right)^{1/p'}<\infty,\quad j=0,1,\ldots \]

For \(p=1,\infty\) the converse assertion is also true.

§ 5. Some remarks.

5.1. In the case \(r_w(S)=0\), the possibility of spectral synthesis for the operator \(S^*\) is equivalent to the single-cell property (linear similarity) of the operators \(S\) and \(T\). Therefore Theorem 3 and the corollary give a new strengthening (for \(p=\infty\), a final one) of the results on basicity and the single-cell property for operators \(T=S\Lambda\) obtained in \((^2)\).

5.2. As early as 1952 L. Carleson proved \((^3)\) that \(\Gamma \overset{\alpha}{\Rightarrow} \mathrm{B}\) (the uniqueness theorem: if \(N(f)\) is infinite and the conditions \(\alpha,\Gamma\) are fulfilled, then \(f\equiv 0\)). By virtue of Theorem 1 this assertion means that every \(S^*\)-invariant subspace is finite-dimensional. We also note that the presence of spectral synthesis for the operator \(S^*\) (for example under the conditions given in Theorem 1) leads to a simple description of the cyclic (\(\equiv\) weakly invertible) elements of the operator \(S\): \(f\in l_A^p(w)\) is cyclic (i.e. \(\{S^n f\}_{n=0}^{\infty}\) is complete in \(l_A^p(w)\)) \(\Longleftrightarrow\) \(f(\lambda)\neq 0\) for every \(\lambda\).

Fig. 1

5.3. The existence of continual chains of invariant subspaces corresponding to the “boundary zero” (see Theorem 2, item 2) was discovered by A. Beurling \((^4)\) in 1949 for the space \(l^2\). Such chains were later studied in \((^5,^6)\) (for the space \(C_A=\{f\in C(\overline{D}_1): f\text{ is analytic in }D_1\}\)) and \((^7,^8)\) (for the algebras \(L^1(\varphi;0,\infty)\) on the half-axis \([0,\infty)\); in particular, \((^8)\) gives a complete description of the primary ideals in such algebras in the non-quasianalytic case).

5.4. The assertion \(\Gamma \overset{\alpha}{\Longleftrightarrow} \mathrm{A}\) is derived from the theorem of R. Salinas \((^9)\)—M. M. Dzhrbashyan—B. I. Korenblum \((^{10})\)* on the quasianalyticity of the classes
\[ D\{A_n\}=\{f\in C(\overline{D}_1):\|f^{(n)}\|\leqslant K_f A_n,\ n\geqslant 0\}. \]

5.5. For the proof of the inclusion \(\mathrm{B}\Rightarrow \mathrm{A}\), certain assumptions on the regularity of the sequence \(\{w_n\}_{n=0}^{\infty}\) (for example \(\beta\)) are necessary, whereas \(\mathrm{A}\Rightarrow \mathrm{B}\) is valid without additional restrictions.

5.6. The main difficulty in the proof of Theorem 1 is the assertions \(\mathrm{B}\Rightarrow \mathrm{A}\) and \(\mathrm{B}\Rightarrow \Gamma\), which mean that without condition \(\Gamma\) the spaces \(l_A^p(w)\) contain “bad” functions, more precisely, the functions from items 1)—2) of Theorem 2. The estimate of the Taylor coefficients of these functions (\(\psi_\alpha m\) and \(Bm\), \(B\) being the infinite Blaschke product \((^1)\)) is carried out by deforming the contour in the integral

\[ (\psi_\alpha m)_k=\frac{1}{2\pi i}\int_{\partial D_1} \frac{\psi_\alpha(\zeta)m(\zeta)}{\zeta^{k+1}}\,d\zeta = \frac{1}{2\pi i}\int_{L_k} \frac{\psi_\alpha(\zeta)m(\zeta)}{\zeta^{k+1}}\,d\zeta, \]

where \(L_k\) is a curve composed of a part of the circle \(\{z:|z|=r_k\}\), \(r_k>1\), and a part of the curve \(L\) (see Fig. 1), the order of tangency of which with the circle \(\partial D_1\) is regulated by the condition

\[ \sum_{0}^{\infty}\frac{\log w_n}{n^{3/2}+1}<\infty. \]

* It should be noted that the theorem mentioned was found in 1955 (independently and almost simultaneously) by the first two authors, and was then rediscovered in \((^{10})\).

5.7. Condition \(\gamma\) in Theorem 1 can be replaced by a Wiener-type condition for \(l_A^p(w)\): if \(f \in l_A^p(w)\) and \(\hat f(\lambda)\ne 0\) for all \(\lambda\), then \(f\) is a weakly invertible element in \(l_A^p(w)\).

5.8. The Borel transform (11) maps the space \(l_A^p(w)\) into the collection \(B_p\) of all entire functions \(f\) of exponential type for which \(\{f^{(n)}(0)\}_0^\infty \in l^p(w)\). Under this correspondence the operator \(S\) corresponds to the integration operator \(J\),

\[ (Jf)(z)=\int_0^z f(t)\,dt,\qquad f\in B_p. \]

The description of the cyclic elements of \(S\) (see Remarks 1 and 2) now leads to theorems on completeness (and basicity) of systems of successive primitives \(\{J^n f\}_{n=0}^\infty\) in the spaces \(B_p\) (see also \((^2,{}^{12})\)).

5.9. Weight sequences \(\{w_n\}_0^\infty\) with

\[ w_n=ecn^\delta,\quad w_n=\exp\!\left(cn^\delta(\log\cdots\log n)^{\varepsilon_1}\right), \]

\[ w_n=\exp\!\left(cn^\delta(\log\cdots\log n)^{\varepsilon_1}(\log\cdots\log n)^{\varepsilon_2}\right) \]

and so on \((c>0,\ 0\leqslant \delta<1,\ \varepsilon_i>0,\ s_i\text{ are natural numbers})\) satisfy all the conditions \(\alpha\)–\(\gamma\), and for them all assertions A–Γ are equivalent.

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

Received
18 II 1969

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