<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we consider Toeplitz operators defined on the Bergman space 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>L</mi><mi>a</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mo>ℂ</mo><mo>+</mo></msub></mrow><mo>)</mo></mrow></math><tex-math>\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$</tex-math></alternatives></inline-formula>
 of the right half plane and obtain Schatten class characterization of these operators. We have shown that if the Toeplitz operators 𝕿<italic>φ</italic> on 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>L</mi><mi>a</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mo>ℂ</mo><mo>+</mo></msub></mrow><mo>)</mo></mrow></math><tex-math>\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$</tex-math></alternatives></inline-formula>
 belongs to the Schatten class <italic>Sp</italic>, 1 ≤<italic>p &lt; ∞,</italic> then 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>φ</mi><mo>˜</mo></mover><mo>∈</mo><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mrow><msub><mo>ℂ</mo><mo>+</mo></msub><mo>,</mo><mi>d</mi><mi>ν</mi></mrow><mo>)</mo></mrow></math><tex-math>\msbm=MTMIB$\tilde \phi \in L^p \left( {{\msbm C}_+ ,d\nu } \right)$</tex-math></alternatives></inline-formula>
, where 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>φ</mi><mo>˜</mo></mover><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>〈</mo><mrow><mi>φ</mi><msub><mi>b</mi><mover><mi>w</mi><mo>¯</mo></mover></msub><mo>,</mo><msub><mi>b</mi><mover><mi>w</mi><mo>¯</mo></mover></msub></mrow><mo>〉</mo></mrow></math><tex-math>$\tilde \phi \left( w \right) = \left\langle {\phi b_{\bar w} ,b_{\bar w} } \right\rangle $</tex-math></alternatives></inline-formula><italic>w ∈</italic> ℂ+ and 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_006.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mover><mi>w</mi><mo>¯</mo></mover></msub><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mi>π</mi></msqrt></mrow></mfrac><mfrac><mrow><mn>1</mn><mo>+</mo><mi>w</mi></mrow><mrow><mn>1</mn><mo>+</mo><mover><mi>w</mi><mo>¯</mo></mover></mrow></mfrac><mfrac><mrow><mn>2</mn><mi>Rew</mi></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>s</mi><mo>+</mo><mi>w</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></math><tex-math>$b_{\bar w} (s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + \bar w}}{{2 Rew} \over {\left( {s + w} \right)^2 }}$</tex-math></alternatives></inline-formula>
. Here 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>ν</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mrow><mo>|</mo><mrow><mi>B</mi><mo>(</mo><mover><mi>w</mi><mo>¯</mo></mover><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>|</mo></mrow><mi>d</mi><mi>μ</mi><mo>(</mo><mi>w</mi><mo>)</mo></math><tex-math>$d\nu (w) = \left| {B(\bar w,w)} \right|d\mu (w)$</tex-math></alternatives></inline-formula>
, where <italic>dμ</italic> (<italic>w</italic>) is the area measure on ℂ+ and 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_008.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>(</mo><mover><mi>w</mi><mo>¯</mo></mover><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mrow><msub><mi>b</mi><mover><mi>w</mi><mo>¯</mo></mover></msub><mo>(</mo><mover><mi>w</mi><mo>¯</mo></mover><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn></msup></math><tex-math>$B(\bar w,w) = \left( {b_{\bar w} (\bar w)} \right)^2 $</tex-math></alternatives></inline-formula>
: Furthermore, we show that if <italic>φ ∈ Lp</italic> (ℂ+,<italic>dv),</italic> then 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_009.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>φ</mi><mo>˜</mo></mover><mo>∈</mo><msup><mi>L</mi><mi>p</mi></msup><mo>(</mo><msub><mo>ℂ</mo><mo>+</mo></msub><mo>,</mo><mi>d</mi><mi>ν</mi><mo>)</mo></math><tex-math>\msbm=MTMIB$\tilde \phi \in L^p ({\msbm C}_+ ,d\nu )$</tex-math></alternatives></inline-formula>
and 𝕿<italic>φ</italic> ∈ <italic>Sp</italic>. We also use these results to obtain Schatten class characterizations of little Hankel operators and bounded operators defined on the Bergman space 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_010.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msubsup><mi>L</mi><mi>a</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mo>ℂ</mo><mo>+</mo></msub></mrow><mo>)</mo></mrow></math><tex-math>\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$</tex-math></alternatives></inline-formula></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$L_a^2 \left( {{\msbm C}_+ } \right)$

Schatten Class Operators in 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2017-0017_eq_001.png"/><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>ℒ</mi><mrow><mo>(</mo><mrow><msubsup><mi>L</mi><mi>a</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><msub><mo>ℂ</mo><mo>+</mo></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></math><tex-math>\msbm=MTMIB${\cal L}\left( {L_a^2 \left( {{\msbm C}_+ } \right)} \right)$</tex-math></alternatives></inline-formula>

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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...

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

Published Online: Dec 29, 2017

Page range: 91 - 117

Received: Feb 15, 2017

Accepted: May 18, 2017

DOI: https://doi.org/10.1515/awutm-2017-0017

Keywords
Right half plane, Bergman space, Schatten class, Toeplitz operators, little Hankel operators

© 2017 Namita Das et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Dec 29, 2017

Page range: 91 - 117

Received: Feb 15, 2017

Accepted: May 18, 2017

DOI: https://doi.org/10.1515/awutm-2017-0017

KeywordsRight half plane, Bergman space, Schatten class, Toeplitz operators, little Hankel operators

© 2017 Namita Das et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Keywords
Right half plane, Bergman space, Schatten class, Toeplitz operators, little Hankel operators