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\section{On the complexity of the description of classes of non self-adjoint
operators}\label{sec:3.2}
\markright{3.2. 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{ern}, 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-self-adjoint 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 a $*$-algebra
$\mathfrak A ={\mathbb C} \langle x,x^* \mid P_j(x,x^*)=0 ,\,
j=1,\dots,m\rangle $. If the class of operators is given by 
non-polynomial relations, then the corresponding $*$-algebra is given
in a more complicated way.
  Each representation $\pi$ of the $*$-algebra $\mathfrak A$
determines the bounded operators $X=\pi(x)$ and $X^*=\pi(x^*)$
such that
\begin{equation}\label{rel:pxx}
           P_j(X,X^*)=0, \qquad j=1, \dots , 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 $\mathfrak A$.

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

For such algebras,  we  estimate the complexity of the
corresponding problem of the $*$-representations theory, i.e., the
 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{sec:3.2.1}). Then we
study complexity of the unitary description of
partial isometries,  weakly centered
operators, and
 algebraic operators (Section \ref{sec:3.2.2}).
In Section \ref{sec:3.2.3}, we speak about the complexity of description
of classes of operators which are defined not by polynomial equalities
but rather by conditions similar to inequalities, or other non-algebraic conditions;
namely, we consider hyponormal operators and pairs of commuting completely
non-unitary isometries.

\subsection{Classes of non self-adjoint operators singled out by a
quadratic or a cubic relation} \label{sec:3.2.1}

A normal operator $X$ is an operator such that $XX^*=X^*X$.
Normal operators make the most studied region in the terrain of
bounded linear operators. Irreducible normal operators are
one-dimensional. The spectral theorem gives a procedure for assembling
any normal operator from irreducible ones.
 
\medskip\noindent\textbf{1.}
Let us have a pair of operators $X$ and $X^*$ which satisfy a
quadratic relation of the form 
\begin{equation}\label{cond}
P_2(X,X^*)=P_2^*(X,X^*)=0.
\end{equation}
A common form of such a relation is the following:
\begin{gather}
P_2(X,X^*) =a(X^2+(X^*)^2)+i^{-1}b(X^2-(X^*)^2)+c[X,X^*] \notag
\\ 
{} + d \{X,X^*\}+e(X+X^*)+i^{-1}f(X-X^*)+gI=0;
\label{polynom2}
\end{gather}
 here $a$, $b$, $c$, $d$, $e$, $f$, $g\in\mathbb{R}$.  We now give a
criterion for the relation
(that is the corresponding $*$-algebra)
 to be $*$-wild  in terms of the coefficients.

\begin{theorem}\label{quadr2}
  The corresponding $*$-algebra is   $*$-wild if and only if 
one of the following conditions hold\textup:
     \begin{itemize}
   \item[$1.$]
$ a=b=c=d=e=f=g=0$\textup;

\item[$2.$]
   $ \displaystyle \Bigl(g-\frac {e^2}{2(a+d)}\Bigr)(a+d)<0$,\quad
  $d-a=b=c=f=0$\textup;

\item[$3.$]
  $ \displaystyle \Bigl (g-\frac {f^2}{2(d-a)}\Bigr)(d-a)<0$,\quad
  $a+d=b=c=e=0$\textup;

\item[$4.$]
   $\displaystyle b^2=(d^2-a^2) \ne 0,
  \quad
  (a+d)\Bigl(g-\frac{e^2}{2(a+d)}\Bigr)<0$,

$
\displaystyle    \frac{e^2}{(a+d)}=\frac{f^2}{(d-a)} $,
\quad   $c=0$.
   \end{itemize}
    \end{theorem}

This theorem follows from Theorem~\ref{quadr1}, by the
change of variables $X=A+iB$, $X^*=A-iB$. It is easy to see that
the relation satisfied by the coefficients is the following: $\alpha =a+d$,
$\beta=d-a$, $\gamma =2b$, $q =2c$, $\epsilon=2f$, $\chi=g$.

\medskip\noindent\textbf{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 the change of variables, $X=A+iB$ and
$X^*=A-iB$. Let the self-adjoint operators $A$ and $B$ satisfy a
cubic semi-linear relations (linear in $B$).  A usual form of
such a relation with the condition $P_3(A,B)=P^*_3(A,B)$ is the
following:
\begin{gather}
P_3(A,B)= \alpha B + 2\beta
\{A,B\} +\epsilon \{A^2,B\} + 2\mu ABA \nonumber 
\\
\label{qpline} 
{}+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^*)&=i^{-1}a(X-X^*) + i^{-1}b(X^2-(X^*)^2) \notag 
\\ 
&{}+i^{-1}(c+d)(X^3-
 (X^*)^3)\notag
\\
& +i^{-1}(c-d)(XX^*X-X^*XX^*)
 \notag
 \\
 & + i^{-1}d\bigl(\{X^2,X^*\}+\{X,(X^*)^2\}\bigr) \notag
\\
&{} +
f [ X,X^*  ]+g\bigl( [ X^*,X^2  ]   +  [(X^*)^2,X  ]\bigr)  =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 from
Theorem~\ref{th:cub1}.

\begin{theorem}
Relation \eqref{qupline} with the conditions  $c\geq0$, $c^2+d^2+g^2
\ne 0$
is $*$-wild if and only if one of the following cases \textup{1)--3)}
holds\textup:
\begin{itemize}
\item[$1)$]
$f=g=0$, and  one of the following holds\textup:
 \begin{itemize}
 \item[$(a)$]
 $I_1>0$, $I_2>0$, $I_3<0$,
 \item[$(b)$]
 $I_2<0$, $I_3=0$,
 \item[$(c)$]
 $I_1>0$, $I_2<0$, $I_3\ne 0$,
 \item[$(d)$]
 $I_2=0$, $I_3=0$, $ b^2-2ac>0 $,
 \item[$(e)$]
 $I_2=0$, $I_3\ne 0$.
 \end{itemize}
\item[$2)$]
  $c=d=b=a=0$, \textup(then $g\ne 0$\textup).
\item[$3)$]  $g(c^2+d^2) \ne 0$ and  one of the following
conditions holds\textup:
 \begin{itemize}
  \item[$(a)$]
$c > 0$, $d=0$, $gb -2cf=0$, $2ag^2-cf^2 < 0$,
  \item[$(b)$]
  $d\ne 0, I_4 \ne 0$,  $ac^2-I_3-(df-gb-2I_4)(2fc^2/g^2)=0$.
 \end{itemize}
\end{itemize}
\end{theorem}

\noindent\textbf{3.}
A known class of quasi-normal non self-adjoint operators
(\cite{91}, see also \cite{halm2})
 is a class of operators $X$ which commute with
$X^*X$. These are representation operators of the $*$-algebra
$\mathfrak K={\mathbb C}\langle x, x^* \mid xx^*x=x^*xx\rangle$.
It follows from the 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 $X=X^*=0$, or $\lambda > 0$ and $X=
{e^{i\phi}}{\sqrt \lambda}$,
or   $X/{\sqrt \lambda}$ is a unilateral shift.
There exists a corresponding structure  theorem.

\medskip\noindent\textbf{4.}
Now we consider another similar class of non self-adjoint 
operators $X \in L(H)$ such that $[X^2, X^*]=0$, i.e.,
\[
X^2X^* = X^* X^2.
\]
Taking the adjoints, we get $
(X^*)^2 X = X (X^*)^2$.
Let $X = A + iB$, $A= A^*$, $B= B^*$. Then the operators in this class 
are selected by the following relation
\begin{equation}\label{a2b}
[A^2,B] = [B, A^2]=0.
\end{equation}
Irreducible representations of a pair $A$, $B$ which satisfies
relation \eqref{a2b}, are one- and two-dimensional. These 
representations, up to a unitary equivalence, are the following: 
one-dimensional $A = a$, $B = b$, $a$, $b\in \mathbb{R}$; 
two-dimensional $A = a \bigl(\begin{smallmatrix}1&0\\0&-1 \end{smallmatrix}
\bigr)$, 
$B = b\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$, $a>0$, 
$b>0$.

\medskip\noindent\textbf{5.}
In Section~\ref{sec:3.1.5} we considered the algebra $\mathfrak{B}_2 = 
\mathbb{C} \langle x,y \mid xyx = yxy\rangle$. Introduce an involution
by setting
$x^\star = y$; then representations of the arising $*$-algebra are related to 
the class of operators $X$ such that
\[
XX^*X = X^*XX^*.
\]
Let $X = UC$ ($U$ is a partial isometry, $C \ge 0$, $\ker U  = \ker 
C$) be the polar decomposition of the operator $X$. Then
\[
UC^3 = C^3 U^*, \quad U^* C^3 = C^3 U,
\]
which implies that $X$ is a quasi-normal operator, and therefore, 
self-adjoint, $X^* = X$. Irreducible $*$-representations of the 
algebra $\mathfrak{B}_2$ equipped with the involution $x^\star = y$ are 
all one-dimensional, $X = \lambda$, $\lambda \in \mathbb{R}$.

\subsection{Partial isometries, weakly centered operators and algebraic
operators}\label{sec:3.2.2}

\textbf{1.}
Passing to relations of degree four, 
we consider only the following $*$-algebras (classes of
operators): $\mathfrak W ={\mathbb C} \langle x,x^* \mid
[xx^*,x^*x] =0\rangle$ (weakly centered operators),
$\mathfrak P ={\mathbb C}\langle x,x^*\mid (x^*x)^2=x^*x\rangle$
(partial isometries),
$\mathfrak {WP}={\mathbb C}
\langle x,x^* \mid[xx^*,x^*x] =0,\, (x^*x)^2=x^*x\rangle$ (weakly
centered operators which are partial isometries).

The following theorem holds.

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

\begin{proof}
Define a homomorphism $\psi\colon\mathfrak W \to M_{3}(\mathcal F_2)$
by
\[
 \psi(x)= \begin{pmatrix} 0&0&2e \\ (1/2)e& (\sqrt
{3}/2)v & 0 \\ (\sqrt {3}/2)u & - (1/2)uv & 0 \end{pmatrix}.
\]

To show that this is a homomorphism, we
 calculate  that
$$
\psi(x)\,\psi(x^*)=
\begin{pmatrix} 2e&0&0 \\ 0&e&0 \\ 0&0&e
\end{pmatrix}, \quad \psi(x^*)\,\psi(x)= \begin{pmatrix}
e&0&0 \\ 0&e&0 \\ 0&0&2e \end{pmatrix} .
$$
 Therefore
$[\psi(x)\,\psi(x^*),\psi(x^*)\,\psi(x)] =0$.

The homomorphism $\psi
$ induces the functor ${{F}}_\psi\colon \rep C^*({\mathcal {F}}_2)
\to \rep
\mathfrak W$.
  We will show that $F_{\psi}$ is full.

It  follows from ${ C}X^*X=\hat X^* \hat X { C}$
that
$$
{ C} = \begin{pmatrix} C_{11}&C_{12}&0 \\
C_{21}& C_{22}& 0 \\
0&0&C_{33}\end{pmatrix}.
$$
From the relations  ${ C}X=\hat X  { C}$, ${
C}X^*= \hat X^* { C}$, we have that $C_{12}=C_{21}=0$,
$C_{11}=C_{22}= C_{33}=\tilde C$, and $\tilde C U=\hat
U\tilde C$, $\tilde CV=\hat V\tilde C$.  Hence,
we can conclude that the functor $F_ {\psi} $ is full.
Therefore, the algebra $\mathfrak W$ is $*$-wild.
\end{proof}

Therefore, the problem of unitary description
of weakly centered operators is
$*$-wild.

\medskip\noindent\textbf{2.}
For partial isometries, the following theorem holds.

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

\begin{proof}
We will show that $\mathfrak P \succ C^*(\mathcal { F}_2)$.
The homomorphism  
$
\psi\colon\mathfrak P \to M_3(C^*(\mathcal { F}_2))
$
is constructed by
$$ 
\psi(x)= \begin{pmatrix} \sqrt{3}/4\,u
& \sqrt{3}/2\,e&0 \\ 3/4\,v & -1/2\,vu^*&0 \\ 1/2\,e &0&0
\end{pmatrix}.
$$
It is easy to verify  that 
$$
\psi(x^*)\,\psi(x)=
\begin{pmatrix} e&0&0 \\ 0&e&0 \\ 0&0&0
\end{pmatrix}.
$$ 
Therefore
$(\psi(x^*)\,\psi(x))^2=\psi(x^*)\,\psi(x)$.

The induced functor $F_{\psi}\colon
\rep C^*(\mathcal { F}_2) \to \rep\mathfrak P$
 is full.
\end{proof}

Therefore, the problem of description of partial isometries up to a unitary
equivalence is $*$-wild.

\medskip\noindent\textbf{3.} The following theorem holds.

\begin{theorem}\label{th:wcpi}
The $*$-algebra $\mathfrak W\mathfrak P$ is  $*$-wild.
\end{theorem}

\begin{proof}
 We will again shown that $\mathfrak W\mathfrak  P
\succ C^*(\mathcal { F}_2)$.
Define a homomorphism  $\psi\colon \mathfrak W\mathfrak P \to M_4(\mathcal
{F}_2)$
as follows:
$$
\psi(x)= \begin{pmatrix}
\sqrt{3}/4\,u & \sqrt{3}/2\,e&0&0 \\ 3/4\,v & -1/2\,vu^*&0 &0 \\
1/2\,e &0&0&0 \\ 0&0&e&0 \end{pmatrix}.  
$$
It is easy to show  that the corresponding functor
$F_{\psi} \colon \rep C^*(\mathcal { F}_2)  \to \rep \mathfrak {WI}$ is
full.
\end{proof}

Thus, the problem of description of partial isometries, which are weakly
centered operators, is $*$-wild.

\begin{remark}
For $n<\infty$, consider the  $*$-algebra
\begin{align*}
\mathfrak{WP}_n = \mathbb{C} \bigl\langle
x, x^* &\mid \text{$x$ is a partial
isometry, and }
\\
&\quad [x^j{x^*}^j, {x^*}^k x^k] = 0,\, \forall k, j =1,\dots, n
\bigr\rangle.
\end{align*}
$*$-Algebra of weakly centered operators is $\mathfrak{WP} = \mathfrak{WP}_1$.
In \cite{besp_mfat},
Theorem~\ref{th:wcpi} was extended to these algebras:
it was proved that the $*$\nobreakdash-algebra
$\mathfrak{WP}_n$ is $*$-wild for any $n<
\infty$.

On the other hand,  the
$*$-algebra of centered operators is nuclear (see
Section~\ref{sec:2.5.2}), and the $*$-algebra of centered partial isometries
is of type $I$ and admits a
complete description of its $*$-representations (Section~\ref{sec:2.1.3})
\end{remark}

\noindent\textbf{4.}
We also consider the complexity problem of unitary description
for algebraic operators, i.e.,
representations of the $*$-algebra 
$$
\mathfrak {A}_{R_n}={\mathbb C} \langle
x,x^*\mid R_n(x)=0\rangle,
$$ 
where $R_n$ is a polynomial in one
variable with complex coefficients. For brevity, we assume that the
polynomial does not have multiple roots.

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

\begin{proof}
        Define a homomorphism $\psi \colon \mathfrak{A}_{R_3} \to
\mathcal{Q}_{2,\perp}$ as follows:
$
        \psi(x) = \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.
\end{proof}

\begin{corollary}
If $R_n$  is a polynomial with three and more distinct roots then
the $*$-algebra $\mathfrak{A}_{R_n}$ is $*$-wild.
 \end{corollary}

\subsection{Hyponormal
operators and pairs of commuting completely non-unitary
isometries}\label{sec:3.2.3}

Now we will consider  classes of non self-adjoint operators
that are  given  by a non-polynomial equality, e.g.,
an inequality, or other non-algebraic conditions.  For such
operators we will study the complexity of the problem to
describe the operators up to unitary equivalence.

\medskip\noindent\textbf{1.} Let Z be a hyponormal operator,  i.e.,
 $ZZ^*-Z^*Z \ge 0$.  We will show that the problem to describe  the
class of hyponormal operators contains the description problem for one
non self-adjoint operator, $X\in L(H)$,  which
does not satisfy any relations, or the
same for a pair of self-adjoint operators. To do that, we  use Wogen's
construction (see \cite{wog}). We consider operators $Z\in
L(\bigoplus_1^{\infty}H)$ of the following form:
\begin{gather*}
Z=
\begin{pmatrix}
0     &       &       &    &      &   \\
I     & 0     &       &    &\smash{\text{\Large 0}}      &    \\
X     & 2I    & 0     &    &      &   \\
      & 0     & 3I    & 0  &      &    \\
      &       & 0     & 3I & 0    &    \\
      & \smash{\text{\raise5pt\hbox{\Large
0}}}      &       &  \ddots  &\ddots &\ddots
\end{pmatrix} , \qquad \|X\| \le 1/2.
 \end{gather*}
The operator $Z$ is a hyponormal operator.
 It is easy to prove the following proposition.

\begin{proposition}
 An operator ${\mathcal Y}$
is intertwining for the pairs $Z$, $Z^*$ and \/$ \tilde Z$, $\tilde Z^*$
if and only if \/ ${\mathcal Y}= Y\otimes I_{\infty}$, where 
$Y$ is an intertwining operator for $X$, $X^*$ and\/ $\tilde
X$, $\tilde X^*$, correspondingly \textup(that is $YX=\tilde
{X}Y$, $YX^*=\tilde X^*Y $\textup).
  \end{proposition}

\noindent\textbf{2.}  Let $S_1$, $S_2$ be isometries without unitary
parts, and $[S_1,S_2]=[S_1^*,S_2^*]=0$.
We will show that this description problem contains the
description problem for a  pair of unitary operators\/ $U$, $V\in
L(H)$.  Define the operators $S_1$, $S_2\in
L(\bigoplus_1^{\infty}H)$ in the following way:
\begin{gather*} S_1=
\begin{pmatrix} 0 & 0 &  &  &  &  &  \\ 
0 & 0      & &     &  & \smash{\text{\raise5pt\hbox{\Large
0}}} & \\ 
I & 0      & 0 & 0      &  &  & \\
 0     & I & 0 & 0      &  &  &   \\ 
     & & I     & 0 & 0 & 0 & \\
      &       & 0     & I      &0 & 0 & \\
    &\smash{\text{\raise5pt\hbox{\Large
0}}} &  &        & \ddots&&\ddots
\end{pmatrix}, 
\\*
S_2= \begin{pmatrix} 
0    &2^{-1/2}I   &  &  &\smash{\text{\lower5pt\hbox{\Large
0}}}   \\ 
0    &-2^{-1/2}VU^* &  &  &  \\ 
2^{-1/2}U & 0 & 0 & 2^{-1/2}I &   \\ 
2^{-1/2}V & 0 & 0 & -2^{-1/2}VU^*& \\ 
            &  &  2^{-1/2}U & 0   &\ddots \\ 
            &  &  2^{-1/2}V  & 0  & \ddots  \\ 
  & \smash{\text{\raise5pt\hbox{\Large
0}}}     & \ddots&\ddots& \ddots 
\end{pmatrix}.
\end{gather*}

\allowbreak{}
It is easy to prove the following proposition.

\begin{proposition}
An operator\/ ${\mathcal Y}$ is
intertwining for the pairs $S_i$, $S_i^*$ and $\tilde S_i$,
$\tilde S_i^*$, $i= 1$, $2$, correspondingly, if and only if \/
${\mathcal Y}= Y\otimes I_{\infty}$, where $Y$ is an
intertwining operator for $U$, $V$ and $\widetilde
U$, $\widetilde V$, correspondingly \textup(that is $YU=\tilde
{U}Y$, $YU^*=\tilde U^*Y$,  $YV=\tilde VY $,
$YV^*=\tilde V^*Y$\textup).
\end{proposition}

In Section \ref{sec:3.2.3} we proved the
complexity of the description of some operators classes.
  This proof is similar  to the proof of $*$\nobreakdash-wildness of the
$*$-algebras above in this chapter. But the operators $Z$ and $S_1$, $S_2$
do not belong  to $\mathfrak{K}\otimes L(H)$.
 It seems that the definitions of majorization and $*$-wildness
 (see Sections \ref{sec:3.1.1} and \ref{sec:3.1.2})
  are restrictive in the case
  $n=\infty$. For the unbounded operator the case $n=\infty$ is
  essential (see, e.g., \cite{sam_tur}).
Therefore, probably, these definitions should be extended.

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