Abstract
Full Text
L. A. MARKUSHEVICH
THE STRUCTURE OF RINGS OF CONTINUOUS FUNCTIONS ON A CIRCLE WITH TWO GENERATORS
(Presented by Academician I. N. Vekua on February 29, 1960)
Consider on the circle \(|\zeta|=1\) the ring \([\varphi,f]\) of continuous functions with two generators \(\varphi(\zeta)\) and \(f(\zeta)\), closed with respect to uniform convergence. Suppose that \(\varphi\) and \(f\) separate the points of the circle, i.e., for every pair of points \(\zeta_1\ne \zeta_2\), \(|\zeta_i|=1\) \((i=1,2)\), either \(\varphi(\zeta_1)\ne \varphi(\zeta_2)\) or \(f(\zeta_1)\ne f(\zeta_2)\). We shall determine what conditions the generators of the ring \(\varphi\) and \(f\) must satisfy in order that the ring \([\varphi,f]\) not coincide with the ring \(C\) of all continuous functions. Wermer, in papers \((^1,^2)\), established what the generator \(f\) must be in order that \([\varphi,f]\ne C\) when \(\varphi(\zeta)\) has a special form: \(\varphi(\zeta)=\zeta\), or \(\varphi(\zeta)=\zeta^2\). An analogous result was obtained by us in the note \((^3)\), when \(\varphi(\zeta)=\zeta^n\), \(n\) being any natural number. In that note we studied a system of equations which is also used in the present work. A similar system is used by Wermer in \((^4,^5)\).
In papers \((^4,^5)\) Wermer, abandoning the special form of one of the generators, imposes rather strong conditions already on both generators. Namely, he requires that:
A. Both functions \(\varphi(\zeta)\) and \(f(\zeta)\) be analytic on the circle \(|\zeta|=1\) and \(\varphi'(\zeta)\ne 0\), \(|\zeta|=1\).
Under this assumption Wermer, without loss of generality, could regard the following condition as satisfied:
B. On the circle there is only a finite set \(M\) of points such that for every \(\zeta_1\in M\) there is another point \(\zeta_2\in M\) for which \(\varphi(\zeta_1)=\varphi(\zeta_2)\).
In the present work we abandon Wermer’s restriction A, replacing it by the condition:
A′. The function \(\varphi(\zeta)\) has a derivative satisfying a Hölder condition, and \(f(\zeta)\) satisfies a Hölder condition.
In order not to make the exposition cumbersome, we shall everywhere assume that condition B is also satisfied; however, it can be replaced by a more general one. For example, we may allow the point \(\varphi(\zeta)\) to run a finite number of times along an arc of the curve \(\gamma:\lambda=\varphi(\zeta)\), when the point \(\zeta\) describes the circle \(|\zeta|=1\) once, although the support of the curve still has a finite number of self-intersection points. Exactly the same arguments apply to certain kinds of curves whose supports have an infinite number of self-intersection points and under more general smoothness assumptions.
Theorem 1. If \(\varphi(\zeta)\) and \(f(\zeta)\) satisfy conditions A′ and B, then \([\varphi,f]\ne C\) if and only if the curve in the space of two complex variables \(\Gamma: z_1=\varphi(\zeta),\ z_2=f(\zeta),\ |\zeta|=1,\) bounds a piece of an analytic surface.
Theorem 2. Let \(\varphi(\zeta)\) and \(f(\zeta)\) satisfy conditions \(A'\) and \(B\). Then \([\varphi, f]\ne C\) if and only if
\[ \int_{|\zeta|=1} \varphi^m(\zeta) f^n(\zeta)\varphi'(\zeta)\,d\zeta=0,\qquad n,m\geqslant 0 . \]
Theorems 1 and 2 are a generalization of Wermer’s theorems, in which he assumes that \(\varphi\) and \(f\) satisfy condition \(A\)*.
In the proof, an important role is played by the integral representation we found for functions connected with the solution of the above-mentioned system of equations (Theorem 4). This representation makes it possible to establish Theorem 3, which, from the form of the curve \(\gamma\), in a number of cases permits one to assert that \([\varphi,f]=C\) for any \(f\).
Theorem 3. Let \(\varphi(\zeta)\) satisfy condition \(B\) and have a continuous derivative, and let the component \(D_0\) of the complement of \(\gamma\) that contains \(\infty\) be such that the complement of \(\overline{D}_0\) is disconnected. Then \([\varphi,f]=C\) for any \(f(\zeta)\) satisfying Helder’s condition.
We outline the course of the proof. Let \([\varphi,f]\ne C\). Then there exists a nontrivial functional on \(C\) that vanishes on \([\varphi,f]\), i.e., there is a measure \(d\mu(\zeta)\ne0\) for which
\[ \int_{|\zeta|=1} f^n(\zeta)\varphi^m(\zeta)\,d\mu(\zeta)=0,\qquad n,m\geqslant 0 . \]
The function \(\lambda=\varphi(\zeta)\) maps the circle onto the curve \(\gamma\). Then \(f(\zeta)=f(\varphi^{-1}(\lambda))=\widetilde f(\lambda)\) is a function of \(\lambda\), single-valued and continuous and satisfying Helder’s condition everywhere except at the points \(\widetilde M\) of self-intersection of \(\gamma\), where \(\widetilde f(\lambda)\) is multivalued. But if \(\alpha\) is a simple arc of \(\gamma\) with endpoints belonging to the set \(\widetilde M\), then \(\widetilde f(\lambda)\) is continuous and satisfies Helder’s condition on the closed arc \(\alpha\). Denote the measure
\[ \prod_{\zeta_j\in M}\bigl(\varphi(\zeta)-\varphi(\zeta_j)\bigr)\,d\mu(\zeta) \]
by \(d\mu_1(\zeta)\); evidently,
\[ \int_{|\zeta|=1} f^n(\zeta)\varphi^m(\zeta)\,d\mu_1(\zeta)=0,\quad n,m\geqslant 0, \]
and from the property of separability of the circle by the functions \(f\) and \(\varphi\) it is not difficult to derive that \(d\mu_1(\zeta)\ne0\). Put \(d\widetilde\mu(\lambda)=d\mu_1(\varphi^{-1}(\lambda))\). Then
\[ \int_{\gamma} \widetilde f^{\,n}(\lambda)\lambda^m\,d\widetilde\mu(\lambda)=0,\qquad m,n\geqslant 0 \tag{1} \]
(in what follows we shall omit the tilde sign \(\sim\)). Consider the function
\[ \Phi^{(n)}(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f^n(\lambda)\,d\mu(\lambda)}{\lambda-z},\qquad n=0,1,2,\ldots \]
It is analytic in the components of the complement of the curve \(\gamma\) and in the component of the complement \(D_0\) that contains \(\infty\); \(\Phi^{(n)}(z)\equiv0\) by virtue of (1). Let \(G_1\) be one of the components of the complement of \(\overline{D}_0\). All components of the complement of \(\gamma\) inside \(G_1\) in which \(\Phi^{(n)}(z)\equiv0\), as well as \(D_0\), will be called domains of zero rank; the union of their closures we denote by \(\Omega_0\). If a component \(D_1\subset G_1\) of the complement of \(\gamma\) has a common boundary arc \(\alpha\) with \(\Omega\), then \(\Phi_1^{(n)}(\lambda)=\pm\mu'(\lambda)f^n(\lambda)\), where \(\Phi_1^{(n)}(\lambda)\) are the boundary values
* Note added in proof. We note that in Wermer’s proofs the analyticity of the generators is repeatedly and essentially used. M. V. Fedoryuk (*), referring to D. A. Anosov, gives a lemma stating that every closed, twice continuously differentiable curve in \(C^n\) projects parallel to a suitable complex \((n-1)\)-dimensional hyperplane onto the complex plane with a finite number of exceptional points. Erroneously supposing that analyticity is used by Wermer only in condition \(B\), Fedoryuk formulates Wermer’s theorems for twice continuously differentiable generators.
to \(\alpha\) of the function \(\Phi^{n}(z)\) in \(D_{1}\). Denoting \(\pm \mu'(\lambda)\) by \(x_{1}(\lambda)\), and \(f^{n}(\lambda)\) by \(y_{1}^{n}(\lambda)\), we shall have
\[ \Phi_{1}^{(n)}(z)=x_{1}(z)y_{1}^{n}(z),\qquad z\in D_{1}, \]
where \(x_{1}(z), y_{1}(z)\) are functions whose boundary values on \(\alpha\) are almost everywhere equal to \(x_{1}(\lambda)\) and \(y_{1}(\lambda)\).
It can be shown analogously that in every component \(D\subset G_{1}\) of the complement to \(\gamma\)
\[ \Phi^{(n)}(z)=x_{1}(z)y_{1}^{n}(z)+\cdots+x_{k}(z)y_{k}^{n}(z), \]
and the equalities \(y_{j}(z)=y_{i}(z)\) occur only for a discrete set of points inside \(D\). We shall say in this case that the domain \(D\) has rank \(k\). The aggregate of the closures of the domains of rank \(k\) will be denoted by \(\Omega_{k}\). We prove that the function
\[ R_{k}^{(n)}(z)=y_{1}^{n}(z)+\cdots+y_{k}^{n}(z) \]
is analytic and bounded by the constant \(kM^{n}\) inside \(\Omega_{k}\), where \(M=\max_{|\zeta|=1}|f(\zeta)|\). Using this, we find an integral representation for \(R_{k}^{(n)}(z)\)
\[ R_{k}^{(n)}(z)=y_{1}^{n}(z)+\cdots+y_{k}^{n}(z)= \frac{1}{2\pi i}\int_{\gamma^{1,0}+\gamma^{2,1}+\cdots+\gamma^{k_{0},k_{0}-1}} \frac{f^{n}(\lambda)\,d\lambda}{\lambda-z}, \qquad z\in \Omega_{k}-(\gamma^{k,k-1}-\gamma^{k+1,k}). \tag{2} \]
Here \(\gamma^{k,k-1}\) is the common boundary of \(\Omega_{k}\) with \(\Omega_{k-1}\), oriented in the positive direction for the domains from \(\Omega_{k}\); \(k_{0}=\max k\).
Using now the boundedness of the integral (2) and condition \(A'\), we establish that the support \(\gamma\) coincides with the support of \(\gamma^{1,0}+\gamma^{2,1}+\cdots+\gamma^{k_{0},k_{0}-1}\). (It follows from this that there are no adjacent domains of the same rank.) The oriented curve \(\gamma\) is either the same as \(\gamma^{1,0}+\gamma^{2,1}+\cdots+\gamma^{k_{0},k_{0}-1}\), or opposite to it, i.e.
\[ \gamma=\pm(\gamma^{1,0}+\gamma^{2,1}+\cdots+\gamma^{k_{0},k_{0}-1}). \]
Hence the validity of Theorem 3 follows: if \([\varphi,f]=C\), then the complement to \(\overline{D}_{0}\) is connected.
It is now easy to obtain Theorem 2.
Indeed, if
\[ \int_{|\zeta|=1} f^{n}(\zeta)\varphi^{m}(\zeta)\varphi'(\zeta)\,d\zeta=0,\qquad n,\ m\ge 0, \]
then this means (since \(\varphi'(\zeta)\ne0\)) that \([\varphi,f]\ne C\). Conversely, let \([\varphi,f]\ne C\). Then
\[ \frac{1}{2\pi i}\int_{|\zeta|=1} \frac{f^{n}(\zeta)\varphi'(\zeta)\,d\zeta}{\varphi(\zeta)-z} = \frac{1}{2\pi i}\int_{\gamma} \frac{f^{n}(\lambda)\,d\lambda}{\lambda-z} \equiv 0,\qquad z\in D_{0}, \]
whence
\[ \int_{|\zeta|=1} f^{n}(\zeta)\varphi^{m}(\zeta)\varphi'(\zeta)\,d\zeta=0,\qquad n,m\ge0. \]
Theorem 4. If \([\varphi,f]\ne C\) and conditions \(A'\) and \(B\) are satisfied, then
\[ \frac{1}{2\pi i}\int_{\pm\gamma}\frac{f^{n}(\lambda)\,d\lambda}{\lambda-z} = R_{k}^{(n)}(z)=y_{1}^{n}+\cdots+y_{k}^{n},\qquad z\in\Omega_{k}-\gamma, \]
where \(|y_{1}(z)|\le M\), \(R_{k}^{(n)}(\lambda)-R_{k-1}^{(n)}(\lambda)=f^{n}(\lambda)\), \(\lambda\in\Omega_{k}\cap\Omega_{k-1}\). Further, if \(D_{k}\) is a component of \(\Omega_{k}-\gamma\), then \(y_{i}(z)\), \(z\in D_{k}\), \(i=1,\ldots,k\), are the roots of the equation
\[ y^{k}+C_{1}(z)y^{k-1}+\cdots+C_{k}(z)=0, \]
whose coefficients \(C_{i}(z)\) are analytic in \(D_{k}\) and continuous in \(\overline{D}_{k}\).
Now we can prove Theorem 1. We show that if \([\varphi,f]\ne C\), then
\[ \Gamma:\quad z_{1}=\varphi(\zeta),\quad z_{2}=f(\zeta),\quad |\zeta|=1, \]
bounds a piece of an analytic surface \(R\) in the space of two complex variables. The converse fact is known \((^{2})\). As points of \(R+\Gamma\) we take the points \((z_{0},y_{i}(z_{0}))\), where \(z_{0}\in\Omega_{k}\), \(k\ge1\), and \(y_{j}(z_{0})\) is a root of the equation corresponding to \(\Omega_{k}\). If \(z_{0}\in\gamma\), and the resultant of the equation
\[ y^{k}+C_{1}y^{k-1}+\cdots+C_{k}=0 \]
is not equal to zero at \(z_{0}\), then in a neighborhood \(U\) of the point \(z_{0}\) one can distinguish \(k\) different analytic-
branches of solutions of the equation. Denote by \(y_j(z)\) the branch that assumes at \(z_0\) the value \(y_j(z_0)\). Then the set of points \((z,y_j(z))\), \(z\in U\), forms a neighborhood of the point \((z_0,y_j(z_0))\in R\), analytically homeomorphic to a disk. If at \(z_0\) the resultant of the equation vanishes, then, since the zeros of the resultant are discrete inside \(\Omega_k\), the point \((z_0,y_j(z_0))\) will be a branch point of order \(p-1\) (if \(y_j(z_0)\) is a root of multiplicity \(p\)). Let now \(\lambda_0\in\gamma\), but \(\lambda_0\notin\widetilde M\), and let \(\lambda_0\) lie on the boundary of two domains \(D_k\) and \(D_{k+1}\) of ranks respectively \(k,k+1\), \(k\ge 0\). Let \(y_1(\lambda_0),\ldots,y_r(\lambda_0)\) be the roots of the equation corresponding to \(D_k\) and not equal to \(f(\lambda_0)\); they coincide with the roots of the equation corresponding to \(D_{k+1}\) and not equal to \(f(\lambda_0)\). Denote by \(3\delta\) the quantity \(\min_{1\le j\le r}|f(\lambda_0)-y_j(\lambda_0)|\). Then, by continuity of the coefficients of the equation in \(\overline D_k\) (and in \(\overline D_{k+1}\)), there exists a neighborhood \(V\) of the point \(\lambda_0\) in which the roots of the equation differ by less than \(\delta\) from \(y_j(\lambda_0)\), \(1\le j\le k\). At every point \(z_1\in V\cap D_k\) at which the resultant of the equation is nonzero, there are exactly \(r\) roots (we shall call them roots of the first group) for which \(|y_j(z_1)-f(\lambda_0)|>2\delta\), \(1\le j\le r\), and \(k-r\) roots for which \(|y_j(z_1)-f(\lambda_0)|<\delta\), \(r<j\le k\). Selecting in a neighborhood of \(z_1\) analytic branches and carrying out analytic continuation inside \(V\cap D_k\) to the point \(z_2\in V\cap D_k\), we again pass from roots of the first group to roots of the first group. Hence it follows that the symmetric functions formed from the roots of the first group,
\(C'_1=-(y_1+\cdots+y_r),\ldots,C'_r=(-1)^r y_1\cdots y_r\), are single-valued analytic bounded functions in \(V\cap D_k\) (and in \(V\cap D_{k+1}\)) and coincide on \(V\cap\gamma\). Consequently they are analytic in \(V\), and the resultant of the equation
\(y^r+C'_1(z)y^{r-1}+\cdots+C'_r(z)=0\) has inside \(V\) a discrete number of zeros. Thus, if \(y_j(\lambda_0)=f(\lambda_0)\), then \((\lambda_0,y_j(\lambda_0))\in\Gamma\); if \(y_j(\lambda_0)\ne f(\lambda_0)\), then \((\lambda_0,y_j(\lambda_0))\) is an algebraic branch point of \(R\) (in particular, a regular point). The case when \(\lambda_0\in\widetilde M\) is investigated similarly. For every \(\lambda\in\gamma\) that is a boundary point of several domains, one of the roots of the equations corresponding to these domains is equal to \(f(\lambda)\); thus the boundary of \(R\) is the whole curve \(\Gamma\). Since \(\Gamma\) is a simple curve, \(R\) is connected. The theorem is proved.
Received
26 II 1960
References
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