Abstract. Let $\varphi $ be a holomorphic map on the open unit disk ${\msbm D}$ such that $\varphi({\msbm D}) \subset{\msbm D}$ and $H({\msbm D})$ be the space of holomorphic functions on ${\msbm D}.$ For a non-negative integer $n,$ we define linear operators $I^n_{\varphi }$ and $J^n_{\varphi }$ as $I^n_{\varphi }f = (f^{(n)} \circ\varphi )$ and $J^n_{\varphi }f = (f^{(n)} \circ\varphi )', f \in H({\msbm D}),$ respectively, where $f^{(n)}$ denotes the $n$-th derivative of $f.$ In this paper, we characterize boundedness and compactness of $I^n_{\varphi }$ and $J^n_{\varphi }$ between Hardy and weighted Bergman spaces. We also compute the essential norms of $I^n_{\varphi }$ and $J^n_{\varphi }$ acting between these spaces.

AMS Subject Classification (1991): 47B33, 46E10, 30D55

Keyword(s): generalized composition operator, Hardy space, Bergman space, Nevanlinna counting function

Received June 29, 2010, and in revised form September 29, 2010. (Registered under 44/2010.)