scholarly journals On a class of unitary operators on the Bergman space of the right half plane

2018 ◽  
Vol 42 (2) ◽  
Author(s):  
NAMITA DAS ◽  
JITENDRA KUMAR BEHERA
Keyword(s):  
Bergman Space ◽  
Half Plane ◽  
Unitary Operators ◽  
The Right

scholarly journals A Berezin-type map and a class of weighted composition operators

2017 ◽  
Vol 4 (1) ◽  
pp. 18-31
Author(s):  
Namita Das
Keyword(s):  
Linear Operator ◽  
Bergman Space ◽  
Half Plane ◽  
Composition Operators ◽  
Weighted Composition Operators ◽  
Unitary Operators ◽  
Area Measure ◽  
Weighted Composition ◽  
The Right

Abstract In this paper we consider the map L defined on the Bergman space $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ of the right half plane ℂ+ by $(Lf)(w) = \pi M'(w)\int\limits_{{{\rm\mathbb{C}}_{\rm{ + }}}} {\left( {{f \over {M'}}} \right)} (s){\left| {{b_w}(s)} \right|^2}d\tilde A(s)$ where ${b_{\bar w}}(s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + w}}{{2{\mathop{Re}\nolimits} w} \over {{{(s + w)}^2}}}$ , s ∈ ℂ+ and $Ms = {{1 - s} \over {1 + s}}$ . We show that L commutes with the weighted composition operators Wa, a ∈ 𝔻 defined on $L_a^2({{\rm\mathbb{C}}_{\rm{ + }}})$ , as ${W_a}f = (f \circ {t_a}){{M'} \over {M' \circ {t_a}}}$ , $f \in L_a^2(\mathbb{C_ + })$ . Here $${t_a}(s) = {{ - ids + (1 - c)} \over {(1 + c)s + id}} , if a = c + id ∈ 𝔻 c, d ∈ ℝ. For a ∈ 𝔻, define ${V_a}:L_a^2({{\mathbb{C}}_{\rm{ + }}}) \to L_a^2({{\mathbb{C}}_{\rm{ + }}})$ by (Vag)(s) = (g∘ta)(s)la(s) where $la(s) = {{1 - {{\left| a \right|}^2}} \over {{{((1 + c)s + id)}^2}}}$ .We look at the action of the class of unitary operators Va, a ∈ 𝔻 on the linear operator L. We establish that Lˆ = L where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over L} = \int\limits_{\mathbb{D}} {{V_a}L{V_a}dA(a)}$ and dA is the area measure on 𝔻. In fact the map L satisfies the averaging condition $$\tilde L({w_1}) = \int\limits_{\mathbb{D}} {\tilde L({t_{\bar a}}({w_1}))dA(a),{\rm{for all }}{w_1} \in {{\rm{C}}_{\rm{ + }}}}$$ where $\tilde L({w_1}) = \left\langle {L{b_{{{\bar w}_1}}},{b_{{{\bar w}_1}}}} \right\rangle$.

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

2017 ◽  
Vol 55 (2) ◽  
pp. 91-117
Author(s):  
Namita Das ◽  
Jitendra Kumar Behera
Keyword(s):  
Bergman Space ◽  
Half Plane ◽  
Toeplitz Operators ◽  
Hankel Operators ◽  
Bounded Operators ◽  
Area Measure ◽  
Schatten Class ◽  
The Right

AbstractIn this paper, we consider Toeplitz operators defined on the Bergman space\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 𝕿φon\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$belongs to the Schatten classSp, 1 ≤p < ∞,then\msbm=MTMIB$\tilde \phi \in L^p \left( {{\msbm C}_+ ,d\nu } \right)$, where$\tilde \phi \left( w \right) = \left\langle {\phi b_{\bar w} ,b_{\bar w} } \right\rangle $w ∈ℂ+and$b_{\bar w} (s) = {1 \over {\sqrt \pi }}{{1 + w} \over {1 + \bar w}}{{2 Rew} \over {\left( {s + w} \right)^2 }}$. Here$d\nu (w) = \left| {B(\bar w,w)} \right|d\mu (w)$, wheredμ(w) is the area measure on ℂ+and$B(\bar w,w) = \left( {b_{\bar w} (\bar w)} \right)^2 $: Furthermore, we show that ifφ ∈ Lp(ℂ+,dv),then\msbm=MTMIB$\tilde \phi \in L^p ({\msbm C}_+ ,d\nu )$and 𝕿φ∈Sp. We also use these results to obtain Schatten class characterizations of little Hankel operators and bounded operators defined on the Bergman space\msbm=MTMIB$L_a^2 \left( {{\msbm C}_+ } \right)$

scholarly journals Analyticity of semigroups on the right half-plane

2017 ◽  
Vol 448 (2) ◽  
pp. 750-766 ◽  
Author(s):  
Mark Elin ◽  
Fiana Jacobzon
Keyword(s):  
Half Plane ◽  
The Right

scholarly journals Abelian theorems for Whittaker transforms

1987 ◽  
Vol 10 (3) ◽  
pp. 417-431 ◽  
Author(s):  
Richard D. Carmichael ◽  
R. S. Pathak
Keyword(s):  
Half Plane ◽  
Complex Variable ◽  
Absolute Value ◽  
Final Value ◽  
The Right ◽  
Abelian Theorems

Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches0or∞in absolute value inside a wedge region in the right half plane.

Asymptotic behaviour of the H-transform in the complex domain

1987 ◽  
Vol 102 (3) ◽  
pp. 533-552 ◽  
Author(s):  
Richard D. Carmichael ◽  
Ram S. Pathak
Keyword(s):  
Asymptotic Behaviour ◽  
Half Plane ◽  
Complex Variable ◽  
Structure Theorem ◽  
Generalized Functions ◽  
Complex Domain ◽  
The Right ◽  
Abelian Theorems

AbstractAbelian theorems for the H-transform of functions and generalized functions are obtained as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. Quasi-asymptotic behaviour (q.a.b.) of the H-transformable generalized functions is defined. A structure theorem for generalized functions possessing q.a.b. is proved and is applied to obtain the asymptotic behaviour of the H-transform of generalized functions having q.a.b. The theorems are illustrated by examples.

scholarly journals Azimuthal and Radial Variation of Wind-Generated Surface Waves inside Tropical Cyclones

2016 ◽  
Vol 46 (9) ◽  
pp. 2605-2621 ◽  
Author(s):  
Paul A. Hwang ◽  
Edward J. Walsh
Keyword(s):  
Wind Speed ◽  
Half Plane ◽  
Wind Wave ◽  
Coverage Area ◽  
Wave Growth ◽  
Local Maxima ◽  
Growth Functions ◽  
Hurricane Wind ◽  
Energy And Momentum ◽  
The Right

AbstractFor wind-generated waves, the wind-wave triplets (reference wind speed, significant wave height, and spectral peak wave period) are intimately connected through the fetch- or duration-limited wave growth functions. The full set of the triplets can be obtained knowing only one of the three, together with the input of fetch (duration) information using the pair of fetch-limited (duration limited) wave growth functions. The air–sea energy and momentum exchanges are functions of the wind-wave triplets, and they can be quantified with the wind-wave growth functions. Previous studies have shown that the wave development inside hurricanes follows essentially the same growth functions established for steady wind forcing conditions. This paper presents the analysis of wind-wave triplets collected inside Hurricane Bonnie 1998 at category 2 stage along 10 transects radiating from the hurricane center. A fetch model is formulated for any location inside the hurricane. Applying the fetch model to the 2D hurricane wind field, the detailed spatial distribution of the wave field and the associated energy and momentum exchanges inside the hurricane are investigated. For the case studied, the energy and momentum exchanges display two local maxima resulting from different weightings of wave age and wind speed. Referenced to the hurricane heading, the exchanges on the right half plane of the hurricane are much stronger than those on the left half plane. Integrated over the hurricane coverage area, the right-to-left ratio is about 3:1 for both energy and momentum exchanges. Computed exchange rates with and without considering wave properties differ significantly.

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