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 ⊂ CnPφg (f) (z) = underover(∫, 0, 1) f (φ (t z)) g (t z) frac(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 B0 to the Bloch-type space Bμ or the little Bloch-type space Bμ, 0, are characterized. In the main results we calculate the essential norm of the operators Pφg : B (or B0) → Bμ (or Bμ, 0) in an elegant way.