Properties of $k^{th}$-order (slant Toeplitz + slant Hankel) operators on $ H^2(\mathbb{T} )$

Commun. Korean Math. Soc. 2020 Vol. 35, No. 3, 855-866 https://doi.org/10.4134/CKMS.c190403 Published online March 17, 2020 Printed July 31, 2020

Anuradha Gupta, Bhawna Gupta Netaji Nagar, University of Delhi; University of Delhi

Abstract : For two essentially bounded Lebesgue measurable functions $ \phi $ and $ \xi $ on unit circle $ \mathbb{T}$, we attempt to study properties of operators $ S_{\mathcal{M}(\phi, \xi)}^k = S_{T_\phi}^k + S_{H_\xi}^k$ on $ H^2(\mathbb{T}) $ ($ k \geq 2 $), where $ S_{T_\phi}^k $ is a $k^{th}$-order slant Toeplitz operator with symbol $\phi $ and $ S_{H_\xi}^k $ is a $k^{th}$-order slant Hankel operator with symbol $\xi $. The spectral properties of operators $ S_{\mathcal{M}(\phi, \phi)}^k $ (or simply $ S_{\mathcal{M}(\phi)}^k $) are investigated on $ H^2(\mathbb{T}) $. More precisely, it is proved that for $ k =2 $, the Coburn's type theorem holds for $ S_{\mathcal{M}(\phi)}^k $. The conditions under which operators $ S_{\mathcal{M}(\phi)}^k $ commute are also explored.

Supported by : Support of CSIR-UGC Research Grant(UGC) [Ref.~No.~21/12/2014(ii) EU-V] to second author for carrying out the research work is gratefully acknowledged