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\section{On complexity of description of classes of non self-adjoint
operators}

The borderline between the theory of operators and the theory of
operator algebras and their representations can be viewed as a river
with numerous two-way bridges joining the banks (see, for example,
[,] and others). One of these bridges is discussed in this section:
we
consider an application of the theory of representations of
$*$-algebras to a study of classes of operators that are singled out
algebraically.

Let   $X$ be a bounded non-selfadjoint operator
acting  in a Hilbert space $H$. The classes of operators
which  satisfy to the polynomial relations  $P_j(X,X^*)=0$
($j=1,\ldots,
m$)
 and more general relations are considered.

For every such class corresponds  the $*$-algebra
$\frak A ={\Bbb C} \langle x,x^* \mid P_j(x,x^*)=0 ,
j=1,\ldots,m\rangle $. If the class of operators  assigned by the
non-polynomial relations then the corresponding $*$-algebra is given
in more complicated way.
  Each representation $\pi$ of the $*$-algebra $\frak A$
determines the bounded operators $X=\pi(x)$ and $X^*=\pi(x^*)$
such that

\begin{equation}
           P_j(X,X^*)=0 \qquad j=1,\ldots , m
 \end{equation}

Conversely, a given operators $X$ and
$X^*$ such that $P_j(X,X^*)=0$ ($j=1,\ldots,m$) uniquely define a
representation of the whole algebra $\frak A$.

To describe the class of operators which satisfying relations
(1), up to a unitary equivalence, is equivalented
 to describing representations of the corresponding $*$-algebra
 $\frak A$.

For such algebras  we  estimate the complication of the
corresponding problem of the $*$-representations theory. It means
 complication of the unitary description of the   corresponding class
 of operators.

We consider classes of operators connected with relations that could
be quadratic, semilinear, cubic, and others (Section 3.2.1). Then we
study complexity of the unitary description of algebraic operators
(Section \ref{S:3.2.2}), partial isometries, and weakly centered
operators
(Section \ref{S:3.2.3}).

\subsection{Classes of non self-adjoint operattors selected by a
quadratic or a cubic relation}

The normal operators $X$, that is operators for which $XX^*=X^*X$.
It is the most studied region of civilization on a territory of
bounded linear operators. The irreducible normal operators are
one-dimantional. The spectral theorem gives an assembly prodedure
of any normal operators from irreducible.

1.Let we have a pair of operators $X$ and $X^*$ which satisfy
to quadratic relations \begin{gather}\label{cond}
P_2(X,X^*)=P_2^*(X,X^*)=0
\end{gather}
The common form of such relations is the following
\begin{eqnarray}\label{polynom2}
P_2(X,X^*)=a(X^2+(X^*)^2)+(b/i)(X^2-(X^*)^2)+c[X,X^*]+
\\ +d \{X,X^*\}+e(X+X^*)+(f/i)(X-X^*)+gI=0
\end{eqnarray}
 (here $a,b,c,d,e,f,g \in{\Bbb R}$) Now we give a
criterion  in the terms of coefficient, that the the relation
  (it's means a corresponding $*$-algebra)
\begin{theorem}\label{quadr2}
  A corresponding $*$-algebra is  a $*$-wild if and only if there
are one of the following conditions:
     \begin{enumerate}
   \item
$ a=b=c=d=e=f=g=0$
\item
   $ (g-\frac {e^2}{2(a+d)})(a+d)<0$,
  $d-a=b=c=f=0$;
  \item
  $   (g-\frac {f^2}{2(d-a)})(d-a)<0$,
  $a+d=b=c=e=0$;
\item
   $b^2=(d^2-a^2) \ne 0,
  \quad
  (a+d)(g-\frac{e^2}{2(a+d)})<0$, $
    \frac{e^2}{(a+d)}=\frac{f^2}{(d-a)} $,
   $c=0$.
   \end{enumerate}
    \end{theorem}

This theorem immediatly follows from theorem(\ref{quadr1}), by
change of variables $X=A+iB$, $X^*=A-iB$ It is easy to see that
the relations between  coefficients is the next $\alpha =a+d$,
$\beta=d-a$, $\gamma =2b$, $\hbar =2c$, $\epsilon=2f$, $\chi=g$.

2. Now we will consider some classes of non self-adjoint
operators which satisfy to the cubic relation. At first we will
go to pair of self-adjoint operators by change  $X=A+iB$ and
$X^*=A-iB$. Let selfadjoint operators $A$ and $B$ satisfy to the
cubic semilinear relations (linear on $B$).  The common form of
such relations with condition $P_3(A,B)=P^*_3(A,B)$ is following

\begin{eqnarray}\label{qpline} P_3(A,B)&=& \alpha B + 2\beta
\{A,B\} +\epsilon \{A^2,B\} + 2\mu ABA +\nonumber \\
&&+i\gamma\left [A,B\right ]+ i\delta\left [A^2,B \right ]
=0 \end{eqnarray} ($\alpha, \beta, \gamma, \delta \in {\Bbb R}^1
$).

It is easy to see that in terms of $X$ and $X^*$ the relation
(\ref{qpline}) have the next form
\begin{eqnarray}\label{qupline}
&&P_3(X,X^*)=(a/i)(X-X^*) + (b/i)(X^2-(X^*)^2) +\nonumber \\ &&+
(c+d)/i(X^3-
 (X^*)^3) +
(c-d)/i(XX^*X-X^*XX^*)+
 \nonumber\\&& (d/i)(\{X^2,X^*\}+\{X,(X^*)^2\}) +
f\left [ X,X^* \right ] +g\left [ X^*,X^2 \right ] +   \nonumber
\\&& + g\left [(X^*)^2,X \right ]  =0
\end{eqnarray}

Where $ a=(\alpha/2), b=\beta, c=\epsilon/4, d=\mu/4,
f=\gamma/2,  g=\delta/4$.
Denotes
$I_1=8c$, $I_2=c^2-4d^2$, $I_3=a(c^2-4d^2)-b^2(c-4d)$
$I_4(2gb-cf)$ and give the theorem.
\begin{theorem}
The relation (\ref{qupline}) with condition  $c\geq0$, $c^2+d^2+g^2
\ne 0$
is $*$-wild if and only if:
\begin{enumerate}
\item
if $f=g=0$, then there are one of the following conditions:
 \begin{enumerate}
 \item
 $I_1>0$, $I_2>0$, $I_3<0$
 \item
 $I_2<0$, $I_3=0$
 \item
 $I_1>0$, $I_2<0$, $I_3\ne 0$
 \item
 $I_2=0$, $I_3=0$, $ b^2-2ac>0 $
 \item
 $I_2=0$, $I_3\ne 0$
 \end{enumerate}
\item
 if $c=d=b=a=0$, then $g\ne 0$.
\item  If $g(c^2+d^2) \ne 0$ then there are one of the following
conditions:
 \begin{enumerate}
  \item
 If $c \geq 0$, $d=0$, $2ag^2-cf^2 < 0$ then
  \item
  $d\ne 0, I_4 \ne 0$,  $ac^2-I_3-(df-gb-2I_4)(2fc^2/g^2)=0$
 \end{enumerate}
\end{enumerate}
\end{theorem}

3. A known class of quasinormal non self-adjoint operators
(seen \cite )  is a class of operators $X$ which commute with
$X^*X$. It is operators of representations of a $*$-algebra
$\frak K={\Bbb C}\langle x, x^* \mid xx^*x=x^*xx\rangle$.
 From relation $[x,x^*x]=0$ and condition
$P(x,x^*)=P^*(x,x^*)$ follows that $[x^*,x^*x]$. Therefore, for
the irreducible representations $X^*X=\lambda I$ ($\lambda \ge
0$).  Then either $X=X^*=0$, either $\lambda \ge 0$ and
$\frac{X}{\sqrt \lambda}= \frac {e^{i\phi}}{\sqrt \lambda}$,
either   $\frac{X}{\sqrt \lambda}$ is a unilateral shift.
There exists a corresponding structure  theorem (see \cite).

4. Now we consider the other class of the non-selfadjoint
operators $X\in L(H)$ such that $[X^2,X^*]=0$, that is
\begin{equation}\label{nonself1}
X^2X^*=X^*X^2
\end{equation}
Then, the conjugate relation is the next
   \begin{equation}\label{nonself2}
(X^*)^2X=X(X^*)^2
\end{equation}
Let $X=A+iB$ ($A=A^*$, $B=B^*$). Then the operators of this class
is selected by the next relations
\begin{equation}\label{selfad1}
[A^2,B]=[B,A^2]=0
 \end{equation}
The irreducible reprethentations of the pair $A,B$, wich satisfy
to the relations (\ref{selfad1}) are
one-dimational and two-dimantional. These representations
up to unitary equivalence have the next form:
 one-dimentional -- $A=a$,
$B=b$ ($a,b\in {\Bbb R}$); two-dimantional:
$A=a\begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}$,
$B=b\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}$
($a>0, \quad b>0$)

5. In Section we were considered the algebra  $\frak
B_3={\Bbb C}\langle x,y \mid xyx=yxy \rangle$. If on this algebra
is defined the involution $x^{\star}=y$ then the representations
of the corredponding $*$-algebra responds the nex class of the
operators $X$ such that
\begin{equation}\label{coso}
XX^*X=X^*XX^*
\end{equation}

\subsection{Partial isometries and weakly centered operators}

We consider only the following $*$-algebras (classes of
operators) $\frak W\frak C ={\Bbb C} \langle x,x^* \mid
[xx^*,x^*x] =0\rangle$ (weakly centered operators);
$\frak P\frak I ={\Bbb C}\langle x,x^*\mid (x^*,x)^2=x^*x$
(partial isometry)
$\frak W\frak P={\Bbb C}
\langle x,x^* \mid[xx^*,x^*x] =0, (x^*x)^2=x^*x\rangle$ (weakly
centered operators which are partial isometry)

\begin{theorem}
 The $*$-algebra $\frak W\frak C$ is $*$-wild.
\end{theorem}

\begin{proof}.
We give the homomorphism $\psi:\frak W\frak C \to M_{3}( F_2)$
as
 $$ \psi(x)= \left (\begin{array}{ccc} 0&0&2e \\ (1/2)e& (\sqrt
{3}/2)v & 0 \\ (\sqrt {3}/2)u & - (1/2)uv & 0 \end{array} \right).
$$

To show that this is a homomorphism, we
 calculate  that
$$
\psi(x)\psi(x^*)=
\left(\begin{array}{ccc} 2e&0&0 \\ 0&e&0 \\ 0&0&e
\end{array}\right), \psi(x^*)\psi(x)= \left(\begin{array}{ccc}
e&0&0 \\ 0&e&0 \\ 0&0&2e \end{array}\right) .$$
 Therefore
$[\psi(x)\psi(x^*),\psi(x^*)\psi(x)] =0$.  The homomorphism $\psi
$ induces the functor ${\mathcal {F}}_\psi: Rep(C^*({\mathcal {F}}_2)
\to Rep
\frak W\frak I$ as follows
\begin{itemize}
\item
if $\rho \in Ob (Rep C^*({\mathcal {F}}_2$):
$\rho(u)=U$ and $\rho(v)=V$, then  $F_{\psi}(\rho)= \rho \otimes
\psi= \pi$,
where
$\pi(x)=X$ and $\pi(x^*)=X^*$;
\item
if $C: \rho \to \hat \rho$ (that is $CU=\hat {U} C,\quad  CV=\hat
{V} C$) , then $F_{\psi}(C)=\mathcal { C}=\left[
\begin{array}{ccc} C&0&0 \\ 0&C&0 \\ 0&0&C \end{array} \right]$
and $\mathcal { C}:\pi \to \hat \pi$ ( that is $\mathcal { C}X=\hat
 X \mathcal { C}$, $\mathcal { C}X^*=\hat X^* \mathcal { C}$).
\end{itemize}  We will show that $F_{\psi}$ is full.

It  follows from $\mathcal { C}X^*X=\hat X^* \hat X \mathcal { C}$
that
$$
\mathcal { C} = \left [\begin{array}{ccc} C_{11}&C_{12}&0 \\
C_{21}& C_{22}& 0 \\
0&0&C_{33}\end{array}\right].
$$
From the relations  $\mathcal { C}X=\hat X \mathcal { C}$, $\mathcal
{
C}X^*= \hat X^* \mathcal { C}$ we have that $C_{12}=C_{21}=0$,
$C_{11}=C_{22}= C_{33}=C$ and $CU=\hat UC$, $CV=\hat VC$.  Hence,
we can conclude that the functor $F_ {\psi} $ is full.
Therefore, the algebra $\frak W\frak C$ is $*$-wild.
\end{proof}

\begin{theorem}
The $*$-algebra $\frak P\frak I$ is $*$-wild.
\end{theorem}

\begin{proof}
We will show that $\frak P\frak I \succ C^*(\mathcal { F}_2$.
The homomorphism  $$\psi:\frak P\frak I \to M_3(\mathcal { F}_2))$$
 is $$ \psi(x)=\left( \begin{array}{ccc} (\sqrt{3}/4)u
& (\sqrt{3}/2)e&0 \\ (3/4)v & -(1/2)vu^*&0 \\ (1/2)e &0&0
\end{array}\right).
$$
It is easy to verify  that $\psi(x^*)\psi(x)=
\left( \begin{array}{ccc} e&0&0 \\ 0&e&0 \\ 0&0&0
\end{array}\right) $. Therefore
$(\psi(x^*)\psi(x))^2=\psi(x^*)\psi(x)$, hence the homomorphism
$\psi$ has been defined correctly.

The proof that the induced functor $F_{\psi}:
Rep C^*(\mathcal { F_2}) \to Rep\frak P\frak I)$
 is full
is similar to the proof in  previous theorem.
\end{proof}

\begin{theorem}
The $*$-algebra $\frak W\frak P$ is  $*$-wild.
\end{theorem}

\begin{proof}
 We will again shown that $\frak W\frak P \succ C^*(\mathcal { F}_2)
$.
We give the homomorphism  $\psi: \frak W\frak P \to M_4(\mathcal
{F}_2)$
as follows:
$$\psi(x)= \left(\begin{array}{cccc}
(\sqrt{3}/4)u & (\sqrt{3}/2)e&0&0 \\ (3/4)v & -(1/2)vu^*&0 &0 \\
(1/2)e &0&0&0 \\ 0&0&e&0 \end{array}\right).  $$
It is easy to show  that the corresponding functor
$F_{\psi}$ :$Rep C^*(\mathcal { F}_2)  \to Rep \frak P\frak I$ is
full.
\end{proof}

\subsection{Algebraic operators}

\begin{corollary} \label{cor7}
	Let $\mathfrak{A}_{R_3} = \mathbb{C} \langle x \mid R_3(x)
	\buildrel {\rm def} \over =
	(x-\alpha_1 e) (x-\alpha_2 e)(x-\alpha_3 e) =0$, 
	$\alpha_1$, $\alpha_2$, $\alpha_3 \in \mathbb{C}$,  $\alpha_k \ne
	\alpha_l \text{ for } k\ne l\rangle$. Then $\mathfrak{A}_{R_3}
	\succ \mathfrak{Q}_{2,\perp}$, and consequently, the
	$*$-algebra $\mathfrak{A}_{R_3}$ is $*$-wild.
\end{corollary}

\begin{proof}
	Define the homomorphism $\psi \colon \mathfrak{A}_{R_3} \to
\mathfrak{Q}_{2,\perp}$ as follows:
\[
	\psi(a) = \alpha_1 q_1 + \alpha_2q_2 +\alpha_3(e-q_1 -q_2).
\]
It is easy to check that the functor $F_\psi$ is full and faithful.
\end{proof}

\begin{remark}
	Corollary~\ref{cor7} is given in \cite{19}. The proof in
	\cite{19}
	actually uses the fact that the problem of unitary
	classification of two orthogonal idempotents is $*$-wild, and
	implicitly contains this proof.
\end{remark}

4)Let X be a hyponormal operator, that is $XX^*-X^*X \ge 0$. Let
$Y$ be a transcendental quasinilpotent  operator, that is
$lim_{n \to \infty} \mid \mid Y^n \mid \mid ^{1/n} \to 0$.
The problems to describe the hyponormal operators and transcendental
quasinilpotent operators are $*$-wild.

\subsection{Connection between algebraic and $*$-wild problems}
%\end{document}