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\AtBeginDocument{{\noindent\small
2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline
{\em Electronic Journal of Differential Equations},
Conference 13, 2005, pp. 101--112.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{101}

\begin{document}

\title[\hfilneg EJDE/Conf/13 \hfil The division method for subspectra]
{The division method for subspectra of self-adjoint differential
vector-operators}

\author[M. S. Sokolov \hfil EJDE/Conf/13 \hfilneg]
{Maksim Sokolov}

\address{Maksim S. Sokolov \hfill\break
 ICTP Affiliated Center, Mechanics and Mathematics Department,
 National University of Uzbekistan,
 Tashkent 700095, Uzbekistan}
\email{sokolovmaksim@hotbox.ru}

\date{}
\thanks{Published May 30, 2005.}
\thanks{Supported by the grants AC-84 and MISC-03/14 from
the Abdus Salam International \hfill\break\indent
Center  for Theoretical Physics}
\subjclass[2000]{34L05, 47B25, 47B37, 47A16}
\keywords{Vector-operator; cyclic vector; spectral representation;
\hfill\break\indent
ordered representation; multiplicity; unitary transformation}

\begin{abstract}
We discuss the division method for subspectra  which appears to be
one of the key approaches in the study of spectral properties of
self-adjoint differential vector-operators, that is operators
generated as a direct sum of self-adjoint extensions on an
Everitt-Markus-Zettl multi-interval system. In the current work we
show how  the division method may be applied to obtain the ordered
spectral representation and Fourier-like decompositions for
self-adjoint differential vector-operators, after which we obtain
the analytical decompositions for the measurable (relative to a
spectral parameter) generalized eigenfunctions of a self-adjoint
differential vector-operator.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defn}[thm]{Definition}
\newcommand{\norm}[1]{\|#1\|^2}

\section{Introduction}

\subsection*{Problem Overview}
We begin with a physical example of a Schr\"odinger
vector-operator.  Gesztesy and Kirsch \cite{shrodinger} in
particular considered a Schr\"odinger operator generated by the
Hamiltonian
\begin{equation}\label{hamiltonian}
  H=-\frac{d^2}{dx^2}+ \big(s^2-\frac14\big)\frac1{\cos^2x}\,,\quad s>0\,.
\end{equation}
Since the potential in Hamiltonian has a countable number of
singularities on a discreet set $X$ in $\mathbb{R}$, leading to
spoiling of the local integrability, it is impossible to apply the
standard methods of the theory of ordinary differential operators.
In order to proceed and build a self-adjoint extension of a
minimal operator generated by (\ref{hamiltonian}) on
$\mathbb{R}\setminus X$, one may take self-adjoint extensions
$T_i$, generated by the same Hamiltonian (\ref{hamiltonian}) in
the coordinate spaces $L^2(-\frac\pi 2+i\pi,\frac\pi 2+i\pi)$,
$i\in\mathbb{Z}$, and then consider the direct sum operator
$\oplus_{i\in\mathbb{Z}}T_i$ in the space
$$\label{space for sh op}
 \oplus_{i\in\mathbb{Z}}L^2\big(-\frac\pi 2+i\pi,\frac\pi 2+i\pi\big)\,.
$$
This direct sum operator appears to be one of the possible
self-adjoint extensions of the minimal operator considered on
$\mathbb{R}\setminus X$. Moreover in the case $s\geqslant 1$ the
minimal on $\mathbb{R}\setminus X$ operator appears to be
essentially self-adjoint and its only self-adjoint extension is a
direct sum operator.

This physical example gave birth to the  theory of general
differential direct sum operators, or in the text below
vector-operators. Beginning from 1992, the theory of differential
vector-operators has been investigated in connection with their
non-spectral properties in a Hilbert space (\cite{everittshubin},
\cite{everittzettl,everittmarkus} and in complete locally convex
spaces \cite{ashuroveveritt,ashuroveveritt1}. The interest in such
a theory is explained by its numerous applications in theoretical
physics and pure mathematics. Thus, physical applications may be
found in a single or a multi-particle quantum mechanics,
especially in problems where a quantum system is split into a
number of disconnected subsystems under the influence of a
potential. For applications in quantum mechanics see also the
respective references in \cite{everittmarkus}.

As it was shown in the fundamental works \cite{everittzettl} and
\cite{everittmarkus}, a differential vector-operator is an object
which resembles an ordinary differential operator by its general
properties, but in fact it has much more complicated structure.

Although the bigger part of studies concerned only non-spectral
properties of differential vector-operators, there has been some
development of their spectral theory recently. Some results
describing position of spectra of Schr\"odinger vector-operators
were presented in 1985 in \cite{shrodinger} and the most recent
results for general quasi-differential vector-operators belong to
Sobhy El-Sayed Ibrahim \cite{ibrahim,ibrahim1}.

The internal spectral structure of abstract self-adjoint
vector-operators was first investigated in \cite{sokolov}, for
which see also \cite{sokolov2, sokolov1}. The structure of
coordinate operators as differential operators played the key role
in \cite{sokolov3} where the unitary transformation making the
ordered representation was described in terms of generalized
eigenfunctions of a differential vector-operator. These
generalized eigenfunctions appear to be only measurable relative
to the spectral parameter, therefore it is an essential problem to
obtain their decomposition over some set of analytical kernels.
This problem is positively solved by Theorem \ref{kernelsdecomp}
of the current work.

\subsection*{Mathematical background}
Basic concepts of quasi-differential operators are well described
in \cite{everittzettl, everittmarkus}. A good reference for
operators with real coefficients is the book of  Naimark
\cite{naimark}.

Let $\Omega$ be a finite or a countable set of indices. On
$\Omega$, we have a multi-interval differential
Everitt-Markus-Zettl system $\{I_i,\tau_i\}_{i\in\Omega}$, where
$I_i$ are arbitrary intervals of the real line and $\tau_i$ are
formally self-adjoint differential expressions of a finite order.
This EMZ system generates a family of Hilbert spaces $\{L^2(I_i)=
L^2_i\}_{i\in\Omega}$ and families of minimal
$\{T_{min,i}\}_{i\in\Omega}$ and maximal
$\{T_{max,i}\}_{i\in\Omega}$ differential operators. Consider a
respective family $\{T_i\}_{i\in\Omega}$ of self-adjoint
extensions. Further, we introduce a system Hilbert space
${\mathbf{L}^2}=\oplus_{i\in\Omega}{ L^2_i}$, consisting of the vectors
$\mathbf{f}=\oplus_{i\in\Omega}{f_i}$ such that $f_i\in
 L^2_i$ and
$$
\norm{\mathbf{f}}=\sum_{i\in\Omega} \|f_i\|_i^2
=\sum_{i\in \Omega}\int_{I_i}|f_i|^2\,dx<\infty\,.
$$
In the space ${\mathbf{L}^2}$ consider the operator
$T:D(T)\subseteq{\mathbf{L}^2}\rightarrow{\mathbf{L}^2}$, defined
on the domain
$$
D(T)=\big\{\mathbf{f}\subseteq{\mathbf{L}^2}:\sum_{i\in\Omega}
\|T_if_i\|_i^2 <\infty\big\}
$$
by $ T\mathbf{f}=\oplus_{i\in\Omega}{T_if_i}$.


The operator $T$ is called a \emph{differential vector-operator}
generated by the self-adjoint extensions $T_i$, or a
\emph{self-adjoint differential vector-operator}. If $\Omega$ is
infinite, the vector-operator $T$ is called \emph{infinite}. The
operators $T_i$ are called \emph{coordinate} operators.

The abstract preliminaries for this work may be found, for
instance, in books \cite{reedsimon, danford}.

Fix $i\in\Omega$. For each $T_i$ there exists a unique resolution
of the identity $E_\lambda^i$ and a unitary operator $U_i$, making the
isometrically isomorphic mapping of the Hilbert space $ L^2_i$
onto the space $L^2(M_i,\mu_i)$, where the operator $T_i$ is represented
as a multiplication operator. Below, we remind the structure of
the mapping $U_i$.

We call $\phi\in L^2_i$ a \emph{cyclic vector} if for each $z\in
L^2_i$ there exists a Borel function $f$, such that
$z=f(T_i)\phi$. Generally, there is no a cyclic vector in $ L^2_i$
but there is a collection $\{\phi^k\}$ of them  in $ L^2_i$, such
that $ L^2_i=\oplus_k{ L^2_i(\phi^k)}$, where $ L^2_i(\phi^k)$ are
$T_i$-invariant subspaces in $ L^2_i$ generated by the cyclic
vectors $\phi^k$. That is $
 L^2_i(\phi^k)=\overline{\{f(T_i)\phi^k\}},
$ for a varying Borel function $f$, such that $\phi^k\in
D(f(T_i))$.

A vector $\phi\in L^2_i$ is called \emph{maximal} relative to the
operator $T_i$, if each measure $(E^i(\cdot)x,x)_i$, $x\in L^2_i$,
is absolutely continuous with respect to the measure
$(E^i(\cdot)\phi,\phi)_i$.

For each Hilbert space $ L^2_i$, there exist a unique (up to
unitary equivalence) decomposition $ L^2_i=\oplus_k
L^2_i(\varphi^k_i)$, where $\varphi^1_i$ is maximal in $ L^2_i$
relative to $T_i$, and a decreasing set of multiplicity sets
$e_k^i$, where $e_1^i$ is the whole line, such that $\oplus_k
L^2_i(\varphi^k_i)$ is equivalent with $\oplus_k
L^2(e^i_k,\mu_i)$, where the measure of the ordered representation
is defined as
$\mu_i(\cdot)=(E^i(\cdot)\varphi_i^1,\varphi_i^1)_i$. A spectral
representation of $T_i$ in $\oplus_k L^2(e^i_k,\mu_i)$ is called
the \emph{ordered representation} and it is unique, up to a
unitary equivalence. Two operators are called \emph{equivalent},
if they create the same ordered representation of their spaces.

For $i\in\Omega$, we introduce a \emph{sliced union} of sets $M_i$
(see also preliminaries) as a set $M$, containing all $M_i$ on
different copies of $\cup_{i\in\Omega} M_i$. The sets $M_i$ do not
intersect in $M$, but they can \emph{superpose}, i.e. two sets
$M_i$ and $M_j$ superpose, if their projections in the set
$\cup_{i\in\Omega} M_i$ intersect.

For $z_i\in  L^2_i$, $i\in\Omega$, define
$\widehat{\mathbf{z_i}}=\{0,\dots,0,z_i,0,\dots,0\}\in{\mathbf{L}^2}$,
where $z_i$ is on the $i$-th place.

For each $i\in\Omega$, let $\delta(T_i)$ denote the
\emph{subspectrum} of the operator $T_i$, i.e.
$\delta(T_i)=\sigma_{pp}(T_i)\cup\sigma_{cont}^*(T_i)$, where
$\sigma_{pp}(T_i)$ is the set of eigenvalues which may be open and
$\sigma_{cont}^*(T_i)$ is the continuous spectrum with a removed
set of spectral measure zero. $\sigma_{cont}^*(T_i)$ may be also
open. Note that $\overline{\delta(T_i)}=\sigma(T_i)$. For
instance, the subspectrum of an operator having the complete
system of eigenfunctions with eigenvalues being the rational
numbers of $[0,1]$ equals to $\mathbb{Q}\cap [0,1]$; the
subspectrum of an operator having the continuous spectrum [0,1] is
assumed to equal to (0,1) without loss of generality. The notion
of the subspetrum arises quite naturally. Indeed, let we are given
a self-adjoint operator $A$ with a simple spectrum
$\sigma(A)=[a,b]$. Choosing any point $\xi\in\sigma(A)$ we can
obtain $\sigma(A)=[a,\xi)\cup[\xi,b]$. If we are interested in
obtaining a formula $A_1\oplus A_2 = A$, where
$\sigma(A_1)=[a,\xi]$ and $\sigma(A_2)=[\xi, b]$, we have to
suppose that $\xi\not\in \sigma_{pp}(A)$. But if we pass to
subspectra, we will not need to care about inessential points
appearing as limit points.

Consider a projecting mapping $P:M\to \cup_{i\in\Omega} M_i$ such
that $P(\delta(T_i))=\delta(T_i)$. Let $\Omega=\cup_{k=1}^K A_k$,
$A_k\cap A_s=\emptyset$ for $k\neq s$ and
\[
A_k=\{s\in\Omega: \forall s,l\in A_k, s\neq l\,,
P(\delta(T_s))\cap P(\delta(T_l))=B_{sl}\,,
\]
where $\norm{E^t(B_{sl})\varphi_t}_t=0$ for any cyclic $
\varphi_t\in L^2_t,\,t=s,l\}$.
 From all such divisions of $\Omega$ we choose and fix the
one, which contains the minimal number of $A_k$. In case when all
the coordinate spectra $\sigma(T_i)$ are simple, we define the
number $\Lambda=\min\{K\}$ as the \emph{spectral index} of the
vector-operator $T$.

The following two lemmas were proved in \cite{sokolov}.

\begin{lem}\label{irequality}
The identity resolution $\{E_\lambda\}$ of the vector-operator $T$
equals to the direct sum of the coordinate identity resolutions
$\{E_\lambda^i\}$, that is $\{E_\lambda\} = \oplus_{i\in\Omega}{\{E_\lambda^i\}} $
\end{lem}
\begin{lem}\label{spectral index}
Let each $T_i$ have a cyclic vector $a_i$ in $ L^2_i$. Then the
vector-operator $T$ has minimum $\Lambda$ cyclic vectors
$\{\mathbf{a}_k\}_{k=1}^\Lambda$, having the form $
\mathbf{a}_k=\sum_{i\in A_k}\widehat{\mathbf{a_i}}. $
\end{lem}

In the next section we will see what a spectral multiplicity of a
vector-operator is. Nevertheless, this notation is intuitively
clear. Running ahead, let us present here two examples, which will
show the difference between the spectral index and the spectral
multiplicity of the vector-operator $T$.

\noindent\textbf{Example 1.} We have a three-interval EMZ system
$\{I_i,\tau_i,1\}_{i=1}^3$ (a kinetic energy, a mirror kinetic
energy, an impulse):
\begin{gather*}
I_1=[0,+\infty),\quad \tau_1=-\big(\frac d{dt}\big)^2,\\
D(T_1)=\{f\in D(T_{max,1}): f(0)+kf'(0)=0,-\infty<k\leqslant\infty\};\\
I_2=[0,+\infty),\quad \tau_2=\big(\frac d{dt}\big)^2,\\
D(T_2)=\{f\in D(T_{max,2}): f(0)+sf'(0)=0,
-\infty<s\leqslant\infty\};\\
I_3=[0,1],\quad \tau_3=\frac 1i\frac d{dt},\\
D(T_3)=\{f\in D(T_{max,3}): f(0)=e^{i\alpha}f(1),\,\alpha\in [0,2\pi]\}.
\end{gather*}
(a) If $k,s\in (-\infty,0]\cup\{+\infty\}$ then
$$
\delta(T_1)=(0,+\infty),\quad \delta(T_2)=(-\infty,0),\quad
\delta(T_3)=\bigcup_{n=-\infty}^\infty (2\pi n -\alpha).
$$
For this system we have: $\{1,2,3\}=\cup_{k=1}^2 A_k$ and
$A_1=\{1,2\},\,A_2=\{3\}$. Thus, here the spectral index does not
coincide with the spectral multiplicity (which is 1) and equals 2.

\noindent (b) The case $0<k,s<+\infty$ leads to
$$
\delta(T_1)=\{-\frac 1 {k^2}\}\cup(0,+\infty), \quad \delta(T_2)=
(-\infty,0)\cup \{\frac 1 {s^2}\},\quad
\delta(T_3)=\bigcup_{n=-\infty}^\infty (2\pi n -\alpha).
$$
 If
$$\alpha\not\in \big[\bigcup_{n=-\infty}^\infty
(2\pi n +\frac1{k^2})\big]\bigcup
\big[\bigcup_{n=-\infty}^\infty (2\pi n-\frac1{s^2})\big],
$$
we have
$A_1=\{1\}$, $A_2=\{2\}$, $A_3=\{3\}$. That is $\Lambda=3$ but
$\oplus_{i=1}^3T_i$ has a simple spectrum.

\noindent\textbf{Example 2.}
Let us have a vector-operator
$\oplus_{i=1}^3T_i$ with
$$
\delta(T_1)=\bigcup_{n\in\mathbb{Z}, n\geqslant 0} n,\quad
\delta(T_2)=\bigcup_{n\in\mathbb{Z}, n\leqslant 0} n,\quad
\delta(T_3)=\bigcup_{n\in\mathbb{Z}, n\neq 0} n.
$$
The spectral index is 3 but spectral multiplicity is 2.

\begin{defn} \rm
A vector-operator $T=\oplus_{i\in\Omega}T_i$ with the simple
coordinate spectra $\sigma(T_i)$ is called \emph{distorted} if its
spectral index does not equal its spectral multiplicity.
\end{defn}

Generally it is not possible to build a spectral representation
for a distorted vector-operator without applying the division
method, but in some cases (Example 1) Theorem \ref{spectral index}
may lead to the construction of a spectral representation even for
some distorted vector-operators. If we want to obtain an ordered
spectral representation for any self-adjoint vector-operator, only
implementation of the division method can achieve this.


\section{The division method for subspectra (DMS)}

Below we present the three theorems (\ref{ordered representation},
\ref{eigexpthm} and \ref{corollaryeigexp}) without their complete
proofs. Only the structural parts of the proofs essential for the
demonstration of the DMS are presented. The complete proof of
Theorem \ref{ordered representation} may be found in
\cite{sokolov1, sokolov2} and refer to \cite{sokolov3} for the
proofs of Theorems \ref{eigexpthm} and \ref{corollaryeigexp}.

\begin{thm}\label{ordered representation}
If $\theta_i$ and $\{e_n^i\}_{n=1}^{m_i}$ are measures  and
multiplicity sets of ordered representations for coordinate
operators $T_i$, $i\in \Omega$, then there exist processes $Pr_1$
and $Pr_2$, such that the measure
$$
\theta =Pr_1(\{\theta_i\}_{i\in\Omega})
$$
is the measure of an ordered representation and the sets
$$
s_n =Pr_2(\{e_k^i\}_{i\in\Omega;\, k=\overline{1,m_i}})
$$
are the canonical multiplicity sets of the ordered representation
of the operator $T$. Thus, the unitary representation of the space
${\mathbf{L}^2}$ on the space $\oplus_n L^2(s_n,\theta)$ is the
ordered representation and it is unique up to unitary equivalence.
\end{thm}

\begin{proof}
We divide the proof into units for convenience. Parts \textbf{(A)}
and \textbf{(B)} represent the DMS.

\noindent\textbf{(A)} Let $a_i$ be maximal vectors relative to the
operators $T_i$ in $L^2_i$. We want to find a maximal vector
relative to the vector-operator $T$. We know, that the vector
$\oplus_{i\in\Omega} a_i$ does not give a single measure, if a set
$P(\delta(T_i))\cap P(\delta(T_j))$ has a non-zero spectral
measure for $i\neq j$. Consider restrictions $T_i|_{
L^2_i(a_i)}=T_i'$. Since all the operators $T_i'$ have single
cyclic vectors $a_i$, we can divide $\Omega$ into $A_k$,
$k=\overline{1,\Lambda}$ and apply Lemma \ref{spectral index} for
the operator $\oplus_{i\in\Omega} T_i'$. Then we have derive
$\Lambda$ vectors $\mathbf{a}^k = \oplus_{j\in A_k}a_j$, which are
maximal in the respective spaces
${\mathbf{L}^2}(\mathbf{a}^k)=\oplus_{j\in A_k}L^2_j(a_j)$.

\noindent\textbf{(B)} Let now $1<\Lambda<\infty$. Define
$T^k=\oplus_{j\in A_k}T_j'$. For any two operators $T^k$ and
$T^s$, $k\neq s$, let us introduce the sets
$\delta_{k,s}=P(\delta(T^k))\cap
P(\delta(T^s))\,\,\,\mbox{and}\,\,\,\delta_{k}=P(\delta(T^k))\setminus
\delta_{k,s}$. There exist unitary representations
$$U^k:{\mathbf{L}^2}(\mathbf{a}^k)\rightarrow
L^2(\mathbb{R},\mu_{\mathbf{a}^k}).$$ Consider measures $\mu_k$
and $\mu_{k,s}$, defined as
$$
\mu_{k,s}(e)=\mu_{\mathbf{a}^k}(e\cap\delta_{k,s})
$$
and $\mu_k(e)=\mu_{\mathbf{a}^k}(e\cap\delta_k)$, for any
measurable set $e$. For any operator $T^k$ (with respect to
$T^s$), measures $\mu_k$ and $\mu_{k,s}$ are mutually singular and
$\mu_k+\mu_{k,s}=\mu_{\mathbf{a}^k}$; therefore,
$$
L^2(\mathbb{R},\mu_{\mathbf{a}^k})=L^2(\mathbb{R},\mu_k)\oplus
L^2(\mathbb{R},\mu_{k,s}).
$$
This means that (according to our designations):
$$
{U^k}^{-1}:L^2(\mathbb{R},\mu_{\mathbf{a}^k})\longrightarrow
\mathbf{L}^2(\mathbf{a}^k_k)\oplus
\mathbf{L}^2(\mathbf{a}^k_{k,s})
$$
and
\begin{equation}\label{maxvector}
\mathbf{a}^k=\mathbf{a}^k_k\oplus \mathbf{a}^k_{k,s},
\end{equation}
where $\mathbf{a}^k_k$ and $\mathbf{a}^k_{k,s}$ form the measures
$\mu_k$ and $\mu_{k,s}$ respectively. Define also as $\max\{w,
\psi\}$ the vector, which is maximal of the two vectors in the
brackets (Note that this designation is valid only for vectors,
considered on the same set. In order not to complicate the
investigation we assume here that any two vectors are comparable
in this sense. In order to achieve this, it is enough to decompose
each coordinate operator $T_i$ into the direct sum $T_i^{pp}\oplus
T_i^{cont}$, where the operators have respectively pure point and
continuous spectra. Then after redesignation we obtain the
equivalent vector-operator to the initial vector-operator
$\oplus T_i$).

Consider first two operators $T^1$ and $T^2$. It is clear, that
the vector
$$\mathbf{a}^{1\oplus 2}=\mathbf{a}^1_1\oplus \mathbf{a}_2^2\oplus\max\left\{\mathbf{a}^1_{1,2},\mathbf{a}^2_{2,1}\right\}$$ is
maximal in
$\mathbf{L}^2(\mathbf{a}^1)\oplus\mathbf{L}^2(\mathbf{a}^2)$. Note
that $\mathbf{a}^1_1$ and $\mathbf{a}_2^2$ and they both may equal
zero. The maximal vector in
$\mathbf{L}^2(\mathbf{a}^1)\oplus\mathbf{L}^2(\mathbf{a}^2)\oplus\mathbf{L}^2(\mathbf{a}^3)\
$ will have the form:
$$\mathbf{a}^{1\oplus 2\oplus 3}=\mathbf{a}^{1\oplus 2}_{1\oplus
2}\oplus \mathbf{a}_3^3\oplus\max\left\{\mathbf{a}^{1\oplus
2}_{{1\oplus 2},3},\mathbf{a}^3_{3,{1\oplus 2}}\right\},$$ where
$\mathbf{a}^{1\oplus 2}_{1\oplus 2}$ is the narrowed vector
$\mathbf{a}^{1\oplus 2}$, corresponding to the set which is free
from the superposition with $\delta(T_3)$, as shown in
(\ref{maxvector}).

Continuing this process, we obtain a maximal vector in the main
space ${\mathbf{L}^2}$:
\begin{equation}\label{max-vec}
\mathbf{a}^{1\oplus \cdots \oplus
\Lambda}=\mathbf{a}^{1\oplus\cdots \oplus\Lambda-1}_{1\oplus\cdots
\oplus\Lambda-1}\oplus
\mathbf{a}_\Lambda^\Lambda\oplus\max\left\{\mathbf{a}^{1\oplus\cdots
\oplus\Lambda-1}_{{1\oplus\cdots
\oplus\Lambda-1},\Lambda},\mathbf{a}^\Lambda_{\Lambda,{1\oplus\cdots
\oplus\Lambda-1}}\right\}.
\end{equation}
Let $\Lambda=\infty$. We obtain $\mathbf{a}^{1\oplus \cdots \oplus
\Lambda}$ as a vector which satisfies the following equality:
\begin{equation}\label{convmaxv}
\left\|[\oplus_{i\in\Omega}{E^i(\cdot)}]\,\mathbf{a}^{1\oplus \cdots \oplus
\Lambda}\right\|^2=\lim_{L\to\infty}
\left\|[\oplus_{j=1}^L{E^j(\cdot)}]\,\mathbf{a}^{1\oplus \cdots
\oplus L}\right\|^2,
\end{equation}
since the limit on the right side exists.

\noindent\textbf{(C)} The next step is to build the measure of the ordered
representation for the vector-operator. From Lemma
\ref{irequality} and the reasonings above, it follows that such a
measure will be
$$
\theta(\cdot)=\left([\oplus_{i\in\Omega}{E^i(\cdot)}]\,\mathbf{a}^{1\oplus \cdots
\oplus \Lambda},\mathbf{a}^{1\oplus \cdots \oplus \Lambda}\right).
$$
\textbf{(D)} The canonical multiplicity sets $s_n$ of the
vector-operator have the form:
\begin{equation}\label{mult-set}
    s_n= \Big[
\bigcup_iP(e_n^i)\Big] \bigcup \Big[\bigcup_{\sum
m_i\geqslant n}\bigcap P\left(e^i_{m_i}\backslash
e^i_{m_i+1}\right)\Big].
\end{equation}
\end{proof}

\noindent\textbf{Example 3}. Let us have a vector-operator
$T=\oplus_{i=1}^5T_i$, generated by five coordinate operators with
simple continuous spectra:
\begin{equation*}
\delta(T_1)=(0,2),\quad \delta(T_2)=(1,3),\quad \delta(T_3)=(2,4),
\quad \delta(T_4)=(3,6),\quad \delta(T_5)=(0,4).
\end{equation*}
Divide $\Omega = \{1,2,3,4,5\}$ into  $A_k$:
$$
A_1=\{1,3\},\quad A_2=\{2,4\},\quad A_3=\{5\}.
$$
The spectral index  $\Lambda$ is 3 and we pass to the three
operators
$$
T^1=T_1\oplus T_3,\quad T^2=T_2\oplus T_4,\quad T^3=T_5,
$$
with maximal elements respectively:
$$
\mathbf{a}^1=a_1\oplus a_3,\quad \mathbf{a}^2=a_2\oplus a_4,\quad
\mathbf{a}^3=a_5.
$$
Consider first two sub-vector-operators $T^1$ and $T^2$. Find elements
$\mathbf{a}^1_1$, $\mathbf{a}^1_{1,2}$, $\mathbf{a}^2_2$,
$\mathbf{a}^2_{2,1}$. Since the spectra are continuous, we may
assume that
$$
\max\{\mathbf{a}^1_{1,2},\mathbf{a}^2_{2,1}\}=\mathbf{a}^1_{1,2}.
$$
We derive a maximal vector $\mathbf{a}^{1\oplus 2}$ in the space
$\oplus_{i=1}^4 L^2_i$:
$$
\mathbf{a}^{1\oplus 2}=\mathbf{a}^1_1\oplus\mathbf{a}^1_{1,2}\oplus
\mathbf{a}^2_2=\mathbf{a}^1\oplus\mathbf{a}^2_2.
$$
Apply the DMS to the vectors $\mathbf{a}^{1\oplus 2}$ and $\mathbf{a}^3$. We
obtain the vecotrs
$$
\mathbf{a}^{1\oplus 2}_{1\oplus 2},\,\, \mathbf{a}^{1\oplus
2}_{1\oplus 2, 3},\,\, \mathbf{a}^3_{3,1\oplus
2}=\mathbf{a}^3,\,\,\mbox{è}\,\, \mathbf{a}^3_{3}=0.
$$
Eventually, we find the maximal element in
$\oplus_{i=1}^5 L^2_i=\oplus_{k=1}^3 L^2(\mathbf{a}^k)$:
\[
\mathbf{a}^{1\oplus 2\oplus 3}=
\mathbf{a}^{1\oplus 2}_{1\oplus 2}\oplus \max\{\mathbf{a}^{1\oplus
2}_{1\oplus 2, 3}, \mathbf{a}^3\}\oplus 0
= \mathbf{a}^{1\oplus
2}_{1\oplus 2}\oplus \mathbf{a}^{1\oplus 2}_{1\oplus 2, 3}\oplus
0=\mathbf{a}^{1\oplus 2}\oplus
0=\mathbf{a}^1\oplus\mathbf{a}^2_2\oplus 0.
\]
It is easy to see that the multiplicity sets for the initial
vector-operator are: $s_1=\mathbb{R}$, $s_2=(0,4)$, $s_3=(1,4)$.

Let us return to Examples 1 and 2. For the distorted
vector-operator $T_1\oplus T_2\oplus T_3$ from Example 1, a
spectral measure will be constructed on the maximal vector
$\mathbf{a}^{1\oplus 2\oplus 3}$. The multiplicity sets $s_i$,
$i\geqslant2$ have measures zero. For the vector-operator from
Example 2 two spectral measures are constructed on
$\mathbf{a}^{1\oplus 2\oplus 3}$ and
$$
\min\{a_{1,2}^1,a_{2,1}^2\}\oplus
\min\{a_{2,3}^2,a_{3,2}^3\}\oplus \min\{a_{3,1}^3,a_{1,3}^1\},
$$
where the sense of the minimums is clear. The multiplicity set
$s_2$ will be
$$
[P(\delta(T_1))\cap P(\delta(T_2))]\cup [P(\delta(T_2))\cap
P(\delta(T_3))]\cup [P(\delta(T_3))\cap P(\delta(T_1))].
$$
Now the term 'distorted vector-operator' is clearly explained by
the form of the cyclic vectors for such an operator.

Let $I=\bigvee_{i\in\Omega}I_i$ denote the sliced union of
intervals $I_i$.  Similarly, $I^k =\bigvee_{j\in A_k}I_j$. If
$x_i$ are variables on $I_i$, then $\vee x_i$ will designate a
variable either on $I$ or $I^k$ depending on the context. This
notation shows, that a vector-function $$z=\{z_1(x_1),\dots,
z_n(x_n),\dots\}$$ on $I$ or $I^k$ may be written as $z(\vee
x_i)$. In particular, we may also write $\mathbf{z}(\vee x_i)$
instead of $\mathbf{z}=\oplus_{i\in\Omega} z_i$.

Let us introduce the space $\oplus_{i\in\Omega} L^{\infty}(I_i^{n})$. Here,
$\mathbf{z}(\vee x_i) \in \oplus_{i\in\Omega} L^{\infty}(I_i^{n})$ means that
$$
\sup_{i\in\Omega} \big\{\mathrm{ess} \sup_{x_i\in I_i^{n}}
|z_i(x_i)|\big\}<\infty,
$$
where for each $i$, families
$\{I_i^{n}\}_{n=1}^\infty$ represent compact subintervals of $I_i$,
such that $\cup_{n=1}^\infty I_i^{n} = I_i$.
In \cite[Lemma 2.1]{ashuroveveritt}, it was shown that
$\oplus_{i\in\Omega} L^{\infty}(I_i^{n})=(\oplus_{i\in\Omega}{L^1(I_i^n)})^*$, where
 the space of Lebesgue-integrable
vector-functions $\oplus_{i\in\Omega}{L^1(I_i^n)}$ is defined analogously to
$\mathbf{L}^2$.

We also need to introduce a symbolic integral $\int_{\bigvee J_i}
f(\vee x_i)\, d(\vee x_i)$ defined by:
$$
\int_{\bigvee J_i} f(\vee x_i)\, d(\vee x_i)
= \oplus_i \int_{J_i} f_i(x_i)\,dx_i,
$$
where $f(\vee x_i)$ is understood to be measurable relative to
$d(\vee x_i)$, if  $f_i(x_i)$ are measurable relative to Lebesgue
measures $dx_i$.

\begin{thm}\label{eigexpthm}
Let $T$ be a self-adjoint vector-operator, generated by an EMZ
system $\{I_i,\tau_i\}_{i\in\Omega}$. Let $U$ be an ordered
representation of the space
$\mathbf{L}^2=\oplus_{i\in\Omega}L^2(I_i)$ relative to $T$ with
the measure $\theta$ and the multiplicity sets $s_k,\,
k=\overline{1,m}$. Then there exist kernels $\Theta_k(\vee x_i,
\lambda)$, measurable relative to $d(\vee x_i)\times \theta$, such
that $\Theta_k(\vee x_i, \lambda)=0$ for $\lambda\in
\mathbb{R}\setminus s_k$ and
$(\oplus_{i\in\Omega}\tau_i-\lambda)\Theta_k(\vee x_i, \lambda)=0$
for each fixed $\lambda$. Moreover for any bounded Borel set
$\Delta$,
\begin{gather}\label{essbound1}
\int_\Delta |\Theta_k(\vee x_i, \lambda)|^2\,d\theta(\lambda)\in
\oplus_{i\in \Omega} L^{\infty}(I_i^{n})\quad \forall n\in
\mathbb{N}.\\
\label{untrans1}
    (U \mathbf{w})^k(\lambda)=\lim_{n\to\infty}\int_{I^{n}}\,
\mathbf{w}(\vee x_i)\,\overline{\Theta_k(\vee x_i, \lambda)}\,d(\vee
    x_i), \quad \mathbf{w}\in \mathbf{L}^2,
\end{gather}
where the limit exists in $L^2(s_k,\theta)$. The kernels
$\{\Theta_k(\vee x_i, \lambda)\}_{k=1}^n$, $n\leqslant m$, are
 linearly independent as vector-functions of
the first variable almost everywhere relative to the measure
$\theta$ on $s_n$.
\end{thm}

\begin{proof}
We again need the DMS to prove this theorem.
Fix $i$. If $\theta_i$ and $\{e^i_p\}_{p=1}^{m_i}$ are
respectively the measure and the multiplicity sets of an ordered
representation for $T_i$, then there exists the decomposition
$L^2_i=\oplus_{p=1}^{m_i}L^2(e_p^i,\theta_i)$, which implies
$T_i=\oplus_{p=1}^{m_i}T_i^p$ and $L^2(e_p^i,\theta_i)$ are
$T_i^p$-invariant. For vector-operator
$(\oplus_{i\in\Omega}\oplus_{p=1}^{m_i}T_i^p)\to\mbox{redesignate}\to
\oplus_sT_s,\, s=\{i,p\}\in\Omega_1$, we may write
$\Omega_1=\cup_{k=1}^\Lambda A_k$.

Let us separate the proof into units for convenience.

\noindent\textbf{(A)} For each $T_j$, $j\in A_k$ and
$k=\overline{1,\Lambda}$, there exists a single cyclic vector
$a_j\in L^2_j$ and
\cite[XII.3, Lemma 9 and XIII.5, Theorem 1(I)]{danford}
a function $W_j(x_j,\lambda)$ defined on $I_j\times e_j$
(note, that for a fixed $i\in\Omega$, $I_j=I_i$ for all
$p=\overline{1,m_i}$) and measurable relative to $dx_j\times
\mu_{a_j}$, such that $W_j(x_j,\lambda)=0,\,\lambda\in
\mathbb{R}\setminus e_j$ and for any bounded $\Delta\subset e_j$:
$\int_\Delta |W_j(x_j,\lambda)|^2\,d\mu_{a_j}(\lambda)\in
L^\infty(I_j^n),\,\,\, n\in\mathbb{N}$. Also
\begin{equation}\label{ran1}
    \left(E^j(\Delta)F_j(T_j)a_j\right)(x_j)=\int_\Delta
    W_j(x_j,\lambda)F_j(\lambda)\,d\mu_{a_j}(\lambda),
\end{equation}
for any $F_j\in L^2(e_j,\mu_{a_j})$. On $I^k=\bigvee_{j\in
A_k}I_j$, we construct the vector-function $$W^k(\vee x_j,
\lambda)=\{W_1(x_1,\lambda),\dots, W_n(x_n,\lambda),\dots\},$$
which is obviously measurable relative to $d(\vee x_j)\times
\sum\mu_{a_j}$. Separate arguments show that this vector-function
is a correctly constructed generalized eigenfunction and thus
satisfies the statement of the theorem within each $A_k$.

Note that since for all $p=\overline{1,m_i}$ there exists the
equality $(\tau_i-\lambda)W_i^p=0$ (see \cite[XIII.5, Theorem
1]{danford}), it is obvious that $(\oplus_{j\in
A_k}\tau_j-\lambda)W^k=0$, where $\tau_j=\tau_i$ for a fixed $i$
and all $p=\overline{1,m_i}$. If $P(\delta(T_i))\cap
P(\delta(T_j))$ has zero spectral measures for all $i,j\in\Omega$,
then $A_k:\Omega_1=\cup_{k=1}^{\Lambda_1} A_k$ may be constructed
such that $A_k$ contains of indices
$\{i,k\},\,i\in\Omega,\,k=\overline{1,\max_i\{{m_i}\}}$.

\noindent\textbf{(B)} Consider the set of indices $\Omega_2=\{j\in\Omega_1:
j=\{i,1\},i\in\Omega\}$. Construct
$A_k:\Omega_2=\cup_{k=1}^{\Lambda_2} A_k$. Apply the reasonings
used in (A), considering everywhere $\Omega_2$ instead of
$\Omega_1$. Hence, for each $A_k$ and we find a vector-function
$W_1^k(\vee x_j,\lambda)$ which is the solution of the equation
$(\oplus_{j\in A_k}\tau_j-\lambda)\mathbf{y}=0$. Consider $W_1^k$
and $W_1^s$ for $s\neq k$. For $\mathbf{a}^k$ there exists the
decomposition $\mathbf{a}^k = \mathbf{a}^k_k\oplus
\mathbf{a}^k_{k,s}$ (see the proof of Theorem \ref{ordered
representation}). This fact induces the decomposition for $W_1^k$:
$W_1^k=W^k_{1,k}\oplus W^k_{1,k,s}$. It is clear that being the
restrictions of $W_{1}^k$, the vector-functions $W^k_{1,k}$ and
$W^k_{1,k,s}$ are also the solutions of the equation
$(\oplus_{j\in A_k}\tau_j-\lambda)\mathbf{y}=0$. They, along with
$\mathbf{a}^k_k$ and $\mathbf{a}^k_{k,s}$ define unitary
transformations $U^k_k$ and $U_{k,s}^k$, such that:
$U^k_k:\mathbf{L}^2(\mathbf{a}^k_{k})\to L^2(\mathbb{R},\mu_{k})$
and $U_{k,s}^k:\mathbf{L}^2(\mathbf{a}^k_{k,s})\to
L^2(\mathbb{R},\mu_{k,s})$ (see the proof of Theorem \ref{ordered
representation}). This implies, that the decomposition
$W^k=W^k_{1,k}\oplus W^k_{1,k,s}$ is correct.

Define as $\max\{W^k_{1,k,s},W^s_{1,s,k}\}$ the vector-function,
which corresponds to the vector
$\max\{\mathbf{a}^k_{k,s},\mathbf{a}^s_{s,k}\}$, respectively
$\min\{W^k_{1,k,s}, W^s_{1,s,k}\}$ as the vector-function which
corresponds to that $\mathbf{a}^k_{k,s}$ or $\mathbf{a}^s_{s,k}$,
which is not maximal of the two.

\noindent\textbf{(C)} Without loss of generality, suppose that $k=1$ and
$s=2$. From the reasonings presented in Part (A) of this
proof, it follows that
$$\Theta_1^{1\oplus 2}= W^1_{1,1}\oplus
W^2_{1,2}\oplus \max\left\{W^1_{1,1,2},W^2_{1,2,1}\right\}
$$ is correctly
constructed vector-function satisfying the statement of the
theorem for the case $T=[\oplus_{j\in A_1}T_j]\oplus [\oplus_{q\in
A_2}T_q]$. Apply the above described process to $\Theta_1^{1\oplus
2}$ and $W_1^3$ to obtain the correctly constructed
vector-function:
$$
\Theta_1^{1\oplus 2\oplus 3}= \Theta^{1\oplus 2}_{1,1\oplus 2}\oplus
W^3_{1,3}\oplus \max\left\{\Theta^{1\oplus 2}_{1,1\oplus
2,3},W^3_{1,3,1\oplus 2}\right\}.
$$
Continuing this process, we finally obtain:
\begin{align*}
\Theta_1(\vee x_i,\lambda)
&=\Theta_1^{1\oplus \cdots\oplus \Lambda_2}\\
&= \Theta^{1\oplus \cdots\oplus \Lambda_2-1}_{1,1\oplus
\cdots\oplus \Lambda_2-1}\oplus W^{\Lambda_2}_{1,\Lambda_2}\oplus
\max\big\{\Theta^{1\oplus \cdots\oplus \Lambda_2-1}_{1,1\oplus
\cdots\oplus
\Lambda_2-1,\Lambda_2},W^{\Lambda_2}_{1,\Lambda_2,1\oplus
\cdots\oplus \Lambda_2-1}\big\},
\end{align*}
where in the case of $\Lambda_2=\infty$, $\Theta_1^{1\oplus
\cdots\oplus \Lambda_2}$ is the function which satisfies
(analogously to (\ref{convmaxv})):
\begin{equation}\label{converge1}
\big\|[\oplus_{i\in\Omega}{E^i(\Delta)}]\,\int_\Delta\Theta_1^{1\oplus
\cdots\oplus
\Lambda_2}d\theta(\lambda)\big\|^2=\lim_{L\to\infty}
\big\|[\oplus_{j=1}^L{E^j(\Delta)}]\,\int_{\Delta}\Theta_1^{1\oplus
\cdots\oplus L}\,d\theta_L(\lambda)\big\|^2,
\end{equation}
 for any bounded Borel set $\Delta$, where
$$
\theta_L(\cdot)=\left([\oplus_{j=1}^L
E^j(\cdot)]\,\mathbf{a}^{1\oplus \cdots \oplus
L},\mathbf{a}^{1\oplus \cdots \oplus L}\right)
$$
is the measure of the ordered representation of the space
$\oplus_{j=1}^L L^2_j$.
The limit on the right side exists and in fact it appears that
$$
\int_{\Delta}\Theta_1^{1\oplus \cdots\oplus
L}\,d\theta_L(\lambda)\to\int_\Delta\Theta_1^{1\oplus \cdots\oplus
\Lambda_2}d\theta(\lambda),
$$
as $L\to \infty$.

\noindent \textbf{(D)} Define
$\Omega_3=\{j\in\Omega_1:j=\{i,2\},i\in\Omega\}$. Construct
$A_k:\Omega_3=\cup_{k=1}^{\Lambda_3} A_k$. Apply processes (B) and
(C) of this proof, substituting everywhere $\Omega_3$ instead of
$\Omega_2$. We obtain a vector-function $ \Theta_2^{1\oplus
\cdots\oplus \Lambda_3}$, which is defined on the set $\cup_i
P(e_2^i)$. But, as we know (see (\ref{mult-set})), the set $s_2$
also includes the sets where there are non-empty superpositions of
$\delta(T_i)$. Therefore, designating
\begin{gather*}
\Theta_2^1=\Theta_2^{1\oplus \cdots\oplus \Lambda_3},\quad
\Theta_2^2=\min\{W^1_{1,1,2},W^2_{1,2,1}\},\dots,\\
\Theta_2^{\Lambda_2+1}=\min\big\{\Theta^{1\oplus \cdots\oplus
\Lambda_2-1}_{1,1\oplus \cdots\oplus
\Lambda_2-1,\Lambda_2},W^{\Lambda_2}_{1,\Lambda_2,1\oplus
\cdots\oplus \Lambda_2-1}\big\},
\end{gather*}
we may again use the process (C) to build the
vector-function $\Theta_2(\vee x_i,\lambda)$ defined on $s_2$ and
$\Theta_2(\vee x_i,\lambda)=0$ for $\lambda\in \mathbb{R}\setminus
s_2$. Using processes (B), (C), (D) and
formula (\ref{mult-set}), we finally obtain
$\Theta_m(\vee x_i,\lambda)$.

\noindent\textbf{(E)} The linear independence is proved by separate
arguments.
\end{proof}


\begin{thm}[Eigenfunction expansions]\label{corollaryeigexp}
For any $\mathbf{w}\in \mathbf{L}^2$, there exists a decomposition
\begin{equation*}\label{eigenfunction expansion}
\mathbf{w}= \sum_{k=1}^m\,\lim_{n\to\infty} \int_{-n}^{+n}(U
\mathbf{w})^k(\lambda)\Theta_k(\vee x_i,
\lambda)\,d\theta(\lambda),
\end{equation*}
\end{thm}

Since the kernels from Theorem \ref{eigexpthm} are only measurable
relative to $\lambda$, the following theorem is important:

\begin{thm}\label{kernelsdecomp}
Each kernel $\Theta_k(\vee x_i,\lambda)$, $k=\overline{1,m}$, may be
decomposed as
\begin{equation}\label{analdec}
\Theta_k(\vee x_i,\lambda)=\sum_{s=1}^{M_k}
\gamma_{sk}(\lambda)\sigma_{sk}(\vee x_i,\lambda),
\end{equation}
where the $M_k$ are finite for each $k$ and $\sigma_{sk}(\vee
x_i,\lambda)$ depend analytically on $\lambda$.
\end{thm}

\begin{proof}
We separate the proof in parts which will correspond to the
analogous parts of the proof of Theorem  \ref{eigexpthm}.

\noindent\textbf{(A*)} Each kernel $W_j(x_j,\lambda)$ from the
part (A) of the proof of Theorem \ref{eigexpthm} may be
decomposed:
$$
W_j(\cdot,\lambda)=\sum_{s=1}^{n_j}\alpha_{js}(\lambda)
\sigma_{js}(\cdot,\lambda),
$$
where $\alpha_{js}$ are supposed to equal zero on
$\mathbb{R}\setminus e_j$, see \cite[p. 1351]{danford}.
Supplementing the defining systems with zeros where necessary, we
obtain:
\begin{align*}
W^k(\vee x_j,\lambda)&= \oplus_{j\in A_k} W_j(x_j,\lambda)\\
& = \oplus_{j\in A_k} \sum_{s=1}^{n_j}\alpha_{js}(\lambda)
 \sigma_{js}(x_j,\lambda)\\
& = \sum_{q=1}^{N_k}\oplus_{j\in A_k}\alpha_{jq}(\lambda)
 \sigma_{jq}(x_j,\lambda) \\
& =\sum_{q=1}^{N_k}\alpha^k_{q}(\lambda)\sigma^k_{q}(\vee x_j,\lambda),
\end{align*}
where
$$
N_k=\max_{j\in A_k} n_j,\quad
\alpha^k_{q}(\lambda)=\sum_{j\in A_k}\alpha_{jq}(\lambda),\quad
 \sigma^k_{q}(\vee x_j,\lambda)=\oplus_{j\in
A_k}\sigma_{jq}(x_j,\lambda).
$$
Since $e_j$ and $e_k$ do not intersect almost everywhere for
$j,k\in\Omega_2$, $j\neq k$, the series $\sum_{j\in
A_k}\alpha_{jq}(\lambda)$ converges almost everywhere on
$\cup_{j\in\Omega_2} P(e_j)$.

\textbf{(B*) and (C*)} Now pass to the part (B). There we obtained
the decompositions $W_1^k=W^k_{1,k}\oplus W^k_{1,k,s}$ and
$W_1^s=W^s_{1,s}\oplus W^s_{1,s,k}$. Let us totally order the set
$\{T^j\}_{j=1}^{\Lambda_2}$ saying that $T^k\preceq T^s$ if
$\max\{W^k_{1,k,s},W^s_{1,s,k}\}=W^k_{1,k,s}$. At that, $T^k\simeq
T^s$ if and only if $T^k\preceq T^s$ and $T^s\preceq T^k$.
According to this, we build $\oplus_{j=1}^{\Lambda_2} T^j$, where
$T^j\preceq T^{j+1}$, $j=\overline{1,\Lambda_2-1}$ if
$\Lambda_2\geqslant 2$. The obtained vector-operator is obviously
equivalent to the initial vector-operator (comprising unordered
operators). Note that
$$
W_1^k(\vee x_i,\lambda)=\sum_{q=1}^{N_k}\alpha^k_{1q}(\lambda)\sigma^k_{1q}(\vee
x_j,\lambda)
$$
and analogously
$$
W_1^s(\vee x_i,\lambda)=\sum_{p=1}^{N_s}\alpha^s_{1p}(\lambda)\sigma^s_{1p}(\vee
x_j,\lambda).
$$
All the above leads to the following equality:
\begin{align*}
\Theta_1^{1\oplus 2}&= W^1_{1,1}\oplus W^2_{1,2}\oplus
\max\left\{W^1_{1,1,2},W^2_{1,2,1}\right\} = W_1^1\oplus W^2_{1,2}\\
&= \Big(\sum_{q=1}^{N_1}\alpha^1_{1q}(\lambda)\sigma^1_{1q}(\vee
x_j,\lambda)\Big)\oplus
\Big(\sum_{p=1}^{N_2}\alpha^2_{1p}(\lambda)\chi_{\delta_{2}}(\lambda)\sigma^2_{1p}(\vee
x_j,\lambda)\Big)\\
&= \sum_{s=1}^{N^{1\oplus 2}}\alpha^{1\oplus
2}_{1s}(\lambda)\sigma^{1\oplus 2}_{1s}(\vee x_j,\lambda)\,,
\end{align*}
where $N^{1\oplus 2}=\max\{N_1,N_2\}$; $\alpha^{1\oplus
2}_{1s}(\lambda)=\alpha^1_{1s}(\lambda)
+\alpha^2_{1s}(\lambda)\chi_{\delta_{2}}(\lambda)$,
$$
\sigma^{1\oplus 2}_{1s}(\vee x_j,\lambda)=\sigma^1_{1s}(\vee
x_j,\lambda)\oplus\big(\sigma^2_{1s}(\vee
x_j,\lambda)\chi_{\delta_{2}}(\lambda)\big),
$$
$s=\overline{1,N^{1\oplus 2}}$.

Continuing this process until the finite $\Lambda_2$, we obtain:
\begin{equation}\label{def1}
 \Theta_1(\vee x_i,\lambda)=\Theta_1^{1\oplus \cdots\oplus
\Lambda_2}=
\sum_{s=1}^{N^{1\oplus\cdots\oplus\Lambda_2}}\alpha^{1\oplus\cdots\oplus\Lambda_2}_{1s}(\lambda)\sigma_{1s}^{1\oplus\cdots\oplus\Lambda_2}(\vee
x_j,\lambda),
\end{equation}
where $N^{1\oplus \cdots\oplus \Lambda_2}=\max\{N_1,N_2,\dots
N_{\Lambda_2}\}$ and for $s=\overline{1,N^{1\oplus \cdots\oplus
\Lambda_2}}$:
\begin{equation}\label{coef1}
\begin{gathered}
\alpha^{1\oplus \cdots\oplus
\Lambda_2}_{1s}(\lambda)=\alpha^1_{1s}(\lambda)+\sum_{i=2}^{\Lambda_2}\alpha^i_{1s}(\lambda)\chi_{\delta_{i}}(\lambda);
\\
\sigma^{1\oplus \cdots\oplus \Lambda_2}_{1s}(\vee
x_j,\lambda)=\sigma^1_{1s}(\vee
x_j,\lambda)\oplus\left(\oplus_{i=2}^{\Lambda_2}\sigma^i_{1s}(\vee
x_j,\lambda)\chi_{\delta_{i}}(\lambda)\right).
\end{gathered}
\end{equation}
In the case of infinite $\Lambda_2$, $N^{1\oplus \cdots\oplus
\Lambda_2}$ is clearly finite. The series in the right side of
(\ref{coef1}) pointwise converges, since it consists of items
defined on non-intersecting sets. $\sigma^{1\oplus \cdots\oplus
\Lambda_2}_{1s}(\vee x_j,\lambda)$ is defined by induction.


\noindent\textbf{(D*)} Borrowing the designations from (D) and
using processes described in (A*) and (C*), we shall come to the
decomposition of $\Theta_2^{1\oplus \cdots\oplus \Lambda_3}$:
\begin{equation*}
\Theta_2^{1\oplus \cdots\oplus \Lambda_3}=
\sum_{s=1}^{N^{1\oplus\cdots\oplus\Lambda_3}}
\alpha^{1\oplus\dots\oplus\Lambda_3}_{2s}(\lambda)
\sigma_{2s}^{1\oplus\cdots\oplus\Lambda_3}(\vee x_j,\lambda).
\end{equation*}
To obtain $\Theta_2(\vee x_i,\lambda)$, as in (D), we repeat part
(C*) for
$$
\Theta_2^1=\Theta_2^{1\oplus \cdots\oplus \Lambda_3},\quad
\Theta_2^2=W^2_{1,2,1},\quad \dots, \quad
\Theta_2^{\Lambda_2+1}=W^{\Lambda_2}_{1,\Lambda_2,1\oplus
\cdots\oplus \Lambda_2-1}.
$$
Finally, the same way we obtain decompositions for all
$\Theta_k(\vee x_i,\lambda)$, $k=\overline{1,m}$, which will have
the form (\ref{analdec}).
\end{proof}


\subsection*{Acknowledgments}
The author is grateful to Professor R. R. Ashurov for
 his  attention to this work

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