On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball

Abstract

We introduce the following integral-type operator on the space H ( B ) of all holomorphic functions on the unit ball B ⊂ C n P φ g ( f ) ( z ) = ∫ 0 1 f ( φ ( t z ) ) g ( t z ) d t t , z ∈ B , where g ∈ H ( B ) , g ( 0 ) = 0 and φ is a holomorphic self-map of B . The boundedness and compactness of the operator from the Bloch space B or the little Bloch space B 0 to the Bloch-type space B μ or the little Bloch-type space B μ , 0 , are characterized. In the main results we calculate the essential norm of the operators P φ g : B ( or B 0 ) → B μ ( or B μ , 0 ) in an elegant way.