\documentclass[reqno]{amsart}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 75, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/75\hfil An abstract approach]
{An abstract approach to some spectral problems of direct sum
differential operators}

\author[Maksim S. Sokolov\hfil EJDE--2003/75\hfilneg]
{Maksim S. Sokolov}  

\address{Maksim S. Sokolov \newline
 Mechanics and Mathematics Department,
National University of Uzbekistan (Uzbekistan, Tashkent 700095)}
\email{sokolovmaksim@hotbox.ru}
\date{}

\thanks{Submitted April 15, 2003. Published July 10, 2003.}
\thanks{Partially supported by the ICTP AC Grant}
\subjclass[2000]{47B25, 47B37, 47A16, 34L05}
\keywords{Direct sum operators, cyclic vector, spectral representation,
\hfill\break\indent unitary transformation}


\begin{abstract}
  In this paper, we study the common spectral properties of abstract
  self-adjoint direct sum operators, considered in a direct sum
  Hilbert space. Applications of such operators arise in the
  modelling of processes of multi-particle quantum mechanics,
  quantum field theory and, specifically, in multi-interval boundary
  problems of differential equations. We show that a direct sum
  operator does not depend in a straightforward manner on the
  separate operators involved. That is, on having a set of
  self-adjoint operators giving a direct sum operator, we show how
  the spectral representation for this operator depends on the
  spectral representations for the individual operators (the
  coordinate operators) involved in forming this sum operator. In
  particular it is shown that this problem is not immediately solved
  by taking a direct sum of the spectral properties of the
  coordinate operators. Primarily, these results are to be applied
  to operators generated by a multi-interval quasi-differential system
  studied, in the earlier works of Ashurov, Everitt, Gesztezy,
  Kirsch, Markus and Zettl. The abstract approach in this paper
  indicates the need for further development of spectral theory for
  direct sum differential operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}

\section{Preliminaries}

Below, we follow the idea for multi-interval quasi-differential
operators, see \cite{everittzetl, everittmarkus}, to construct and
extend their results to the general case of abstract operators in
Hilbert space. All the information on spectral theory of abstract
linear operators required in this paper may be found, for
instance, in the texts \cite{reedsimon, danford}.

Let $\Omega$ be finite or countable set of indices; designate
$\omega=\texttt{card}\,(\Omega)$. Consider a family of separable
Hilbert spaces $\{\mathcal{H}_i\}_{i\in\Omega}$ and a family of
self-adjoint operators $\{T_i\}_{i\in\Omega}$, such that
$$
T_i:D(T_i)\subseteq\mathcal{H}_i\to \mathcal{H}_i.
$$
We introduce the sum Hilbert space $\mathcal{H}=\oplus_{i\in\Omega}{\mathcal{H}_i}$, consisting of
vectors $\overline{x}=\oplus_{i\in\Omega}{x_i}$, such that $x_i\in\mathcal{H}_i$ and
\begin{equation}
\|\overline{x}\|^2_\mathcal{H}=\sum_{i\in\Omega}\|x_i\|_i^2<\infty,
\end{equation}
where $\|\cdot\|_i^2$ are norms in $\mathcal{H}_i$. In this direct sum space
$\mathcal{H}$ consider the operator
$$T:D(T)\subseteq\mathcal{H}\to \mathcal{H},$$
defined on the domain
\begin{equation}
D(T)=\{\overline{x}\in\mathcal{H}:\sum_{i\in\Omega}\|T_ix_i\|_i^2<\infty\}
\end{equation}
by $T\overline{x}=\oplus_{i\in\Omega}{T_ix_i}$. Clearly, if the operator $T_i$ is
self-adjoint, for all $i\in\Omega$, then and only then is $T$
self-adjoint.

For each $T_i$ there is a unique resolution of the identity $E_\lambda^i$ and a
unitary operator $U_i$, giving an isometric isomorphic mapping of
the space $\mathcal{H}_i$ on to the space $L^2(M_i,\mu_i)$, such that the operator $T_i$ in $\mathcal{H}_i$
is represented as the multiplication operator in $L^2(M_i,\mu_i)$. Below, we present a
more detailed structure of the mapping $U_i$.

Fix $i\in\Omega$. We call $\phi\in\mathcal{H}_i$ a \emph{cyclic vector}, if for each
$z\in\mathcal{H}_i$ there exists a Borel measureable function $f$, such that
$z=f(T_i)\phi$. Generally, such a cyclic vector does not exist in $\mathcal{H}_i$,
but there is a collection $\{\phi^k\}$ of vectors  in $\mathcal{H}_i$, such
that $\mathcal{H}_i=\oplus^k{\mathcal{H}_i(\phi^k)}$, where $\mathcal{H}_i(\phi^k)$ are
$T_i$-invariant subspaces in $\mathcal{H}_i$, generated by cyclic vectors
$\phi^k$, that is
$$
\mathcal{H}_i(\phi^k)=\overline{\{f(T_i)\phi^k\}},
$$
varying Borel function $f$, such that $\phi\in D(f(T_i))$. Then there
exist unitary operators
$$U^k:\mathcal{H}_i(\phi^k)\to L^2(\mathbb{R},\mu^k),$$
where $\mu^k(\Delta)={\|E_\Delta^i\phi^k\|_i^2}$, for any Borel
set $\Delta$. In $L^2(\mathbb{R},\mu^k)$, the operator $T_i$ has
the form of multiplication by $\lambda$, i.e.
$$
\left(U^kT_i|_{\mathcal{H}_i(\phi^k)}{U^k}^{-1}z\right)(\lambda)=\lambda z(\lambda).
$$
Then the operator
$$
U_i=\oplus^k{U^k}:\oplus^k{\mathcal{H}_i(\phi^k)}\to\oplus^k{L^2(\mathbb{R},\mu^k)}
$$
gives the spectral representation of the space $\mathcal{H}_i$ in the space
$L^2(M_i,\mu_i)$, where $M_i$ is a union of nonintersecting copies
of the real line (\emph{sliced union}) and $\mu_i=\sum_k\mu^k$.
That is
$$(U_iT_iU_i^{-1}z)(\lambda)=f(\lambda)z(\lambda),$$
where $z\in U[D(T_i)]$ and $f$ is a Borel function defined almost
everywhere according to the measure $\mu_i$.

\section{Spectral properties of the operator $T$}
In this section it is shown how spectral representation of the direct
sum operator $T$ may depend on spectral representations of the
given operators $T_i$. For this purpose, we first prove some
auxiliary statements.
\begin{defn}\label{definition1} \rm
For $i\in\Omega$, we introduce a \emph{sliced union} of sets $M_i$
(see preliminaries) as a set $M$, containing all $M_i$ on
different copies of $\cup_{i\in\Omega} M_i$. In this set $M$, the
sets $M_i$ do not intersect, but they may \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.
\end{defn}

Separate arguments show that the following auxiliary proposition
is valid
\begin{prop}\label{proposition}
Let us have a set of measures $\mu_i$, $i\in \Omega$, defined on
nonintersecting supports. If
$$\sum_{i\in\Omega} \int_{-\infty}^\infty f(\lambda)\, d{\mu_i(\lambda)}<\infty,$$
 for any borel function $f(\lambda)$, then the following equality is true:
\begin{equation}\label{prop}
\sum_{i\in\Omega}\int_{-\infty}^\infty{f(\lambda)}\,d{\mu_i(\lambda)} =
\int_{-\infty}^\infty{f(\lambda)}\,d{\sum_{i\in\Omega}\mu_i(\lambda)}.
\end{equation}
\end{prop}

Using this proposition, we prove the following lemmas:
\begin{lem}\label{irequality}
The resolution of the identity $E_\lambda$ of the system operator
$T$ is given by the direct sum of resolutions of the identity
$E_\lambda^i$ of the operators $T_i$; that is
\begin{equation}
E_\lambda = \oplus_{i\in\Omega}{E_\lambda^i}
\end{equation}
\end{lem}

\begin{proof}
Consider $\overline{x}\in D(T)$; this holds if and only if
$$
\|T\overline{x}\|^2_\mathcal{H}=\sum_{i\in\Omega}\|T_ix_i\|_i^2
=\sum_{i\in\Omega}\int_{-\infty}^\infty {\lambda^2}\,d \|E_\lambda^i x_i\|_i^2<\infty.
$$
Recall that we consider the sets $M_i = \mbox{supp}\{
\|E_\lambda^i x_i\|_i^2\}$ divided in the sense of slicing, so
that they do not intersect. Using Proposition $\ref{proposition}$
implies that the following equality is true:
\begin{equation}\label{walkingsum}
\sum_{i\in\Omega}\int_{-\infty}^\infty{\lambda^2}\,d{\|E_\lambda^i x_i\|_i^2}
=\int_{-\infty}^\infty{\lambda^2}\,d{\sum_{i\in\Omega}
\|E_\lambda^i x_i\|_i^2}.
\end{equation}
In turn this implies that $\overline{x}\in D(T)$, if and only if
$$
\int_{-\infty}^\infty{\lambda^2}\,d{\sum_{i\in\Omega}\|E_\lambda^i x_i\|_i^2}<\infty;
$$
and
$$
\|T\overline{x}\|^2_\mathcal{H}=\int_{-\infty}^\infty{\lambda^2}\,
d{\sum_{i\in\Omega}\|E_\lambda^i x_i\|_i^2}.
$$
Using the  uniqueness property of a resolution of the identity,
these two statements show that the operator $\oplus_{i\in\Omega}{E_\lambda^i}$ is the
resolution of the identity of the system operator $T$, that is, according
to our notations $E_\lambda=\oplus_{i\in\Omega}{E_\lambda^i}$. This completes the proof of the lemma.
\end{proof}
\begin{lem}\label{borel function}
For any Borel function $f$, and any vector $\overline{x}\in
D(f(T))$, the following equality is satisfied:
$f(T)\overline{x}=[\oplus_{i\in\Omega}{f(T_i)}]\overline{x}$.
\end{lem}
\begin{proof}
Let $\overline{x}\in D(f(T))$. Then from
Proposition $\ref{proposition}$ and Lemma $\ref{irequality}$,
we obtain, for any $\overline{y}\in\mathcal{H}$:
\begin{multline*}
(f(T)\overline{x},\overline{y})_\mathcal{H}=\int_{-\infty}^\infty{f(\lambda)}\,d{(E_\lambda
\overline{x},\overline{y})_\mathcal{H}}=
\int_{-\infty}^\infty{f(\lambda)}\,{\sum_{i\in\Omega}(E^i_\lambda x_i,y_i)_i}=\\
=\sum_{i\in\Omega}\int_{-\infty}^\infty{f(\lambda)}\,d{(E^i_\lambda
x_i,y_i)_i}=\sum_{i\in\Omega}(f(T_i)x_i,y_i)_i=([\oplus_{i\in\Omega}{f(T)}]\overline{x},\overline{y})_\mathcal{H}.
\end{multline*}
Since $\overline{y}$ is arbitrary, we have
$f(T)\overline{x}=[\oplus_{i\in\Omega}{f(T_i)}]\overline{x}$. This completes the
proof of the lemma.
\end{proof}
For $\varphi_i\in\mathcal{H}_i$, $i\in\Omega$, define
$$
\overline{\varphi_i}=\{0,\dots,0,\varphi_i,0,\dots,0\}\in\mathcal{H},
$$
where $\varphi_i$ is on $i$-th place.
Consider a projection $P:M\to \cup_{i\in\Omega} M_i$ (see
Definition \ref{definition1}), such that
$P(\sigma(T_i))=\sigma(T_i)$.

\begin{defn} \rm
Divide $\Omega$ into non-intersecting sets
\begin{multline}
A_k=\{s\in\Omega: \forall s,l\in A_k, s\neq l, P(\sigma(T_s))\cap
P(\sigma(T_l))=B_{sl},\\ \mbox{where}\,
\|E^t_{B_{sl}}\varphi_t\|^2_t=0, \,\,\mbox{for any cyclic}\,\,
\varphi_t \in\mathcal{H}_t,\,t=s,l\}.
\end{multline}
 From all possible divisions of this type, we choose and fix the
one which contains the minimal number of the sets $A_k$. With this
notation, we call the number $\Lambda=\min\{k\}$ a \emph{spectral
index} of the direct sum operator $T$.
\end{defn}

\begin{thm}\label{spectral index}
Let each $T_i$ has a cyclic vector $\phi_i$ in $\mathcal{H}_i$. Then the
operator $T$ has minimum $\Lambda$ of cyclic vectors
$\{\overline{\xi}_k\}_{k=1}^\Lambda$, having the form
\begin{equation}
\overline{\xi}_k=\sum_{i\in A_k}\overline{\phi_i}.
\end{equation}
\end{thm}

\begin{proof}
First consider the case of two operators. Let $s,l\in \Omega$; then, in
order to obtain one cyclic vector in $\mathcal{H}_s\oplus\mathcal{H}_l$ having the
form $\phi_s\oplus\phi_l$, for any $\overline{x}=x_s\oplus x_l\in
\mathcal{H}_s\oplus\mathcal{H}_l$ it is necessary to find a Borel function $f$, such that
$$
\overline{x}=f(T_s\oplus T_l)[\phi_s\oplus\phi_l].
$$
 From Lemma $\ref{borel function}$ it follows that
$$
\overline{x}=[f(T_s)\oplus f(T_l)][\phi_s\oplus\phi_l].
$$
On the other hand we must determine
each space $\mathcal{H}_p\, (p=s,l)$ by closing the set
$\{f_p(T_p)\phi_p\}$, where $f_p$ varies over all Borel functions
such that $\phi_p\in D(f_p(T_p))$. If $s,l\in A_k$, then supposing
that $f=f_p$ on $P(\sigma(T_p))$, we obtain the required function
$f$, since functions in the isomorphic space $L^2$ are considered
equal on any set of measure zero. Hence, it is clear, that for all
$i\in A_k$, we may construct a single cyclic vector of the form
$$
\overline{\xi}_k=\oplus_{i\in A_k}\phi_i = \sum_{i\in A_k}\overline{\phi_i},
$$
using the process described above, considering pairs of operators.

We recall that we have a minimal number of the sets $A_k$. Consider the
Hilbert space
\begin{equation}\label{2hs}
[\oplus_{i\in A_k}\mathcal{H}_i]\oplus[\oplus_{j\in A_q}\mathcal{H}_j]
\,\,\mbox{with}\,\, k\neq q.
\end{equation}
It follows then that the set
$$
[\cup_{i\in A_k}P(\sigma(T_i))]\cap [\cup_{j\in A_q}P(\sigma(T_j))] = B_{kq}
$$
has a non-zero spectral measure. From the above results it follows
that, by joining cyclic vectors $\overline{\xi}_k=\oplus_{i\in
A_k}\phi_i$ and $\overline{\xi}_q=\oplus_{j\in A_q}\phi_j$ into
one
$$
\overline{\xi}_k+\overline{\xi}_q=\sum_{i\in A_k}\overline{\phi_i}
+ \sum_{j\in A_q}\overline{\phi_j},
$$
we have to obtain the Hilbert space ($\ref{2hs}$), by closing the set
$$
\{f_k(\oplus_{i\in A_k}T_i)\overline{\xi}_k\}\oplus\{f_q(\oplus_{j\in A_q}T_j)
\overline{\xi}_q\},
$$
with varying Borel functions $f_k$ and $f_q$,
which coincide on $B_{kq}$. This is not possible, since the set of
such functions is not dense in the isomorphic space $L^2$
(isomorphism is understood under spectral representation of the
space ($\ref{2hs}$)). Hence, we have $\Lambda$ cyclic vectors
$$
\overline{\xi}_k=\sum_{i\in A_k}\overline{\phi_i}\in \mathcal{H},\,k=\overline{1,\Lambda}.
$$
This completes the proof of the theorem.
\end{proof}

\begin{cor}
Let each $T_i$ has a single cyclic vector. Then \\ 1. $\Lambda=1$
if and only if the operators $T_i, i\in \Omega$, have almost
everywhere (relatively to the spectral measure) pairwise
non-superposing spectra. \\2. a) $\omega<\aleph_0$.
$\Lambda=\omega$, if and only if all the operators $T_i$ have
pairwise superposing spectra. b) $\omega=\aleph_0$.
$\Lambda=\infty$, if and only if all the operators $T_i$ have
pairwise superposing spectra, except maybe a finite number of these operators.
\end{cor}

\begin{proof}
The proof directly follows from the results given in the proof of
Theorem $\ref{spectral index}$.
\end{proof}

\begin{rem} \rm
Note, that these two cases are contiguous. Case 2 is the most
natural for applications, in particular for the direct sum of differential operators.
\end{rem}

Now suppose that each operator $T_i$ has $m_i$ cyclic
vectors. Then, there exists a decomposition
\begin{equation}\label{decomposition}
T=\oplus_{i\in\Omega}{T_i}=\oplus_{i\in\Omega}{\oplus_{k=i}^{m_i}T_i^k}=\oplus_{s}T_s,
\end{equation}
where each $T_s$ has a single cyclic vector. For the operator $T$,
decomposed as above, we apply Theorem $\ref{spectral index}$. This
implies that we can find spectral index $\Lambda$ for the operator
$T$, decomposed as in ($\ref{decomposition}$). It is clear, in
this case, that there exists an estimate for the spectral index
given by
\begin{equation}
\Lambda\geqslant \max\{m_i\}.
\end{equation}
As it has been stated above, for each operator $T_i$ there exists
a unitary operator $U_i$, such that $U_i:\mathcal{H}_i\to
L^2(M_i,\mu_i)$. Hence
$$
\oplus_{i\in\Omega}{U_i}:\oplus_{i\in\Omega}{\mathcal{H}_i}\to\oplus_{i\in\Omega}{L^2(M_i,\mu_i)}.
$$
In the general case (i.e. when there are $T_i$ with more then one
cyclic vector),
$$
\oplus_{i\in\Omega}{U_i}:\oplus_{i\in\Omega}{\oplus_{k=1}^{m_i}\mathcal{H}_i^k}\to
\oplus_{i\in\Omega}{\oplus_{k=1}^{m_i} L^2(\mathbb{R},\mu^k_i)}.
$$
From Theorem $\ref{spectral index}$ follows that there exists a unitary
operator
\begin{equation}\label{uov}
V:\oplus_{i\in\Omega}{\oplus_{k=1}^{m_i} L^2(\mathbb{R},\mu^k_i)}= \oplus_s
L^2(\mathbb{R},\mu_s)\to\oplus_{q=1}^\Lambda
L^2\left(\mathbb{R},\sum_{j\in A_q}\mu_j\right).
\end{equation}
This implies that for any direct sum operator $T$, there exists a
unitary operator $V\oplus_{i\in\Omega}{U_i}$, which represents the space $\mathcal{H}$ in
the space $L_2$:
\begin{equation}
V\oplus_{i\in\Omega}{U_i}:\mathcal{H}\to L^2(N,\mu),
\end{equation}
where $N$ is a sliced union of $\Lambda$ copies of $\mathbb{R}$ and
$$
\mu=\sum_{q=1}^\Lambda\sum_{j\in A_q}\mu_j,
$$
according to
the symbols in $(\ref{uov})$. Furthermore we know, that for each $T_s$ (see
$(\ref{decomposition}))$, there exists a real-valued almost
everywhere finite function $f_s$ on $\mathbb{R}$, such that

1) $\psi_s\in D(T_s)$ if and only if $f(\cdot)(U_s\psi)(\cdot)\in
L^2(\mathbb{R},\mu_s)$;

2) if $\phi_s\in U_s[D(T_s)]$, then
$(U_sT_sU_s^{-1}\phi_s)(m)=f_s(m)\phi_s(m)$.

Defining $f=f_s\chi_s$, where $\chi_s=1$ on $s$-th copy of
$\mathbb{R}$, and zero elsewhere, according to the above notations, we obtain

\begin{thm}\label{spectral representation}
If the unitary operators $U_i$ give spectral representations of
Hilbert spaces $\mathcal{H}_i$ onto the spaces $L^2(M_i,\mu_i)$, then the unitary
operator $$W=V\oplus_{i\in\Omega}{U_i}$$ gives a spectral representation of the
space $\mathcal{H}$ onto the space $L^2(N,\mu)$. According to this representation,
there exists a real-valued almost everywhere finite function $f$
on $N$, such that

1) $\overline{\psi}\in D(T)$ if and only if
$f(\cdot)(W\overline{\psi})(\cdot)\in L^2(N,\mu)$;

2) if $\overline{\phi}\in W[D(T)]$, then
$(WTW^{-1}\overline{\phi})(m)=f(m)\overline{\phi}(m)$.
\end{thm}

These abstract results appear to be the foundation for our further
works where we shall build an ordered representation for a direct
sum operator and consider matters connected with eigenvalue
expansions for self-adjoint direct sum differential operators.

\subsection*{Acknowledgements} The author is grateful to
Professor R. R. Ashurov for his helpful advice and continuing
attention to the progress of the research work that led to the
preparation of this paper;
to the Abdus Salam International Center for Theoretical Physics
for their Affiliated Center Grant which greatly helped to prepare
this work;
and to the anonymous referee for his/her thorough reading of the original
manuscript and making corrections.



\begin{thebibliography}{99}

\bibitem{everittzetl} W. N. Everitt and A. Zettl,
{\em Quasi-differential operators generated by a countable number of
expressions in Hilbert spaces},
Proc. London Math. Soc. (3) 64 (1992), 524-544.

\bibitem{everittmarkus} W. N. Everitt and L. Markus,
{\em Multi-interval linear ordinary boundary value
problems and complex symplectic algebra},
 Memoirs of the American Mathematical Society, 151 (2001), no. 715.

\bibitem{shrodinger} F. Gesztezy, W. Kirsch,
{\em One dimensional Sr\"odinger
 operators with interactions singular on a discrete set},
 Jurnal f\"ur Mathematik 362 (1985), 28-50.

\bibitem{reedsimon} M. Reed and B. Simon,
{\em Methods of Modern Mathematical Physics}, Volume 1: Functional Analysis.
Russian issue, Moscow, "Mir"-press, 1977.

\bibitem{danford} N. Danford and J.T. Schwartz,
{\em Linear Operators}, Volume 2: Spectral Theory.
Russian issue, Moscow, "Mir"-press, 1966.


\end{thebibliography}


\end{document}