It was recently proved that in some special cases asymmetric truncated Toeplitz operators can be characterized in terms of compressed shifts and rank-two operators of special form. In this paper we show that such characterizations hold in all cases. We also show a connection between asymmetric truncated Toeplitz operators and asymmetric truncated Hankel operators. We use this connection to generalize results known for truncated Hankel operators to asymmetric truncated Hankel operators.

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In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind.

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.