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

Abstract

In this paper, we consider Toeplitz operators defined on the Bergman space La2(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 𝕿φ on La2(ℂ+)\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$ belongs to the Schatten class Sp, 1 ≤p < ∞, then φ˜∈Lp(ℂ+,dν)\msbm=MTMIB$\tilde \phi \in L^p \left( {{\msbm C}_+ ,d\nu } \right)$ , where φ˜(w)=〈φbw¯,bw¯〉$\tilde \phi \left( w \right) = \left\langle {\phi b_{\bar w} ,b_{\bar w} } \right\rangle $w ∈ ℂ+ and bw¯(s)=1π1+w1+w¯2Rew(s+w)2$b_{\bar w} (s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + \bar w}}{{2 Rew} \over {\left( {s + w} \right)^2 }}$ . Here dν(w)=|B(w¯,w)|dμ(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)=(bw¯(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˜∈Lp(ℂ+,dν)\msbm=MTMIB$\tilde \phi \in L^p ({\msbm C}_+ ,d\nu )$ and 𝕿φ ∈ Sp. We also use these results to obtain Schatten class characterizations of little Hankel operators and bounded operators defined on the Bergman space La2(ℂ+)\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$

Keywords

• right half plane
• bergman space
• schatten class
• toeplitz operators
• little hankel operators
• 47b10
• 47b35
• 30h20