\input amstex \documentstyle{amsppt} \magnification=1200 \hcorrection{.25in} \advance\voffset.25in \advance\vsize-.5in \topmatter \title Canonical systems and finite rank perturbations of spectra \endtitle \author Alexei G. Poltoratski\endauthor \thanks Research at MSRI is supported in part by NSF grant DMS-9022140.\endthanks \address MSRI, 1000 Centennial Drive, Berkeley, California 94720\endaddress \email agp\@msri.org \endemail \abstract We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or unitary operators. In the n-dimensional case, we prove that for any cyclic self-adjoint operator $A$, operator $A_\lambda= A + \Sigma_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$ iff operator $A_\eta=A+\eta(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\eta\in\Bbb R$ for $k=1,2,...,n$. We also show that if $A_\lambda$ is pure point for a.e.\ $\lambda\in \Bbb R^n$ then $A_\lambda$ is pure point for a.e.\ $\lambda\in \gamma$ for any analytic curve $\gamma\in\Bbb R^n$. \endabstract \subjclass 47A55, 30E20\endsubjclass \keywords finite rank perturbations of self-adjoint and unitary operators, canonical systems of measures \endkeywords \tolerance=1000 \font\twelvecmmi=cmmi12 \endtopmatter \document \heading Introduction \endheading This note analyzes the spectral properties of finite rank perturbations of self-adjoint and unitary operators. In Section 1 we will try to put some well-known results on rank one perturbations into a different prospective. The new approach will allow us to shorten some of the existing proofs. Section 2 deals with rank n perturbations and contains the main results of this paper. Our approach in Section 1 will be based on the notion of canonical systems of measures. This notion was introduced by Kolmogorov and Rokhlin [R], and, independently, by Ambrose, Halmos and Kakutani [A-H-K, H1-2] more than half a century ago. It was originally used to study partitions, homeomorphisms and factor spaces of various measure spaces. The term ``canonical'' belongs, most likely, to Kolmogorov. We will first discuss examples of canonical systems of measures naturally appearing in certain problems of Function Theory in the unit disk and Perturbation Theory. We will then use Rokhlin's uniqueness Theorem to show that different problems yield the same families of measures. In our first example of canonical systems (see Example 1 in Section 1) we are going to discuss families of measures on the unit circle generated by inner functions in the unit disk. Such families first appeared in [C] and then were further studied in [A1-5], [P1-3] and [S]. Our second example (see Example 2 in Section 1) deals with families of spectral measures of rank one perturbations of self-adjoint or unitary operators. These families were extensively studied by many authors. We refer to [A], [A-D] and [D] for the basic results on this subject, and to [J-L-R-S], [R-M-S], [P3] and [Si] for the latest developments and further references. As was proven by D. Clark in [C], any family of measures generated by an inner function (as in Example 1) is a family of spectral measures of all rank one perturbations of a model contraction. In Section 1 we will show how Rokhlin's Theorem can help establish this connection for families of spectral measures of cyclic self-adjoint or unitary operators. Section 2 is devoted to n-dimensional analogies of Simon-Wolff Theorem on the pure point spectrum of a random rank one perturbation. The one-dimensional result was applied in [S-W] to prove the the existence of Anderson localization for the wave propagation described by the discrete Schr\"odinger operator with random potential in one dimension. It is still an open question if Anderson localization takes place in dimension more than one. As we will find out, the n-dimensional case adds certain new and somewhat unexpected features to the general picture. First, using the observations made in Section 1, we will prove that for any cyclic self-adjoint operator $A$ and its cyclic vectors $\phi_1,...,\phi_n$, operator $A_\lambda=A+\sum_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$ iff operator $A_\eta=A+\eta(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\eta\in\Bbb R$ for $k=1,2,...,n$. I. e. random perturbation $A_\lambda$ is pure point almost surely iff it is pure point almost surely on the skeleton $S$ (the union of coordinate axes) in $\Bbb R^n$. Using the terminology of Function Theory, this result means that $S$ is, in some sense, a {\it sampling set} for this perturbation problem. We will also prove that if $A_\lambda$ is pure point for a.e.\ $\lambda\in \Bbb R^n$ then $A_\lambda$ is pure point for a.e.\ $\lambda\in \gamma$ for any analytic curve $\gamma\in\Bbb R^n$. This statement reveals a certain fine structure of the set $$E=\{\lambda\in\Bbb R^n| A_\lambda \text {is not pure point }\}.$$ Such a property of $E$ may seem surprising, since various examples of perturbations of singular spectra show that $E$ can be almost ``arbitrarily bad''. For instance, as follows from the results of [R-M-S] and [G], if the spectrum of $A$ contains an interval, then, even if $A_\lambda$ is pure point a. s., $E$ is everywhere dense and $G_\delta$ on every line parallel to the coordinate axes. The results of Section 2 allow some infinite-dimensional generalizations which are to be published elsewhere. \remark{\bf Acknowledgments} This work was done during my stay at the Mathematical Sciences Research Institute, and I would like to thank its administration and staff for their hospitality and support. \endremark \heading 1. Canonical systems of measures in function theory and functional analysis \endheading Let us start with the following \definition{Definition} Consider a measure space $(X, \Cal A, \mu)$ where $X$ is a set of points, $\Cal A$ is a $\sigma$-algebra of subsets of $X$ and $\mu$ is a finite measure on $\Cal A$. Space $(X, \Cal A,\mu)$ is called Lebesgue space if there exist a 1-1 measure preserving map from $(X, \Cal A, \mu_c)$ onto the unit interval with Lebesgue measure. \enddefinition Here $\mu_c$ denotes the continuous part of $\mu$ and the term "measure preserving" means in particular that the map induces a 1-1 correspondence between $\Cal A$ and Lebesgue $\sigma$-algebra on the unit interval. Let $\xi$ be a measurable partition of a Lebesgue space $(X, \Cal A, \mu)$: $\xi=\{S_\alpha\}_{\alpha\in Y}$ where $Y$ is some set of parameters, $S_\alpha\in\Cal A$, $S_\alpha\cap S_\beta=\emptyset$ if $\alpha\neq\beta$ and $\bigcup_Y S_\alpha=X$. Then we can consider the factor space $(X/\xi, \Cal A_\xi, \mu_\xi)$ where $\sigma$-algebra $\Cal A_\xi$ and measure $\mu_\xi$ are induced by $\Cal A$ and $\mu$ via the factor map. Suppose each set $S_\alpha$ from our partition $\xi$ is itself a Lebesgue space $(S_\alpha,\Cal A_\alpha, \mu_\alpha)$ for some $\sigma$-algebra $\Cal A_\alpha$ and measure $\mu_\alpha$. \definition{Definition} The system of measures $\{\mu_\alpha\}_{\alpha\in Y}$ is called canonical if for any $A\in\Cal A$ we have $A\cap S_\alpha\in\Cal A_\alpha$ for $\mu_\xi$-a.e.\ $\alpha$ and $$\mu(A)=\int_{X/\xi}\mu_\alpha(A\cap S_\alpha) d\mu_\xi(\alpha). \tag 1$$ \enddefinition Note that the factor space $X/\xi$ can be naturally identified with the set $Y$ of parameters $\alpha$. This justifies the expression ``$\mu_\xi$-a.e.\ $\alpha$'' and integration over $d\mu_\xi(\alpha)$ in (1). Suppose we have a measurable partition $\xi$ of $(X,\Cal A,\mu)$. Does there exist a corresponding canonical system of measures? If $\xi$ is countable the answer is obviously yes. In this case $\mu_\xi$ in (1) is discrete and $\mu_\alpha$ can be chosen as the restriction of $\mu$ on $S_\alpha$. However when $\xi$ is uncountable, most of $S_\alpha$ have measure 0 and the canonical system can not be constructed so easily. Nevertheless we have the following \proclaim{Theorem 1 (Rokhlin [R])} For any measurable partition $\xi$ of Lebesgue measure space $(X, \Cal A, \mu)$ there exists a canonical system $\{\mu_\alpha\}$ satisfying (1). Such a system is essentially unique i. e. for any two such canonical systems $\{\mu_\alpha\}$ and $\{\mu'_\alpha\}\ \ \ $ $\mu_\alpha=\mu'_\alpha$ for $\mu_\xi$-a.e.\ $\alpha$. \endproclaim In the rest of this section we are going to use the uniqueness part of this result to show how different problems in function theory and functional analysis yield the same families of measures. Note that condition $\bigcup S_\alpha=X$ can be replaced by a weaker condition $$\mu(\bigcup S_\alpha)=\mu(X)\tag 2$$ in all the above definitions and in Theorem 1. Our first example of a canonical system is related to Analytic Function Theory in the unit disk. \example{Example 1} Let $\theta$ be an inner function in the unit disk $\Bbb D$. We will denote by $M_+(\Bbb T)$ the set of all finite positive Borel measures on the unit circle $\Bbb T=\partial \Bbb D$. For each $\alpha\in\Bbb T$ function $\frac{\alpha+\theta}{\alpha-\theta}$ has positive real part in $\Bbb D$. Therefore for each $\alpha\in\Bbb T$ there exists $\mu_\alpha\in M_+(\Bbb T)$ such that its Poisson integral satisfies $$P\mu_\alpha=\int_{\Bbb T}\frac{1-|z|^2}{|\xi-z|^2} d\mu_\alpha(\xi) =Re\frac{\alpha+\theta}{\alpha-\theta} .\tag 3$$ We will denote by $M_\theta(\Bbb T)$ the family $\{\mu_\alpha\}_{\alpha\in\Bbb T}$ of all such measures corresponding to $\theta$. Note that since $Re\frac{\alpha+\theta}{\alpha-\theta}=0$ a.e.\ on $\Bbb T$, all measures $\mu_\alpha$ are singular. If $\theta$ is defined in the upper half plane $\Bbb C_+$ then one can replace the Poisson integral in (3) with the Poisson integral in $\Bbb C_+$ and define an analogous family $M_\theta(\hat\Bbb R)$ on the completed real line $\hat\Bbb R=\Bbb R\cup\{\infty\}$ (see [P3]). Families $M_\theta$ possess many interesting properties (see [A1-5], [C], [P1-3] and [Sa]). Our next task is to show that all such families are {\it canonical systems}. First let us notice that the defining formula (3) implies that measures $\mu_\alpha$ are concentrated on disjoint sets $S_\alpha=\{\xi\in\Bbb T | \theta(\xi)=\alpha\}$ (all analytic functions in this paper are defined on the boundary by their nontangential boundary values). Also if $\Cal P_z$ is a Poisson kernel for some $z\in\Bbb D$ then $$\int_{\Bbb T}\int_{\Bbb T} \Cal P_zd\mu_\alpha dm(\alpha)=\int_{\Bbb T} Re\frac{\alpha+\theta(z)}{\alpha-\theta(z)}dm(\alpha)=$$ $$=1=\int_{\Bbb T} \Cal P_z dm.\tag 4$$ Note that since linear combinations of Poisson kernels are dense in the space of all continuous functions on $\Bbb T$, one can replace $\Cal P_z$ in (4) with an arbitrary continuous function. Further, by taking the limit over continuous functions, one can replace $\Cal P_z$ in (4) with a characteristic function of any Borel set $B\subset \Bbb T$ to obtain the following analogy of formula (1) (see [A1]): $$\int_{\Bbb T} \mu_\alpha(B)dm(\alpha)=m(B).\tag 5$$ Therefore family $M_\theta$ defined by formula (3) is indeed a canonical system on $(\Bbb T, \Cal B, m)$ for the partition $S_\alpha=\{\theta=\alpha\}$ where $\Cal B$ is Borel $\sigma$-algebra on $\Bbb T$. The same argument holds in the case of the upper half plane for the families $M_\theta(\hat\Bbb R)$. \endexample Our second example concerns Perturbation Theory of self-adjoint and unitary operators. \example{Example 2} Let $A_0$ be a singular bounded cyclic self-adjoint operator acting on a separable Hilbert space, $\phi$ its cyclic vector. Consider the family of rank one perturbations of $A_0$: $$A_\lambda=A_0+\lambda(\cdot,\phi)\phi\ \ \ \ \ \lambda\in\Bbb R.$$ Let $\mu_\lambda$ be the spectral measure of $\phi$ for $A_\lambda$. We will denote by $M_{A_0,\phi}$ the family $\{\mu_\lambda\}_{\lambda\in\Bbb R}$. For basic results and references on families $M_{A_0,\phi}$ the reader can consult [A], [A-D], [D] or [S]. Here we will show that family $M_{A_0,\phi}=\{\mu_\lambda\}_{\lambda\in\Bbb R}$ is a {\it canonical system}. Note that Cauchy integral of $\mu_\lambda$ satisfies: $$K\mu_\lambda(z)=\int_{-\infty}^{\infty}\frac{d\mu_\lambda(t)} {t-z}=((A_\lambda-z)^{-1}\phi,\phi).\tag 6$$ Also for the resolvents of operators $A_\lambda$ we have: $$(A_0-z)^{-1}-(A_\lambda-z)^{-1}=\left[\lambda(\cdot,\phi) (A_\lambda-z)^{-1}\phi\right](A_0-z)^{-1}.\tag 7$$ Combining (6) and (7) we obtain $$K\mu_\lambda(z)=\frac{K\mu_0(z)}{1+\lambda K\mu_0(z)}\tag 8$$ (see [A]). Let us observe that, as follows from (6), measures $\mu_\lambda$ are concentrated on the sets $S_\lambda=\{x\in\Bbb R| ((A_\lambda-x)^{-1}\phi,\phi)= \infty\}$. Via (7) $S_\lambda$ can be redefined as $$S_\lambda= \{x\in\Bbb R| ((A_0-x)^{-1}\phi,\phi)= -\frac1\lambda\}$$ which shows that $S_\lambda$ are pairwise disjoint. To prove the integral formula notice that if $\Cal P_z=\frac y{(x-t)^2+y^2}$ is the Poisson kernel for $z=x+iy\in\Bbb C_+$ then $$\int_{-\infty}^\infty\int_{-\infty}^{\infty}\Cal P_z(t)d\mu_\lambda(t)d\lambda= \int_{-\infty}^\infty Re \frac{K\mu_0(z)}{1+\lambda K\mu_0(z)}d\lambda.$$ Since $\mu_\lambda$ is a positive measure, $\omega=-1/(K\mu_0(z))$ belongs to $\Bbb C_+$. Therefore $$\int_{-\infty}^\infty\int_{-\infty}^{\infty}\Cal P_z(t)d\mu_\lambda(t)d\lambda= \int_{-\infty}^{\infty}\Cal P_\omega(\lambda)d\lambda =\pi= \int_{-\infty}^{\infty}\Cal P_z(\lambda)d\lambda.$$ Since every continuous $L^1(\Bbb R)$ function can be approximated by linear combinations of Poisson kernels, in the same way as in the previous example we can conclude that $$\int_{-\infty}^{\infty}\mu_\lambda(B)d\lambda=|B|\tag 9$$ for any Borel set $B$ (see [S]). Therefore family $M_{A_0,\phi}$ is a canonical system on $(\Bbb R, \Cal B, dx)$ for the partition $S_\lambda=\{((A_0-x)^{-1}\phi,\phi)=-\frac 1\lambda\}$, where $\Cal B$ denotes Borel $\sigma$-algebra. An analogous argument can be applied in the case of unitary perturbations. Let $U_1$ be a unitary cyclic singular operator, $v, ||v||=1$ its cyclic vector. Consider the family of unitary rank one perturbations of $U_1$ : $$U_\alpha= U_1+(\alpha-1)(\cdot , U_1^{-1}v)v, $$ $\alpha\in\Bbb T$. For the resolvents we have $$(U_1-z)^{-1}-(U_\alpha-z)^{-1}= (U_\alpha-z)^{-1}[(\alpha-1)(\cdot , U_1^{-1}v)v] (U_1-z)^{-1} .$$ Denote by $\nu_\alpha$ the spectral measure of $v$ for $U_\alpha$. Let $ K\nu_\alpha$ be the Cauchy integral of measure $\nu_\alpha$ in the unit disk $\Bbb D$: $$K\nu_\alpha=\int_{\Bbb T}\frac{1}{1-\overline\xi z}d\nu_\alpha(\xi).$$ Since $$ K\nu_\alpha=((U_\alpha-z)^{-1}v , U_\alpha^{-1}v)$$ for $z\in\Bbb D$, we have that $$ K\nu_\alpha=\frac{\alpha K\nu_1} {1+ (\alpha-1) K\nu_1} .\tag 10 $$ Using (10) in the same way as we used (7) we can obtain that family $M_{U_1,v}=\{\nu_\alpha\}_{\alpha\in\Bbb T}$ is a canonical system on $(\Bbb T, \Cal B, m)$ for the partition $$ S_\alpha=\{((U_1-z)^{-1}v , U_1^{-1}v)=\frac 1{1-\alpha}\}.$$ \endexample Now we will use Rokhlin's Theorem to show that every family of spectral measures of rank one perturbations (Example 2) is in fact a family generated by an inner function (Example 1) and vice versa. Indeed, if we consider the following inner function in $\Bbb C_+$: $$\theta(z)=\frac {1-i((A_0-z)^{-1}\phi,\phi)}{1+i((A_0-z)^{-1}\phi,\phi)} \tag 11$$ then the partitions from Examples 1 and 2 coincide: $S_\alpha=S_\lambda$ for $\alpha=\frac {\lambda+i}{\lambda-i}$. Hence by Rokhlin Theorem canonical system $M_\theta$ essentially coincides with canonical system $M_{A_0,\phi}$. But in both systems the measures depend on parameter continuously (with respect to the $*$-weak topology of the space of measures). Since the systems coincide {\it almost everywhere} and are continuous, they must coincide {\it everywhere}: $$M_\theta=M_{A_0,\phi}.\tag 12$$ Conversely, for each inner $\theta$ we can choose $A_0$ to be the operator of multiplication by $z$ in $L^2(\mu_1), \ \ \mu_1\in M_\theta$ and $\phi=1\in L^2(\mu_1)$. Then $\theta$, $A_0$, and $\phi$ will satisfy (12). In a similar way, for unitary operators, if we choose $$\theta(z)=\frac 1{K\nu_1(z)}-1$$ for any $z\in\Bbb D$ then we will have $$M_\theta=M_{U_1,v}.\tag 13$$ with $\mu_\alpha=\nu_\alpha$ for any $\alpha\in\Bbb T$, $\mu_\alpha\in M_\theta (\Bbb T)$ and $\nu_\alpha\in M_{U_1,v}$ (cf. [C]). The families of measures from Examples 1 and 2 are not the only natural examples of canonical systems. For instance, in [D] one can find two more families of operators: the family of all self-adjoint extensions of a given symmetric operator with deficiency indices (1,1) and the family of all self-adjoint extensions associated with a given limit-point Stourm-Liouville problem. Applying the argument, similar to the one we used here, one can prove that the corresponding families of spectral measures are canonical systems generated by inner functions. If $A_0$ and $\phi$ are as in Example 2, then $A_i=A_0+i(\cdot,\phi)\phi$ is a dissipative operator. Its Cayley transform $T$ is a completely nonunitary $C_0$ contraction. Sz.Nagy-Foias model theory provides $T$ with a characteristic function $I$ in the unit disk. Function $I$ is related to our function $\theta$, defined by (11), via the equation $I(z)=\theta(\omega(z))$ where $\omega$ is the standard conformal map from the unit disk onto the upper half-plane. \heading 2. Finite rank perturbations of singular spectra \endheading In this section we will need the following Lemma, which can be obtained directly from the definition of families $M_\theta$: %and also immediately follows from the results of [C]. \proclaim{Lemma 2} Let $\theta$ be an inner function in $\Bbb D$ and $\xi\in\Bbb T$. Then 1) measure $\mu_\alpha\in M_\theta$ has a point mass at $\xi$ iff functions $\theta$ and $\theta'$ have non-tangential limits $\theta(\xi)$ and $\theta'(\xi)$ at $\xi$ and $\theta(\xi)=\alpha$; 2) function $\theta'$ has a non-tangential limit $\theta'(\xi)$ at $\xi$ iff there exists $\mu_\beta\in M_\theta$ such that $$\int_\Bbb T\frac{d\mu_\beta(\omega)}{|\xi-\omega|^2}<\infty;\tag 14$$ 3) measure $\mu_\alpha\in M_\theta$ has a point mass at $\xi$ iff the non-tangential limit $\theta(\xi)$ is equal to $\alpha$ and (14) holds for any $\beta\in\Bbb T,\ \beta\neq\alpha$. \endproclaim An analogous statement holds true for families $M_\theta$ on the real line. Suppose $\theta'(\xi)$ (the non-tangential limit of $\theta'$ at $\xi$) exists everywhere on $\Bbb T$ except for $\xi\in E, \ E\subset\Bbb R$. If $m(E)=0$ then, by formula (5), for measures from $M_\theta$ we have $\mu_\alpha(E)=0$ for almost every $\alpha\in\Bbb T$. Since $\mu_\alpha$ is concentrated on $\{\theta=\alpha\}$, statement 1) from Lemma 2 implies that $\mu_\alpha$ is pure point for a.e.\ $\alpha$. Now we can use statement 2) from Lemma 2 and connections with operator theory discussed in the previous section (formula (12)) to obtain the following \proclaim{Theorem 3 [S-W]} Let $A$ be a cyclic self-adjoint operator, $\phi$ its cyclic vector and $\mu$ the spectral measure of $\phi$ for $A$. Then operator $$A_\lambda=A+\lambda(\cdot,\phi)\phi$$ is pure point for a.e.\ $\lambda\in\Bbb R$ iff $$\int_\Bbb R\frac{d\mu(x)}{|x-y|^2}<\infty\tag 15$$ for a.e.\ $y\in\Bbb R$. \endproclaim The purpose of this section is to find an n-dimensional analogy of this result. We will prove the following \proclaim{Theorem 4} Let $A$ be a cyclic self-adjoint operator, $\phi_1,\phi_2,...,\phi_n$ its cyclic vectors. The following two conditions are equivalent: 1) operator $$A_\lambda=A+\sum_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$$ is pure point for a.e.\ $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$. 2) operator $$A_\lambda=A+\lambda(\cdot,\phi_k)\phi_k$$ is pure point for a.e.\ $\lambda\in\Bbb R$ for $k=1,2,...,n$; i. e. $$\int_\Bbb R\frac{d\mu_k(x)}{|x-y|^2}<\infty$$ for a.e.\ $y\in\Bbb R$ for $k=1,2,...,n$, where $\mu_k$ is the spectral measure of $\phi_k$ for $A$. \endproclaim The theory of unitary rank n perturbations is slightly more complicated due to the fact that a sum of two unitary perturbations may not be unitary. Because of that, we will have to define the rank n perturbation of a unitary operator $U$ corresponding to vectors $\phi_1,...,\phi_n$ recursively. I. e., let $\alpha=(\alpha_1,...,\alpha_n)\in\Bbb T^n$. Denote $$\alpha^1=\alpha_1\in\Bbb R,\ \alpha^2=(\alpha_1,\alpha_2)\in\Bbb R^2,...$$ $$...,\alpha^{n-1}=(\alpha_1,...,\alpha_{n-1})\in\Bbb R^{n-1}, \ \ \text {and}\ \ \alpha^{n}=\alpha=(\alpha_1,...,\alpha_{n})\in\Bbb R^{n}.$$ Put $$U_{\alpha^1}=U+(\alpha_1-1)(\cdot,U^{-1}\phi_1)\phi_1,$$ and $$U_{\alpha^k}=U_{\alpha^{k-1}}+(\alpha_k-1)(\cdot,U_{\alpha^{k-1}} ^{-1}\phi_k)\phi_k$$ for $k=2,3,...,n$. Finally, denote $U_\alpha=U_{\alpha^n}$. If vectors $\phi_1,\phi_2,...,\phi_n$ are pairwise orthogonal, then our definition yields $$U_\alpha=U+\sum_{k=1}^n\alpha_k(\cdot,U^{-1}\phi_k)\phi_k.$$ One can show that the following Theorem is equivalent to Theorem 4: \proclaim{Theorem 4'} Let $U$ be a cyclic unitary operator, $\phi_1,\phi_2,...,\phi_n$ its cyclic vectors. Then the following two conditions are equivalent: 1) operator $U_\alpha$ (defined as above) is pure point for a.e.\ $\alpha\in\Bbb T^n$; 2) operator $U_\alpha=U+\alpha(\cdot,U^{-1}\phi_k)\phi_k$ is pure point for a.e.\ $\alpha\in\Bbb T$ for $k=1,2,...,n$. \endproclaim Since it is more convenient for us to operate in the unit disk, we will prove the result in the form of Theorem 4' rather than Theorem 4. Our proof will be based on the following approach. Let $U_1$ be a cyclic singular unitary operator. Suppose $\phi_1$ and $\phi_2$ are its cyclic vectors, $||\phi_1||=||\phi_2||=1$. Let $\mu_\alpha$ be the spectral measure of $\phi_1$ for $$U_\alpha=U_1+(\alpha-1)(\cdot,U_1^{-1}\phi_1)\phi_1.\tag 16$$ Then, as was shown in Section 1, there exists an inner function $\theta, \ \theta(0)=0$ such that $\{\mu_\alpha\}_{\alpha\in\Bbb T}=M_\theta$. By the Spectral Theorem, each operator $U_\alpha$ is unitarily equivalent to the operator $Y_\alpha$ of multiplication by $z$ in $L^2(\mu_\alpha)$. We will denote by $\theta^*(H^p), p>0$ the invariant subspace of the backward shift operator in $H^p$ corresponding to $\theta$: $$\theta^*(H^p)=\overline{Span}_{H^p} \left[ \left\{\frac{\theta(\lambda)-\theta(z)}{\lambda-z} \right\}_{\lambda\in\Bbb D}\right].$$ In the case $p=2$ we have $\theta^*(H^2)=H^2\circleddash \theta H^2$. The following operator $V_\alpha$ acting from $L^2(\mu_\alpha)$ to $\theta^*(H^2)$ was studied in [A2], [C] and [P1]: $$V_\alpha f=\frac{Kf\mu_\alpha}{K\mu_\alpha}.$$ We will need the following \proclaim{Theorem 5 [C]} For each $\alpha\in\Bbb T$ operator $V_\alpha$ maps $L^2(\mu_\alpha)$ onto $\theta^*(H^2)$ unitarily. Operator $V_\alpha$ sends operator $Y_\alpha$ of multiplication by $z$ in $L^2(\mu_\alpha)$ into the following operator $T_\alpha$ in $\theta^*(H^2)$: $$T_\alpha= V_\alpha Y_\alpha V^*_\alpha f=z\left(f-\left(f,\frac\theta z\right)\frac\theta z\right) +\left(f,\frac\theta z\right)\alpha.\tag 17$$ \endproclaim Operators $T_\alpha$ are all possible unitary rank one perturbations of the model contraction $T_\theta=P_\theta S$, where $S$ is the operator of multiplication by $z$ in $\theta^*(H^2)$ and $P_\theta$ is the orthogonal projector from $H^2$ to $\theta^*(H^2)$ (see [C]). Also note, that in our situation $\theta(0)=0$, otherwise (17) would not be valid. The conjugate operator $V_\alpha^*$ in (17) can also be defined quite naturally: \proclaim{Theorem 6 [P1]} For each function $f\in\theta^*(H^2)$ and each $\alpha\in\Bbb T$ the non-tangential boundary values of $f$ exist $\mu_\alpha$-a.e.\ Function $V_\alpha^*f\in L^2(\mu_\alpha)$ coincides with the non-tangential boundary values of $f$ $\mu_\alpha$-a.e.\ \endproclaim Via the original Fourier transform and formula (17), each unitary rank one perturbation $U_\alpha$ given by (16) is identified with an operator in $\theta^*(H^2)$. Vector $\phi_1$ corresponds to $1\in\theta^*(H^2)$ and vector $\phi_2$ corresponds to some function $f \in\theta^*(H^2)$. As $\mu_\alpha$ was chosen to be the spectral measure of $\phi_1$ for $U_\alpha$, the measure $|f|^2\mu_\alpha$ is the spectral measure of $\phi_2$ for $U_\alpha$ (note that by Theorem 6 the non-tangential boundary values of $|f|^2$ exist $\mu_\alpha$-a.e., therefore the notation $|f|^2\mu_\alpha$ makes sense). Lemma 2 and Theorem 3 imply that operator $U_{(\alpha,\beta)}=U_\alpha+\beta(\cdot,U_\alpha^{-1} \phi_2)\phi_2$ is pure point for a.e.\ $\beta\in\Bbb T$ iff the (non-tangential boundary values of the) derivative of $K|f|^2\mu_\alpha$ exist a.e.\ on $\Bbb T$. We will use this fact in the proof of Theorem 4'. By one of the properties of $\theta^*(H^2)$, if $f\in\theta^*(H^2)$ is equal to 0 at 0, then the function $\hat f$ such that $\hat f=\theta \overline f$ a.e.\ on $\Bbb T$ also belongs to $\theta^*(H^2)$. We will need the following \proclaim{Lemma 7} Let $f\in\theta^*(H^2),||f||=1, f_0=f-f(0)$. Then 1) there exist $g, h\in \cap_{0\frac 12\tag 30$$ in $\Bbb D$ since fraction $$\frac{\overline {I_1(\xi)} g(z)+ h(z)} {1-\overline{I_1(\xi)}\theta(z)}\tag 31$$ is the Cauchy integral of a probability measure (namely it is the Cauchy integral of the spectral measure of $\phi_2$ for $U+I_1(\xi)(\cdot,U^{-1}\phi_1)\phi_1$, see formula (19)). Now if we fix $z\in\Bbb D$ then fraction (31) is a bounded antianalytic function of $\xi$ in $\Bbb D$. Hence (30) must hold true for every $\xi\in\Bbb D$ because it holds a.e.\ on $\Bbb T$. This implies that for any fixed $z\in\Bbb D$ the denominator $$\overline{I_2(\xi)}+(1-\overline{I_2(\xi)}) \frac{\overline {I_1(\xi)} g(z)+ h(z)} {1-\overline{I_1(\xi)}\theta(z)}$$ on the right hand side of (29) is an antianalytic function in $\Bbb D$ whose absolute value is bounded away from 0. Hence for every $z\in\Bbb D$ the whole fraction on the right hand side of (29) is a bounded antianalytic function of $\xi$ in $\Bbb D$. Thus $$\int_\Bbb T\int_\Bbb T \Cal K_z(\omega) d\nu_{(I_1(\xi),I_2(\xi))}(\omega)dm(\xi)= \frac{\overline{I_1(0)}\frac{g(z)+ h(z)}{1-\overline{I_1(0)}\theta(z)}} {\overline{I_2(0)}+(1-\overline{I_2(0)}) \frac{\overline {I_1(0)} g(z)+ h(z)} {1-\overline{I_1(0)}\theta(z)}}=\tag 32$$ $$=\varphi(z)=\int_{\Bbb T} \Cal K_z\varphi dm.$$ Inequality (30) implies that $|\varphi|\leq\frac 1{1-|I_2(0)|}$. Proceeding as in Examples 1 and 2 from the previous section, we can replace $\Cal K_z$ in (32) with the characteristic function of any Borel set $B\subset \Bbb T$ to obtain (26). $\blacktriangle$\enddemo We are now ready to generalize $1)\Rightarrow 2)$ part of Theorems 4 and 4' in the following way: \proclaim{Theorem 9} Let $A$ be a cyclic self-adjoint operator, $\phi_1,\phi_2,...,\phi_n$ its cyclic vectors. Let $\gamma$ be an analytic curve in $\Bbb R^n$. Suppose that operator $$A_\lambda=A+\sum_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$$ is pure point for a.e.\ $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$. Then $A_{\gamma(t)}$ is pure point for a.e.\ $t\in\Bbb R$. \endproclaim \proclaim{Theorem 9'} Let $U$, $\phi_1,\phi_2,...,\phi_n$ and $U_\alpha$ be the same as in Theorem 4'. Let $\gamma$ be an analytic curve in $\Bbb T^n$. Suppose that operator $U_\alpha$ is pure point for a.e.\ $\alpha=(\alpha_1,\alpha_2,...,\alpha_n) \in\Bbb T^n$. Then $U_{\gamma(\xi)}$ is pure point for a.e.\ $\xi\in\Bbb T$. \endproclaim \demo{Proof of Theorem 9' for n=2} If $\nu_{(I_1(\xi),I_2(\xi))}$ is the spectral measure of $\phi_2$ for $U_{(I_1(\xi),I_2(\xi))}$ then, as we showed in the previous proof, its Cauchy integral must satisfy $$K\nu_{(I_1(\xi),I_2(\xi))}= \frac{\frac{\overline{I_1(\xi)}g+ h}{1-\overline{I_1(\xi)}\theta}} {\overline{I_2(\xi)}+(1-\overline{I_2(\xi)}) \frac{\overline {I_1(\xi)} g+ h} {1-\overline{I_1(\xi)}\theta}}\tag 33$$ By Lemma 2 to show that $\nu_{(I_1(\xi),I_2(\xi))}$ is pure point it is enough to show that function $$\frac{\overline {I_1(\xi)} g+ h} {1-\overline{I_1(\xi)}\theta}\tag 34$$ has a non-tangential derivative $\nu_{(I_1(\xi),I_2(\xi))}$-a.e.\ Denote by $E$ the subset of $\Bbb T$ where the non-tangential derivative of (34) does not exist. 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