Schatten Class Operators in ℒ(La2(ℂ+))\msbm=MTMIB${\cal L}\left( {L_a^2 \left( {{\msbm C}_+ } \right)} \right)$

Abstract

Abstract In this paper, we consider Toeplitz operators defined on the Bergman space L a 2 ( ℂ + ) \msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$ of the right half plane and obtain Schatten class characterization of these operators. We have shown that if the Toeplitz operators 𝕿 φ on L a 2 ( ℂ + ) \msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$ belongs to the Schatten class Sp , 1 ≤p < ∞, then φ ˜ ∈ L p ( ℂ + , d ν ) \msbm=MTMIB$\tilde \phi \in L^p \left( {{\msbm C}_+ ,d\nu } \right)$ , where φ ˜ ( w ) = 〈 φ b w ¯ , b w ¯ 〉 $\tilde \phi \left( w \right) = \left\langle {\phi b_{\bar w} ,b_{\bar w} } \right\rangle $ w ∈ ℂ+ and b w ¯ ( s ) = 1 π 1 + w 1 + w ¯ 2 Rew ( s + w ) 2 $b_{\bar w} (s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + \bar w}}{{2 Rew} \over {\left( {s + w} \right)^2 }}$ . Here d ν ( w ) = | B ( w ¯ , w ) | d μ ( w ) $d\nu (w) = \left| {B(\bar w,w)} \right|d\mu (w)$ , where dμ (w) is the area measure on ℂ+ and B ( w ¯ , w ) = ( b w ¯ ( w ¯ ) ) 2 $B(\bar w,w) = \left( {b_{\bar w} (\bar w)} \right)^2 $ : Furthermore, we show that if φ ∈ Lp (ℂ+,dv), then φ ˜ ∈ L p ( ℂ + , d ν ) \msbm=MTMIB$\tilde \phi \in L^p ({\msbm C}_+ ,d\nu )$ and 𝕿 φ ∈ Sp . We also use these results to obtain Schatten class characterizations of little Hankel operators and bounded operators defined on the Bergman space L a 2 ( ℂ + ) \msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$