Communications of the
Korean Mathematical Society CKMS

Many scholars studied the boundedness of Ces\`{a}ro operators between $Q_K$ spaces and Bloch spaces of holomorphic functions in the unit disc in the complex plane, however, they did not describe the compactness. Let $0<\alpha<+\infty$, $K(r)$ be right continuous nondecreasing functions on $(0,+\infty)$ and satisfy $$\int_{0}^{1\over e}K(\log{1\over r})rdr<+\infty.$$Suppose $g$ is a holomorphic function in the unit disk. In this paper, some sufficient and necessary conditions for the extended Ces\`{a}ro operators $T_g$ between $\alpha$-Bloch spaces and $Q_K$ spaces in the unit disc to be bounded and compact are obtained.

Keywords: extended ces\`{a}ro operators, $\alpha$-Bloch spaces, $Q_K$ spaces, boundedness, compactness

MSC numbers: Primary 30H25, 30H30, 47B33, 47B38