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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,
\cite{??}, cite{??} 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 $H$. We consider classes of operators
which  satisfy  polynomial relations  $P_j(X,X^*)=0$
($j=1,\ldots,
m$)
 and more general relations.

 For every such class of relations there corresponds  $*$-algebra
$\frak A ={\Bbb C} \langle x,x^* \mid P_j(x,x^*)=0 ,
j=1,\ldots,m\rangle $. If the class of operators is assigned by 
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, \quad j=1,\ldots , m
.
\end{equation}

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

The problem to describe the class of operators which satisfy relations
\eqref{??}, up to a unitary equivalence, is equivalent
 to the one of describing representations of the corresponding $*$-algebra
 $\frak A$.


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

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


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

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

 1. Let we have a pair of operators $X$ and $X^*$ which satisfy
quadratic relation of the form 
\begin{equation}\label{cond}
P_2(X,X^*)=P_2^*(X,X^*)=0
\end{equation}
The common form of such relation is the following
\begin{align}
P_2(X,X^*)& =a(X^2+(X^*)^2)+(b/i)(X^2-(X^*)^2)+c[X,X^*] \notag
\\ 
& + d \{X,X^*\}+e(X+X^*)+(f/i)(X-X^*)+gI=0
\label{polynom2}
\end{align}
 (here $a$, $b$, $c$, $d$, $e$, $f$, $g\in\mathbb{R}$) Now we give a
criterion  in terms of the coefficients, for the relation
(that is the corresponding $*$-algebra)
 to be $*$-wild.

\begin{theorem}\label{quadr2}
  The corresponding $*$-algebra is  a $*$-wild if and only if 
one of the following conditions hold:
     \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 relation between  coefficients is the following: $\alpha =a+d$,
$\beta=d-a$, $\gamma =2b$, $\hbar =2c$, $\epsilon=2f$, $\chi=g$.

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

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


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

where $ a=(\alpha/2)$, $b=\beta$, $c=\epsilon/4$, $d=\mu/4$,
$f=\gamma/2$,  $g=\delta/4$.

Write
$I_1=8c$, $I_2=c^2-4d^2$, $I_3=a(c^2-4d^2)-b^2(c-4d)$
$I_4(2gb-cf)$; then the following theorem follows.

\begin{theorem}
Relation \eqref{qupline} with conditions  $c\geq0$, $c^2+d^2+g^2
\ne 0$
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
(see \cite{??})  is a class of operators $X$ which commute with
$X^*X$. They are operators of representations of a $*$-algebra
$\frak K={\Bbb C}\langle x, x^* \mid xx^*x=x^*xx\rangle$.
It follows from relation $[x,x^*x]=0$ and condition
$P(x,x^*)=P^*(x,x^*)$, that $[x^*,x^*x]=0$. Therefore, for
irreducible representations we have $X^*X=\lambda I$, $\lambda \ge
0$.  Then either $X=X^*=0$, either $\lambda > 0$ 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 another similar class of non self-adjoint 
operators $X \in L(H)$, for which $[X^2, X^*]=0$, that is
\begin{equation}
X^2X^* = X^* X^2.
\end{equation}
Taking adjoint, we get
\begin{equation}
(X^*)^2 X = X (X^*)^2
\end{equation}
Let $X = A + iB$ ($A= A^*$, $B= B^*$). Then the operators of this calss 
are selected by th,e following relation
\begin{equation}\label{a2b}
[A^2,B] = [B, A^2]=0.
\end{equation}
Irreducible representations of the pair $A$, $B$, which satisfy 
relation \eqref{a2b}, are one-dimensional and two-dimesional. These 
representations, up to a unitary equivalence, are the following: 
one-dimensional $A = a$, $B = b$, $a$, $b\in \mathbb{R}$; 
two-dimensional $A = a \bigl(\begin{smallmatrix}1&0\\0&-1 \end{smallmatrix}
\bigr)$, 
$B = b\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$, $a>0$, 
$b>0$.

5. In Section~\ref{??} we considered the algebra $\mathfrak{B}_3 = 
\mathbb{C} \langle x,y \mid xyx = yxy\rangle$. Introduce involution 
$x^\star = y$; then representations of arising $*$-algebra are related to 
the class of operators $X$ such that
\begin{equation}
XX^*X = X^*XX^*
\end{equation}
Let $X = UC$ ($U$ being a partial isomety, $C \ge 0$, $\ker Cu  = \ker 
C$) be right polar decomposition of the operator $X$. Then
\begin{equation}
UC^3 = C^3 U^*, \quad U^* C^3 = C^3 U,
\end{equation}
which implies that $X$ is a quasi-normal operator, and therefore, 
self-adjoint one, $X^* = X$. Irreducible $*$-representations of of the 
algebra $\mathfrak{B}_3$ equipped with the involution $x^\star = y$ are 
all one-dimensional, $X = \lambda$, $\lambda \in \mathbb{R}$.

\subsection{Partial isometries and weakly centered operators}

Passing to the fourth degree relations, 
we consider the following $*$-algebras (classes of
operators) ${\mathfrak M} = \mathbb{C}\langle x,x^* \mid  [xx^*,x^*x]
=0\rangle$ (weakly centered operators); ${\mathfrak P}=\mathbb{C} \langle
x,x^* \mid (x^*x)^2=x^*x\rangle$ (partial isometries); 
$\mathfrak{MP}=\mathbb{C} \langle
x,x^* \mid[xx^*,x^*x] =0, (x^*x)^2=x^*x\rangle$ (weakly centered
 operators which are partial isometries).
 
 
 1. The followinf theorem holds.
 
 \begin{theorem} The $*$-algebra
$\mathfrak M$ is $*$-wild.
\end{theorem}

\begin{theorem}
If $n<\infty$ then $\frak C_n$ is $*$-wild.
\end{theorem}

	 Thus, we will study  the  unitary classification
 of  representations of
the $*$-algebras
$\frak A =\langle x,x^* \mid [xx^*,x^*x] =0\rangle$,
$\frak B=\langle y,y^* \mid (y^*y)^2=y^*y\rangle$
and  $\frak C=\langle x,x^* \mid[xx^*,x^*x] =0, (x^*x)^2=x^*x\rangle
$.

\begin{theorem}
The $*$-algebra $\frak B_1$ is p-wild.
\end{theorem}

{\bf Proof}. We have to show that $\frak A \succ \frak U$, where
$\frak U=  \langle u,v,u^*,v^*\mid uu^*=
u^*u=e,\quad  vv^*=
v^*v=e\rangle$ is known to be p-wild (of Example 2).

We give the representation $\phi:\frak A \to \frak U \otimes M_3$
of the $*$-algebra $\frak A$ over the $*$-algebra $\frak U$ as
$$
\phi(x)= \left [\begin{array}{ccc} 0&0&2e \\
(1/2)e& (\sqrt {3}/2)v & 0 \\
(\sqrt {3}/2)u & - (1/2)uv & 0
\end{array} \right ].
$$

To show that this is a representation, we
 calculate  that
$$
\phi(x)\phi(x^*)=
\left[\begin{array}{ccc} 2e&0&0 \\ 0&e&0 \\ 0&0&e \end{array}\right],
\phi(x^*)\phi(x)=
\left[\begin{array}{ccc} e&0&0 \\ 0&e&0 \\ 0&0&2e \end{array}\right]
.$$
Therefore $[\phi(x)\phi(x^*),\phi(x^*)\phi(x)] =0$.
The representation $\phi $ induces the
functor $\Phi_\phi: \frak R(\frak U) \to \frak R (\frak A)$
as follows
\begin{itemize}
\item
if $\rho \in Ob (\frak R(\frak U$)):
$\rho(u)=U$ and $\rho(v)=V$, then  $\Phi_{\phi}(\rho)= \rho \otimes
\phi=
\pi$,
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 $\Phi_{\phi}(C)={\mathcal C}=\left[
\begin{array}{ccc} C&0&0 \\ 0&C&0 \\ 0&0&C \end{array} \right]$
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} It is evident  that the functor $\Phi_{\phi}$ is
faithful.  We will show that $\Phi_{\phi}$ 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 $\Phi_ {\phi} $ is full.
Therefore, the algebra $\frak A$ is p-wild.  \hfill Q.e.d.
\begin{theorem}
The $*$-algebra $\frak B_2$ is $*$-wild.
\end{theorem}

We will show that $\frak B_2 \succ \frak U$.
The representation $\phi:\frak B_2 \to \frak U \otimes M_3$
is
$$
\phi(y)=\left[ \begin{array}{ccc} (\sqrt{3}/4)u & (\sqrt{3}/2)e&0 \\
(3/4)v & -(1/2)vu^*&0 \\ (1/2)e &0&0
\end{array}\right].
$$
It is easy to verify  that $\phi(y^*)\phi(y)=
\left[ \begin{array}{ccc} e&0&0 \\ 0&e&0 \\ 0&0&0 \end{array}\right]
$. Therefore $(\phi(y^*)\phi(y))^2=\phi(y^*)\phi(y)$, hence the
representation
$\phi$ has been defined correctly.



The proof that the induced functor $\Phi_{\phi}:
\frak R(\frak P) \to \frak R(\frak B_2)$
$(\rho \to \rho \otimes \phi; C \to C \otimes I_3)$ is full and
faithful
is similar to the proof in   theorem 1. Q.e.d.

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

{\bf Proof}. We will again shown that $\frak C \succ \frak U$.
We give the representation  $\phi: \frak C \to \frak P \otimes M_4$
as follows:
$$\phi(z)=
\left[\begin{array}{cccc} (\sqrt{3}/4)u & (\sqrt{3}/2)e&0&0 \\
(3/4)v & -(1/2)vu^*&0 &0 \\ (1/2)e &0&0&0 \\ 0&0&e&0
\end{array}\right].
$$
It is easy to verify that the representation $\phi$
is well defined .
As in theorem 1 we can show that the induced functor $\Phi_{\phi}$
:$\frak R(\frak P) \to \frak R(\frak B)$ is full and faithful.
\hfill Q.e.d.





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}



\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}