A commuting pair of isometries

Question

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.

The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $x_u$ and $x_s$ are unitary and unilateral shift respectively. Indeed $x_u$ is the restriction of $x$ to $H_u=\bigcap x^nH$ and $x_s$ is the restriction of $x$ to $H_s$ where,
$$H_s:=H\ominus H_u=\bigoplus_{n\geq0} (x^nH\ominus x^{n+1}H)$$

For $x\in B(H)$, let us denote $e_x$ by the projection onto $xH$.

Q. I am looking for a commuting pair of isometries $(x_1,x_2)$ in $B(H)$ such that all the following hold:

i) $e_y$ commutes with all projections $e_{x^n}$ for $n\geq1$.

ii) $H_s^{(x)}:=H\ominus (\cap x^nH)$ is not $y$-invariant.

Answer 1

Such a pair $(X,Y)$ is constructed as follows. Consider a Hilbert space $M$ with an orthonormal basis $\{e_n:n\in\mathbb Z\}$ and the bilateral shift $U$ on $M$ such that $Ue_n=e_{n+1}$. Denote by $S$ the restriction of $U$ to the space $N$ generated by $\{e_n:n\geq 0\}$, and construct the space $H=N\oplus M\oplus M\oplus\cdots$, that is, the space of square summable sequences $(m_0,m_1,\dots)$ such that $m_0\in N$ and $m_j\in M$ for $j>0$. Now we define $X,Y$ by setting $$X(m_0,m_1,\dots)=(Sm_0,Um_1,Um_2,\dots),$$ and $$Y(m_0,m_1,\dots)=(0,m_0,m_1,\dots).$$ The range projections are easily calculated: $$YY^*(m_0,m_1,\dots)=(0,P_Nm_1,m_2,\dots),$$ and $$X^nX^{*n}(m_0,m_1,\dots)=(S^nS^{*n}m_0,m_1,\dots).$$ It is trivial to verify that $YY^*$ commutes with $X^nX^{*n}$ for every $n$, but $Y$ does not leave the space $$H\ominus\left(\bigcap_n X^nH\right)=\{(m_0,0,0,\dots):m_0\in N\}$$ invariant.

This construction is based on an idea suggested by Ronald Douglas and formalized in Canonical models for bi-isometries (actual paper, MR review).