the standard quantum-mechanical oscillator, ChK-oscillator can be determined using the ladder operators. The ladder operators $\aspm$ of sectorial oscillator leave invariant the subspaces $\H^{(N)}$, and the ladder operators $\arpm$ of radial oscillator transform the subspaces $\H^{(N)}$ into the subspaces $\H^{(N\pm 2)}$. In addition, operators $\aspm$ and $\arpm$ leave invariant subspaces $\He$ and $\Ho$. Finally, the ladder operators $\acopm$ of boundary oscillator transform the subspace $\H^{(N)}\cap\Hco$ in the subspace$\H^{(N\pm 1)}\cap\Hco$ and equal to zero on $\mathcal{H}^{\bullet,\,0}$, and the ladder operators $\aocpm$ transform the subspace $\H^{(N)}\cap\Hoc$ in the subspace $\H^{(N\pm 1)}\cap\Hoc$ and equal to zero on $\mathcal{H}^{0,\,\bullet}$. Action of these operators is schematically shown in figure 2, where the basic elements $\left\{U_{k,\,l}(z,\overline{z})\right\}_{k,\,l}^{\infty}$ are represented by points of the rectangular lattice.

![Action of ladder operators](pic_2){width="8.0cm"}

### Generalized sectorial oscillator

Taking into account the recurrence relation , we define the “position operators ” $Z$ and $\overline{Z}$ by the relations

$$\begin{aligned}
Z\,U_{k,\,l}(z,\,\z)=U_{k+1,\,l}(z,\,\z)+U_{k,\,l-1}(z,\,\z)+U_{k-1,\,l+1}(z,\,\z),
\label{13-2.1}\\
\overline{Z}\,U_{k,\,l}(z,\,\z)=
U_{k,\,l+1}(z,\,\z)+U_{k-1,\,l}(z,\,\z)+U_{k+1,\,l-1}(z,\,\z). \label{13-2.2}\end{aligned}$$

These relations define the operators $Z$ and $\overline{Z}$ on the dense in $\H$ linear span of all ChK-polynomials $U_{k,\,l}(z,\,\z)$. After closing of these operators they become bounded operators (which we denote by the same symbols) on the whole space $\H$ and satisfy the relations $Z^*=\overline{Z}$, $\overline{Z}^*=Z$.

Following [@N16], we 