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Edit File: Berezan_MAA2007_01.04.08.tex

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%\volume{228}
%\Copyrightyear{2004}
%\DOI{003-0001}
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%\seriesextra{Just an add-on}
%\seriesextraline{This is the Concrete Title of this Book\br H.E. R and S.T.C. W, Eds.}
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%\Volumeandyear{1 (2004)}
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%\commby{inhouse}
%\submitted{March 14, 2003}
%\received{March 16, 2000}
%\revised{June 1, 2000}
%\accepted{July 22, 2000}
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\title[Spectral theory of block Jacobi type matrices]
 {Spectral theory of infinite block Jacobi type normal matrices,
 orthogonal polynomials on complex domain and complex moment problem}
%----------Author 1
\author[Yu. M. Berezansky]{Yu. M. Berezansky}

\address{%
Institute of Mathematics\\
National Academy of Science of Ukraine\\
3 Tereshchenkivs'ka St.\\
01601 Kyiv\\
Ukraine}

\email{berezan@mathber.carrier.kiev.ua}

%\thanks{1}
%----------Author 2
%\author{A Second Author}
%\address{The address of\br
%the second author\br
%sitting somewhere\br
%in the world}
%\email{dont@know.who.knows}
%----------classification, keywords, date
\subjclass{Primary 44A60, 47A57, 47A70}

\keywords{Block Jacobi matrix, generalized eigenvector, orthogonal
polynomials, direct and inverse spectral problems, moment problem}

\date{12.04.2008}
%----------additions
\dedicatory{To the memory of M.G.Krein}
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\begin{abstract}
In this survey we describe the new results concerning spectral
theory of normal block Jacobi matrices and corresponding questions
on complex moment problem and orthogonal polynomials.
\end{abstract}

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\maketitle
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%\tableofcontents
\section{Introduction}
In this talk we propose an analog of the Jacobi matrix related to
complex moment problem and to a system of polynomials orthogonal
with respect to some probability measure on the complex plane. Such
a matrix has a block three-diagonal structure and gives rise to a
normal operator acting on a space of $\ell_2$ type.

Roughly speaking, such results are a generalization of classical
theory of Jacobi Hermitian matrices on the case of normal operators.
They are deeply connected with some works of M.G.Krein
(\cite{33,34}, 1948-1949 years) devoted to spectral approach for the
proof of integral representation of positive definite kernels and
Jacobi matrices with operator-valued elements.

The results of the talk devoted to complex moment problem and are
connected with many numbers of works starting from 1957 year:
Y.Kilpi \cite{32}, N.I. Akhiezer \cite{1}, A.Atzmon \cite{2},
C.Berg, J.P.R.Christensen, P.Ressel \cite{21}, T.M. Bisgaard
\cite{22}, J.Stochel, F.H.Szafraniec \cite{39} and other
mathematicians who are cited in the references of above mentioned
articles.

The results devoted to unitary block Jacobi matrices are connected
with new works on orthogonal polynomials on unit circle, in
particular, with works of M.J. Cantero, L. Moral, L. Vel\'{a}zques
(\cite{23}, 2003), B. Simon (\cite{38}, 2005), L.B. Golinskii
(\cite{30}, 2006).

Some results of this talk were obtained together with M.E. Dudkin.

\section{Classical Jacobi matrices and moment problem, orthogonal polynomials on the
axis} \label{sec:1}

$1^0.$ I remind at first the corresponding classical situation.

In the classical theory it is investigated in the space $\ell_2$ of
sequences $f=(f_n)^\infty_{n=0}, f_n \in \mathbb{C},$ the Hermitian
Jacobi matrix:
\begin{equation}\label{1.1}
J=
\begin{bmatrix}
b_0 & a_0 & 0 & 0 & 0 & \ldots \\
a_0 & b_1 & a_1 & 0 & 0 & \ldots \\
0 & a_1 & b_2 & a_2 & 0 & \ldots \\
\vdots & \vdots & \vdots & \vdots& \vdots & \\
\end{bmatrix}, b_n \in \mathbb{R}, a_n > 0, n \in
\mathbb{N}_0=\{0,1,2,\ldots\}.
\end{equation}
This matrix generates on finite sequences $f \in \ell_{fin}$ the
operator on $\ell_2,$  which is Hermitian with equal defect numbers
and therefore has a selfadjoint extension on $\ell_2.$ Under some
conditions on $J$ (for example,
$\sum\limits^\infty_{n=0}\frac{1}{a_n}=\infty$) the closure
$\widetilde{J}$ of $J$ is selfadjoint.

The direct spectral problem, i.e. the eigenfunction expansion for
$\widetilde{J}$ (or for some selfadjoint extension of $J$), is
constructed in the following way (for simplicity we will assume that
$\widetilde{J}$ is selfadjoint).

We introduce $\forall \lambda \in \mathbb{R}$ the sequence of
polynomials
\begin{equation*}
P(\lambda)=\big(P_n(\lambda)\big)^\infty_{n=0} \in \ell =
\mathbb{C}^\infty
\end{equation*}
as a solution of the equation
\begin{equation}\label{1.2}
\begin{split}
& JP(\lambda)=\lambda P(\lambda), P_0(\lambda)=1,\,\text{i.e.}
\forall n \in \mathbb{N}_0 \\
&
a_{n-1}P_{n-1}(\lambda)+b_nP_n(\lambda)+a_nP_{n+1}(\lambda)=\lambda
P_n(\lambda),\\
& P_{-1}(\lambda)=0, P_0(\lambda)=1.
\end{split}
\end{equation}
The solution of this recurrence exists: it is necessary step by step
go, starting from $P_0(\lambda);$ such procedure is possible,
because all $a_n >0.$

The sequence of polynomials $P(\lambda)$ is a generalized
eigenvector for $\widetilde{J}$ with eigenvalue  $\lambda$ (we use
some quasinuclear rigging of the space $H=\ell_2:$
\begin{equation}\label{1.3}
H_- \supset H_0 \supset H_+, P(\lambda)\in H_-).
\end{equation}
The corresponding Fourier transform $F=\widehat{}$ \, is:
\begin{equation}\label{1.4}
\ell_2 \supset \ell_{fin} \ni f=(f_n)^\infty_{n=0} \mapsto
\widehat{f}(\lambda)=\sum\limits^\infty_{n=0}f_nP_n(\lambda) \in
L^2(\mathbb{R},d\rho(\lambda))=:L^2.
\end{equation}

This mapping is an unitary operator (after closure) between $\ell_2$
and $L^2.$ Image of $\widetilde{J}$ is the operator of
multiplication on $\lambda$ on the space $L^2.$ The polynomials
$P_n(\lambda)$ are orthonormal w.r.t. spectral measure
$d\rho(\lambda):$
\begin{equation}\label{1.5}
\int\limits_{\mathbb{R}}P_j(\lambda)P_k(\lambda)d\rho(\lambda)=\delta_{j,k},\,
j,k \in \mathbb{N}_0.
\end{equation}
Note that \eqref{1.5} is a consequence from Parseval equality which
takes place for mapping \eqref{1.4}:
\begin{equation}\label{1.6}
\forall f,g \in \ell_{fin}\quad
(f,g)_{\ell_2}=\int\limits_{\mathbb{R}}\widehat{f}(\lambda)\overline{\widehat{g}(\lambda)}d\rho(\lambda).
\end{equation}
$2^0.$ The inverse spectral problem in this classical case is
following. Let us have a Borel probability measure $d\rho(\lambda)$
on $\mathbb{R}$ for which all moments $s_n$ exist:
\begin{equation}\label{1.7}
s_n=\int\limits_{\mathbb{R}}\lambda^n d\rho(\lambda),
n\in\mathbb{N}_0
\end{equation}
(and support of $d\rho(\lambda)$ contains an infinite set on finite
interval).

The question is: is it possible to recover corresponding Jacobi
matrix $J$ in such manner, that the initial measure $d\rho(\lambda)$
is equal to spectral measure for $\widetilde{J}$? What way is for
such reconstruction?

The answer is simple: it is necessary to take the sequence of
functions
\begin{equation}\label{1.8}
1, \lambda, \lambda^2, \ldots \in L^2
\end{equation}
(which are linearly independent) and apply to it the classical
procedure of orthogonalization (by Schmidt). As result, we get the
sequence of orthonormal polynomials
\begin{equation}\label{1.9}
P_0(\lambda)=1, P_1(\lambda), P_2(\lambda), \ldots.
\end{equation}
Then the matrix $J$ is reconstructed by formulas: $\forall
n\in\mathbb{N}_0$
\begin{equation}\label{1.10}
a_n=\int\limits_{\mathbb{R}}\lambda
P_n(\lambda)P_{n+1}(\lambda)d\rho(\lambda),
b_n=\int\limits_{\mathbb{R}}\lambda
\big(P_n(\lambda)\big)^2d\rho(\lambda).
\end{equation}
$3^0.$ The classical moment problem. The question is the following:
we have a sequence $s=(s_m)^\infty_{m=0},s_m \in\mathbb{R};$ when a
finite Borel measure $d\rho(\lambda)$ exists such that:
\begin{equation}\label{1.11}
s_m=\int\limits_{\mathbb{R}}\lambda^m d\rho(\lambda),
m\in\mathbb{N}_0 \,(\text{i.e.} \eqref{1.7}).
\end{equation}
The answer: iff $\forall f=(f_j)^\infty_{j=0}\in\ell_{fin}$
\begin{equation}\label{1.12}
\sum\limits^\infty_{j,m=0}s_{j+m}f_j\overline{f_m}\geq 0.
\end{equation}

It is known that this result is deeply connected with spectral
theory of Jacobi matrices (see below, in the second part of talk).

Now we will explain in what manner it is possible to get the
representation \eqref{1.11} if condition \eqref{1.12} is fulfilled.

In accordance with \eqref{1.12} we introduce (quasi) scalar product
\begin{equation}\label{1.13}
(f,g)_S=\sum\limits^\infty_{j,m=0}s_{j+m}f_j\overline{g_m}, f,g\in
\ell_{fin},
\end{equation}
and construct in the usual way corresponding Hilbert space $S.$ The
shift operator:
\begin{equation*}
\forall f\in \ell_{fin}\quad (Tf)_j=f_{j-1}, j\in\mathbb{N}_0
\,(f_{-1}=0)
\end{equation*}
is (as it is easy to understand) Hermitian on $S$ with equal defect
indexes. Therefore this operator has a selfadjoint extension
$\widetilde{T}$ in $S.$

Further we construct the generalized eigenvector expansion of this
operator on the space $S.$ For this aim it is necessary to introduce
the quasinuclear rigging of the space $S$ of type \eqref{1.3}. The
simple calculation shows that now the generalized eigenvector
$P(\lambda),\lambda\in\mathbb{R},$ has the form:
\begin{equation}\label{1.14}
P(\lambda)=(1,\lambda,\lambda^2,\ldots) \in\ell,
\lambda\in\mathbb{R}.
\end{equation}
The Fourier transform is:
\begin{equation}\label{1.15}
S\supset\ell_{fin}\ni
(f_j)^\infty_{j=0}=f\mapsto\widehat{f}(\lambda)=\sum\limits^\infty_{j=0}f_j\lambda^j
\in L^2(\mathbb{R},d\rho(\lambda))=:L^2.
\end{equation}
Here $d\rho(\lambda)$ is the spectral measure of operator
$\widetilde{T}.$ The Parseval equality has the form \eqref{1.6}, as
earlier.

From \eqref{1.15} we conclude that for vector
$\delta_n=(0,\ldots,0,\underset{n}{1},0,0,\ldots)$ the Fourier
transform $\widehat{\delta}_n=\lambda^n.$ Therefore the Parseval
equality \eqref{1.6} gives the required representation \eqref{1.11}:
\begin{equation*}
s_m=(\delta_m,\delta_0)_S=(\widehat{\delta}_m,\widehat{\delta}_0)_{L^2}=\int\limits_{\mathbb{R}}\lambda^m
d\rho(\lambda), m \in\mathbb{N}_0.
\end{equation*}

\section{Complex moment problem}\label{sec:2}
$1^0.$ The transfer from classical Jacobi matrices to the normal
Jacobi type block matrices we will start from corresponding moment
problem.

In the simplest case the problem is following. Instead of sequence
\\ $s=(s_m)^\infty_{m=0}, s_m \in \mathbb{R},$ we have the sequence
\begin{equation*}
s=(s_{m,n})^\infty_{m,n=0}, s_{m,n}\in\mathbb{C}.
\end{equation*}
The question is: under what conditions a finite Borel measure
$d\rho(z)$ on $\mathbb{C}$ exists such that
\begin{equation}\label{2.1}
s_{m,n}=\int\limits_{\mathbb{C}}z^m\overline{z}^nd\rho(z), \,
m,n\in\mathbb{N}_0.
\end{equation}

In this case also some conditions of positiveness of type
\eqref{1.12} play essential role. So, similar to classical moment
problem we use the sequence
\begin{equation*}
f=(f_{j,k})^\infty_{j,k=0}, f_{j,k}\in\mathbb{C};
\end{equation*}
$\ell=(\mathbb{C}^2)^\infty$ is the set of all such sequences,
$\ell_{fin}$ denotes the set of finite sequence from $\ell.$

The condition of positiveness, analogical to \eqref{1.12}, is the
following:
\begin{equation}\label{2.2}
\sum\limits^\infty_{j,k,m,n=0}s_{j+n,k+m}f_{j,k}\overline{f}_{m,n}\geq
0, \, f\in\ell_{fin}
\end{equation}
(note that the disposition of indexes in this sum near $s,f$ and
$\overline{f}$ is essential). As for classical moment problem we
introduce the scalar product, connected with \eqref{2.2}:
\begin{equation}\label{2.3}
(f,g)_S=\sum\limits^\infty_{j,k,m,n=0}s_{j+n,k+m}f_{j,k}\overline{g}_{m,n},
\, f,g \in \ell_{fin},
\end{equation}
and construct the corresponding Hilbert space $S.$

Now we consider two operators $T$ and $T^+$ on the space $S,$ acting
on $f\in\ell_{fin}$ by rules:
\begin{equation}\label{2.4}
(Tf)_{j,k}=f_{j,k-1}, (T^+f)_{j,k}=f_{j-1,k}, \,j,k\in\mathbb{N}_0;
f_{-1,k}=f_{j,-1}=0.
\end{equation}
It is clear that $T$ is formally normal: algebraically
\begin{equation*}
TT^+=T^+T.
\end{equation*}

Under some additional conditions of growth $s_{m,n}, m,n \to
\infty,$ it is possible to assert that the closure $\widetilde{T}$
is a normal operator and we can apply to our situation the scheme
similar to Section \ref{sec:1}.$3^0$ but for normal (instead of
selfadjoint) operator. This way gives the following result.
\begin{thm}\label{thm:1}
Consider the sequence $s=(s_{m,n})^\infty_{m,n=0}, s_{m,n}
\in\mathbb{C}.$ If such finite Borel measure $d\rho(\lambda)$ on
$\mathbb{C}$ exists, that \eqref{2.1}
\begin{equation*}
s_{m,n}=\int\limits_{\mathbb{C}}z^m\overline{z}^nd\rho(z), \, m,n
\in \mathbb{N}_0,
\end{equation*}
then the condition \eqref{2.2} of positiveness is fulfilled.
Conversely, if for $s$  the condition \eqref{2.2} is fulfilled and
\begin{equation}\label{2.5}
\sum\limits^\infty_{p=1}\frac{1}{\sqrt[2p]{s_{2p,2p}}}=\infty,
\end{equation}
then the representation \eqref{2.1} takes place.
\end{thm}

a) Explain that the condition \eqref{2.5} gives the normality (not
only formal normality) of the operator $T.$ For formally normal
operator in general it is impossible to assert that such operator
can be extended to normal operator (unlike to Hermitian operators
with equal defect numbers and and selfadjoint operators). Therefore
the conditions of type \eqref{2.5} it is necessary to assume.

b) The proof of Theorem \ref{thm:1} is analogical to classical
moment problem, but now it is necessary to use the generalized
eigenfunctions expansion for normal operator (instead of selfadjoint
one).

Now instead of \eqref{1.14} we have the following generalized
eigenvector:
\begin{equation}\label{2.6}
P(z)=(z^m\overline{z}^n)^\infty_{m,n=0}\in \ell, \,z\in\mathbb{C} \,
\text{is eigenvalue}.
\end{equation}
The corresponding Fourier transform of type \eqref{1.15} is:
\begin{equation}\label{2.7}
S\supset \ell_{fin}\ni (f_{j,k})^\infty_{j,k=0}=f \mapsto
\widehat{f}(z)=\sum\limits^\infty_{j,k=0}f_{j,k}z^j\overline{z}^k
\in L^2(\mathbb{C},d\rho(z))=:L^2.
\end{equation}

c) In classical situation the sequence $\eqref{1.8}\, 1,\lambda,
\lambda^2, \ldots \in L^2=L^2(\mathbb{R}, d\rho(\lambda))$ is
connected with moments $\eqref{1.7}\,
s_n=\int\limits_{\mathbb{R}}\lambda^nd\rho(\lambda),
n\in\mathbb{N}_0.$ The orthogonalization of functions \eqref{1.8}
gives the orthonormal polynomials $P_n(\lambda),$ connected with
Jacobi matrix \eqref{1.1}. But now instead of sequence \eqref{1.8}
we have a double sequence:
\begin{equation}\label{2.8}
(z^j\overline{z}^k)^\infty_{j,k=0}, \,z\in \mathbb{C};
z^j\overline{z}^k \in L^2.
\end{equation}
The question is: in what manner it is convenient to introduce into
\eqref{2.8} the linear order for the application of
orthogonalization procedure and get the analog of $P_n(\lambda)$ for
"Jacobi" normal matrices?

Answer. The order is such:
\begin{equation}\label{2.9}
\begin{array}{cclclcl}
&&&&& \overline{z}^n & \rightarrow\\
&z^0\overline{z}^0&\textbf{---}&z^0\bar{z}^1&\textbf{---}&z^0\bar{z}^2&\textbf{---}\\
\nearrow&\vline&\nearrow&&\nearrow&&\\
&z^1\bar{z}^0&&z^1\bar{z}^1&&&\\
&\vline&\nearrow&&&&\nearrow\\
z^n&z^2\bar{z}^0&&&&z^j\bar{z}^k&\\
\downarrow&\vline&&&\nearrow&&\\
\end{array}
\end{equation}

\section{Block Jacobi type normal matrices and their spectral
theory}\label{sec:3} The natural problem arises: on what way it is
possible to develop the classical theory for the complex moment
problem? What are the corresponding analog of Jacobi matrices and
orthogonal polynomials on complex plane (or on some set from
$\mathbb{C}$, on unit circle $\mathbb{T}\subset \mathbb{C}$, for
example)?

The previous account gives the following picture. \\
$1^0.$ Direct spectral problem.

Instead of the space $\ell_2=\mathbb{C}\oplus\mathbb{C}\oplus\ldots$
it is necessary to take the space
\begin{equation}\label{3.1}
\mathbf{l}_2=\mathcal{H}_0\oplus\mathcal{H}_1\oplus\mathcal{H}_2\oplus\ldots,\,\mathcal{H}_n=\mathbb{C}^{n+1}
(\mathbb{C}^1=\mathbb{C})
\end{equation}
and instead of Jacobi matrix \eqref{1.1} - the following Jacobi type
block matrix, which acts on the space $\mathbf{l}_2$ \eqref{3.1} at
first on finite vectors $\mathbf{l}_{fin}\subset\mathbf{l}_2:$
\begin{equation}\label{3.2}
J=
\begin{bmatrix}
b_0&c_0&0&0&0&\ldots\\
a_0&b_1&c_1&0&0&\ldots\\
0&a_1&b_2&c_2&0&\ldots\\
\vdots&\vdots&\vdots&\vdots&\vdots&
\end{bmatrix};\,
\begin{array}{lllll}
\text{here} \, a_n,b_n,c_n \,\text{are operators}\\
\text{(finite-dimensional matrices):} \\
a_n: \mathcal{H}_n \to \mathcal{H}_{n+1},\\
b_n: \mathcal{H}_n \to \mathcal{H}_n,\\
c_n: \mathcal{H}_{n+1} \to \mathcal{H}_n.
\end{array}
\end{equation}
The essential conditions $a_n>0$ in \eqref{1.1} now have the form:
$\forall n\in\mathbb{N}_0$
\begin{equation}\label{3.3}
\begin{split}
& a_n=\begin{pmatrix}
a_{n;\,0,0} & a_{n;\,0,1} & \ldots & a_{n;0,n}\\
0 & a_{n;\,1,1} & \ldots & a_{n;1,n}\\
\vdots& \vdots& \ddots & \vdots\\
0& 0& \ldots& a_{n;\,n,n} \\
0& 0& \ldots & 0
\end{pmatrix},\\
& c_n=\begin{pmatrix}
c_{n;\,0,0} & c_{n;\,0,1} & 0&0&\ldots & 0  \\
c_{n;\,1,0} & c_{n;\,1,1} & c_{n;\,1,2}&0&\ldots  & 0 \\
\vdots& \vdots& \vdots&& \ddots & \vdots \\
c_{n;\,n,0}& c_{n;\,n,1}& c_{n;\,n,2}&c_{n;\,n,3} & \ldots &
c_{n;\,n,n+1}
\end{pmatrix};\\
&
a_{n;0,0}>0,a_{n;1,1}>0,\ldots,a_{n;n,n}>0;c_{n;0,1}>0,c_{n;1,2}>0,\ldots,c_{n;n,n+1}>0.
\end{split}
\end{equation}

Under some simple conditions on $a_n,b_n,c_n$ matrix $J$ is formally
normal: $JJ^+=J^+J$ ($J^+$ is the adjoint matrix to $J$). If
$a_n,b_n,c_n$ are uniformly bounded operators then the closure
$\widetilde{J}$ is bounded normal operator on $\mathbf{l}_2$ (we
will speak here for simplicity only about this case).

Let $z\in\mathbb{C}$ belongs to the spectrum of $\widetilde{J}.$
Corresponding generalized eigenvector has the form:
\begin{equation}\label{3.4}
P(z)=\left(P_n(z)\right)^\infty_{n=0};
\end{equation}
here $P_n(z)\in\mathcal{H}_n$ is a vector-valued polynomial w.r.t.
$z,\overline{z}$ of degree $n$ (i.e. its coordinates are some linear
combinations of $z^j\overline{z}^k, j+k\leq n$).

This eigenvector $P(z)$ is a solution of two equations of
\eqref{1.2} type:
\begin{equation}\label{3.5}
JP(z)=zP(z), \, J^+P(z)=\overline{z}P(z).
\end{equation}
The corresponding Fourier transform $\widehat{}$\, has the form:
\begin{equation}\label{3.6}
\mathbf{l}_2\supset\mathbf{l}_{fin}\ni f=(f_n)^\infty_{n=0} \mapsto
\widehat{f}(z)=\sum\limits^\infty_{n=0}(f_n,P_n(z))_{\mathcal{H}}
\in L^2(\mathbb{C},d\rho(z))=:L^2,
\end{equation}
where $d\rho(z)$ is a spectral measure of $\widetilde{J}$ with
compact support.

So, we have the following result.
\begin{thm}\label{thm:2}
Consider on the space $\mathbf{l}_2$ \eqref{3.1} the bounded normal
operator $\widetilde{J}$ which is generated by block Jacobi matrix
\eqref{3.2} with conditions \eqref{3.3}.

The corresponding generalized eigenvectors of the form \eqref{3.4}
are the solutions of equations \eqref{3.5} and give rise to Fourier
transform $\widehat{}$\, \eqref{3.6}. This transform is an unitary
operator  between $\mathbf{l}_2$ and $L^2$ constructed by spectral
measure $d\rho(z)$ with compact support. The polynomials $P_n(z)$
generate the orthonormal basis in the space $L^2.$
\end{thm}

It is necessary to remark that every bounded normal operator in
Hilbert space for which one cyclic vector exists is a unitary
equivalent to the operator $\widetilde{J}$ which is generating in
the space \eqref{3.1} by matrix \eqref{3.2} with conditions
\eqref{3.3}. \\
$2^0.$ Inverse spectral problem.

Let we have the probability Borel measure $d\rho(z)$ on $\mathbb{C}$
with compact support. As earlier, the question is: is it possible to
recover corresponding block Jacobi normal matrix \eqref{3.2},
\eqref{3.3} in such manner, that $d\rho(z)$ is a spectral measure
for $\widetilde{J}$? What way is for such reconstruction?

As earlier, it is necessary to take the sequence \eqref{2.8} of
functions $z^j\overline{z}^k$ and apply to it in $L^2$ the Schmidt
orthogonalization procedure. As result, we get the polynomials
$P_n(z)$ of type \eqref{3.4}. It is necessary to take such measure
$d\rho(z)$ that the functions $z^j\overline{z}^k$ are linearly
independent (for example, support $d\rho(z)$ contains some open
set).

The linear order for the sequence \eqref{2.8} it is necessary to
take as on the picture \eqref{2.9}. So, we have the following order:
\begin{equation}\label{3.7}
z^{0}\overline{z}^{0}; z^{1}\overline{z}^{0}, z^{0}\overline{z}^{1};
z^{2}\overline{z}^{0}, z^{1}\overline{z}^{1}, z^{0}\overline{z}^{2};
\ldots; z^{n}\overline{z}^{0}, z^{n-1}\overline{z}^{1}, \ldots,
z^{0}\overline{z}^{n}; \ldots.
\end{equation}

After orthogonalization we get the following table:
\begin{equation*}
\begin{matrix}
P_{0;0}(z)\equiv 1;&P_{1;0}(z),&P_{2;0}(z),&\ldots&P_{n;0}(z),&\ldots\\
&P_{1;1}(z);&P_{2,1}(z),&\ldots&P_{n;1}(z),&\ldots\\
&&P_{2;2}(z);&\ldots&P_{n;2}(z),&\ldots\\
&&&&\vdots&\\
&&&&P_{n;n}(z);&\ldots
\end{matrix}
\end{equation*}
Now we can construct "generalized eigenvector" \eqref{3.4} setting:
\begin{equation*}
P_n(z)=(\overline{P_{n;0}(z)},\overline{P_{n;1}(z)},\ldots,\overline{P_{n;n}(z)}).
\end{equation*}
On this way it is possible to prove the following result.
\begin{thm}\label{thm:3}
Let $d\rho(z)$ be a probability Borel measure with compact support;
assume that the functions \eqref{3.7} are linearly independent. Then
this measure is the spectral measure for normal bounded operator
$\widetilde{J}$ which is generated on the space $\mathbf{l}_2$
\eqref{3.1} by block Jacobi type matrix \eqref{3.2} with conditions
\eqref{3.3}. Its elements are such:
\begin{equation}\label{3.8}
\begin{split}
&
a_{n;\alpha,\beta}=\int\limits_{\mathbb{C}}z\overline{P_{n;\beta}(z)}P_{n+1;\alpha}(z)d\rho(z),
\, \alpha=0,\ldots,n+1; \beta=0,\ldots,n;\\
&
b_{n;\alpha,\beta}=\int\limits_{\mathbb{C}}z\overline{P_{n;\beta}(z)}P_{n;\alpha}(z)d\rho(z),
\, \alpha,\beta=0,\ldots,n;\\
&
c_{n;\alpha,\beta}=\int\limits_{\mathbb{C}}z\overline{P_{n+1;\beta}(z)}P_{n+1;\alpha}(z)d\rho(z),
\, \alpha=0,\ldots,n; \beta=0,\ldots,n+1.
\end{split}
\end{equation}
\end{thm}

If we start from spectral measure $d\rho(z)$ of bounded normal
operator $\widetilde{J}$ generating by matrix $J$
\eqref{3.2}-\eqref{3.3}, then the formulae \eqref{3.8} give the
elements of this matrix.

\section{Block Jacobi unitary matrices and orthogonal polynomials on the unit
circle} \label{sec:4}
$1^0.$ The theory of orthogonal polynomials on
the unit circle $\mathbb{T}\subset\mathbb{C}$ is intensive developed
approximately in the last 50-60 years. In 2005 B. Simon has
published 2-volumes book \cite{38} in this topic. But the idea of
construct an Jacobi type block matrix for this theory is also new.

It is necessary to say that in 2003 M.J. Cantero, L. Moral and L.
Vel\'{a}sques published the article \cite{23} with a deep connected
area. But  they consider the 5-diagonal matrix on ordinary space
$\ell_2$ and not used the natural Jacobi type block matrix in
corresponding space.

Roughly speaking, the orthogonal polynomials on $\mathbb{T}$ and
corresponding tri\-go\-nometric moment problem is the particular
case of above mentioned theory.

The difference is such: the functions $z^j\overline{z}^k$ from
\eqref{2.8}, when $z \in \mathbb{T},$ are linearly depending in the
space $L^2(\mathbb{T},d\rho(z))$ for arbitrary measure $d\rho(z)$ on
$\mathbb{T}$ because $\forall n \in \mathbb{N}_0$
\begin{equation*}
z^j\overline{z}^k=z^{j+n}\overline{z}^{k+n}.
\end{equation*}

Therefore instead of all functions $z^j\overline{z}^k$ from
\eqref{2.8} it is necessary to take only such functions:
\begin{equation}\label{4.1}
z^{0}\overline{z}^{0}=1; z^{1}\overline{z}^{0}=z^{1},
z^{0}\overline{z}^{1}=z^{-1}; z^{2}\overline{z}^{0}=z^{2},
z^{0}\overline{z}^{2}=z^{-2}; \ldots; z\in\mathbb{T}.
\end{equation}
The linear order for orthogonalization is previous: as in
\eqref{2.9}, \eqref{3.7}, i.e. \eqref{4.1}. For the linearly
independence of functions \eqref{4.1} it is necessary to assume that
the support of $d\rho(z)$ consists from infinite many points.

In this case instead of this space \eqref{3.1} we take the space
$\mathbf{l}_{2,u}\subset\mathbf{l}_2:$
\begin{equation}\label{4.2}
\mathbf{l}_{2,u}=\mathcal{H}_0\oplus\mathcal{H}_1\oplus\mathcal{H}_2\oplus\ldots,
\, \text{where}\, \mathcal{H}_0=\mathbb{C},
\mathcal{H}_1=\mathcal{H}_2=\ldots=\mathbb{C}^2.
\end{equation}
The Jacobi type block matrix $J$ of the form \eqref{3.2} acts now on
the space $\mathbf{l}_{2,u}$ and its blocks have the form, another
as \eqref{3.3}, namely:
\begin{equation*}
\begin{split}
& a_0=\begin{bmatrix}a_{0;0,0}\\0\end{bmatrix},
b_0=\begin{bmatrix}b_{0;0,0}\end{bmatrix},
c_0=\begin{bmatrix}c_{0;0,0}& c_{0;0,1}\end{bmatrix},\\
& a_n=\begin{bmatrix}a_{n;0,0}& a_{n;0,1} \\ 0 & 0 \end{bmatrix},
c_n=\begin{bmatrix}0 & 0 \\ c_{n;1,0} & c_{n;1,1} \end{bmatrix};\\
& a_{0;0,0},c_{0;0,1}, a_{n;0,0}, c_{n;1,1} > 0, \, n=1,2,\ldots.
\end{split}
\end{equation*}

For unitary case it is possible to repeat all constructions from
Sections \ref{sec:2},\ref{sec:3}, including the Theorems
\ref{thm:1}, \ref{thm:2}, \ref{thm:3}.

\section{Some applications to the integration of nonlinear difference equations and concluding
remarks} \label{sec:5} $1^0.$ At first we give some additional
references.

The account of classical spectral theory of Jacobi matrices,
corresponding moment problem and spectral approach to the
representation of positive definite kernels see in \cite{1,4,37,43};
see also \cite{19,8,9,12}. Classic theory of orthogonal polynomials,
including such polynomials on set from complex plane (in particular
on unit circle): \cite{42,29,40,38,31}. The book \cite{41} contains
the results about orthogonal polynomials of two real variables
$x=$Re $z$, $y=$ Im $z.$

The materials of Sections \ref{sec:2}-\ref{sec:4} is an account of
results of Yu. M. Berezansky and M.E. Dudkin which are published
with profs in the articles \cite{13,14,15}. Note also that the main
idea of the transfer from ordinary infinite matrices in the space
$\ell_2$ to the block Jacobi type matrices and corresponding
spectral theory in the space $\mathbf{l}_2$ or $\mathbf{l}_{2,u}$
was contained in the talk of Yu. M. Berezansky on the International
Conference in Munich, Germany, at July 2005 \cite{10}. The article
\cite{14} contains also some conditions for normality of Jacobi type
block matrices.

The order of orthogonalization \eqref{2.9} is actually not new (see
in terms of variables $x,y\in\mathbb{R}$ \cite{3}, Ch. 12,
\cite{41}). For the case under consideration, it is necessary to
take into account that, e.g., for a bounded operator $A$ to be
normal, its parts, Re$A=1/2(A+A^*)$ and Im$A=1/2i(A-A^*),$ must be
selfadjoint and commuting.

Note that the books \cite{3} and \cite{41} contain many interesting
facts connected with Sections \ref{sec:2}-\ref{sec:4}. Our Theorems
\ref{thm:2}, \ref{thm:3} also provide answers to some questions
formulated in \cite{3}, Ch. 12, Subsection 12.3.

It is convenient now to do two remarks concerning results of Section
\ref{sec:2}-\ref{sec:4}.

At first, it is interesting to find the form of five-diagonal matrix
in ordinary space $\ell_2$ if we know that the corresponding
operator on $\ell_2$ is unitary with one cyclic vector. Of course it
is generated by block Jacobi matrix in the space $\mathbf{l}_{2,u}$
but it is necessary to find the formulae for its elements (their
representation using Verblunsky coefficients). This problem is
solved in \cite{24,26}.

Second remark. In the theory of orthonormal polynomials on unit
circle the important formula exists which give a possibility to find
these polynomials step by step (Szeg\"{o} recursion) \cite{42,38}.
Therefore the following question arose: is it possible to find
analogical formula for orthogonal polynomials on complex plane? The
article \cite{17} contains the solution of this problem: in reality
the Szeg\"{o} recursion is equivalent to the two equalities
\eqref{3.5}. They can be rewritten as one recursion which
generalized the Szeg\"{o} recursion on general orthogonal
polynomials on complex plane. \\
$2^0.$ Transfer to the short account of results concerning the
applications of the theory of Section \ref{sec:2}-\ref{sec:4} to the
integration of some nonlinear differential-difference equations.

Consider the classical Toda lattice on semi-axis:
\begin{equation}\label{5.1}
\begin{split}
& \dot{a}_n(t)=\frac{1}{2}a_n(t)(b_{n+1}(t)-b_n(t)), \,\\
& \dot{b}_n(t)=a^2_n(t)-a^2_{n-1}(t); a_{-1}(t)=0;
\,n\in\mathbb{N}_0, t\in [0,T].
\end{split}
\end{equation}
Here $a_n(t)>0, b_n(t)$ are real continuously differentiable
functions, $\cdot=\frac{d}{dt}, T\leq \infty.$ Let us put the Cauchy
problem for \eqref{5.1}: for a given initial data
$a_n(0),b_n(0),n\in\mathbb{N}_0,$ it is necessary to find the
solution $a_n(t),b_n(t),n\in\mathbb{N}_0,$ for $t>0;$ equality
$a_{-1}(t)=0$ is some boundary condition.

To find the solution of this problem it is possible to apply the
following procedure. We construct by $a_n(t), b_n(t)$ for every
$t\in [0,T]$ the Jacobi matrix \eqref{1.1} $J(t).$ This matrix is
Hermitian. If we assume the boundness of this solution then the
corresponding operator $\widetilde{J}(t)$ is selfadjoint on the
space $\ell_2.$ Denote by $d\rho(\lambda;t)$ its spectral measure.
The change of solution $a_n(t),b_n(t)$ w.r.t. $t$ is, of course,
very complicated, but it is possible to prove that change of
$d\rho(\lambda;t)$ is very simple:
\begin{equation}\label{5.2}
d\rho(\lambda;t)=c(t)e^{\lambda t}d\rho(\lambda;0),\, t\in [0,T],
\lambda \in \mathbb{R},
\end{equation}
where $c(t)$ is a normalizing factor (spectral measure $\forall t$
is a probability measure). As result, the procedure of finding the
solution of Cauchy problem for \eqref{5.1} is such: we find the
initial spectral measure $d\rho(\lambda;0)$ of operator
$\widetilde{J}(0),$ then using \eqref{5.2} we calculate the spectral
measure $d\rho(\lambda;t)$ for $t>0$ and, finally, reconstruct the
matrix $J(t)$ using the inverse spectral problem. As result, the
elements of $J(t)$ are the solution of our Cauchy problem. Note,
that the equation \eqref{5.1} is "isospectral": spectrum of
$\widetilde{J}(t)$ is stable w.r.t. $t\in[0,T]$ (see \eqref{5.2}).

Such approach was proposed in \cite{5,6,7}; it is a difference
analog of classical inverse spectral problem method for solution of
Cauchy problem for Korteweg-de Vries differential equation (instead
of Sturm-Liouville equation we use more simple spectral theory of
Jacobi matrices).

Such approach was generalized on more complicated as \eqref{5.1}
equations. In particularly, it was investigated the "nonisospectral"
equations for which the change of spectral measure is more
complicated as \eqref{5.2}. Also it was investigated the non-Abelian
case when $a_n(t),b_n(t)$ are matrices e.t.c. Some of corresponding
and connected results see in \cite{25,16,28,20,27,35} and in the
book \cite{43}.

Two years ago L.B. Golinskii has published the article \cite{30} in
which he applied the approach \cite{5,6,7} with the change of
spectral theory of Jacobi matrices by spectral theory of
five-diagonals unitary matrices in the space $\ell_2$ (using the
results of works \cite{23,38}). Such approach give the possibility
to integrate the another as \eqref{5.1} equations, namely the Shur
flows. This point of view is fruitful: in \cite{18,11} it was shown
that it is possible to apply for integration the spectral theory of
normal (and unitary) block Jacobi matrices, i.e. the results of
Sections \ref{sec:2}-\ref{sec:4}. Now instead of Toda equation
\eqref{5.1} we can investigate the another differential-difference
equations, including non-Abelian one (for example, Polyakov type
systems). Using the results of Sections \ref{sec:2}-\ref{sec:4} it
is possible to investigate also corresponding "nonisospectral"
systems. In the case of unitary operators (described in the Section
\ref{sec:4}) such results were get in \cite{36}.

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