We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

Given two arbitrary sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying

$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$

$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

$c= ({c}_{n} )_{n\geq 0} $

$\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $

${\ell }^{2} ({ \mathbb{Z} }_{+ } )$

$({\lambda }_{j} )_{j\geq 1} $

$({\mu }_{j} )_{j\geq 1} $

$c$

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

$({\lambda }_{j} )_{j\geq 1} $

$({\mu }_{j} )_{j\geq 1} $

$({\rho }_{j} )_{j\geq 1} $

$({\sigma }_{j} )_{j\geq 1} $

$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$

$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

${\ell }^{2} ({ \mathbb{Z} }_{+ } )$

$({\rho }_{j} )_{j\geq 1} $

$({\sigma }_{j} )_{j\geq 1} $

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

We solve a joint similarity problem for pairs of operators of Foias–Williams/Peller type on weighted Bergman spaces. We show that for the single operator, the Hardy space theory established by Bourgain and Aleksandrov–Peller carries over to weighted Bergman spaces, by establishing the relevant weak factorizations. We then use this fact, together with a recent dilation result due to the first author and Rochberg, to show that a commuting pair of such operators is jointly polynomially bounded if and only if it is jointly completely polynomially bounded. In this case, the pair is jointly similar to a pair of contractions by Paulsen’s similarity theorem.

AMS 2000 Mathematics subject classification: Primary 47B35; 47B47