Search

For 0 < p < ∞ , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space A α 2 on the unit ball of C m. The main results of this paper are twofold: (1) For α , β ∈ R , we show that the necessary and sufficient condition for the embedding operator I : A α 2 → A β 2 is in the Schatten p -class is that p (β − α) − 2 m > 0. (2) For a positive Borel measure μ on B m and a real number τ , we prove that the fractional embedding operator I α + τ τ : A α 2 → L 2 (B m , d μ) is in the Schatten p -class if and only if the Berezin transform of μ , denoted by μ ˜ α α + τ , is in L p 2 (B m , d λ) , where d λ (z) = (1 − | z | 2) − m − 1 d v (z) is the Möbius invariant volume measure and d v is the Lebesgue measure on B m , or equivalently, the averaging function of μ , denoted by μ ˆ r α , is in L p 2 (B m , d λ). • For p > 0 , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space on the unit ball. • The range 0 < p < 2 is an open problem left by H. T. Kaptanoğlu (see [5]). • We prove that the fractional embedding operator is in Schatten p -class iff the Berezin transform of the positive Borel measure is in L − space. [ABSTRACT FROM AUTHOR]