Communications of the
Korean Mathematical Society CKMS

In this paper, we introduce Toeplitz operators and Hankel operators with complex Borel measures on the closed unit disk. When a positive measure $mu$ on $(-1,1)$ is a Carleson measure, it is known that the corresponding Hankel matrix is bounded and vice versa. We show that for a positive measure $mu$ on $mathbb{D}$, $mu$ is a Carleson measure if and only if the Toeplitz operator with symbol $mu$ is a densely defined bounded linear operator. We also study Hankel operators of Hilbert--Schmidt class.

Keywords: Toeplitz operators, Hankel operators, densely defined operators, Carleson measures

MSC numbers: Primary 47B35, 47L60, 28A25