We prove a sufficient condition for products of Toeplitz operators T f Tḡ, where f, g are square integrable holomorphic functions in the unit ball in C n , to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators H f H * g is also given.

We apply the theory of de Branges–Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $$H^2$$ H 2 . In particular, we discuss the kernels of the operators $$T_{{\bar{f}}/ f}$$ T f ¯ / f and $$T_{{\bar{I}}{\bar{f}}/ f}$$ T I ¯ f ¯ / f , where f is an outer function in $$H^2$$ H 2 and I is inner such that $$I(0)=0$$ I ( 0 ) = 0 . We also obtain a result on the structure of de Branges–Rovnyak spaces generated by nonextreme functions.

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.