Properties of $k^{th}$-order (slant Toeplitz + slant Hankel) operators on $ H^2(\mathbb{T} )$
Commun. Korean Math. Soc.
Published online March 17, 2020
Anuradha Gupta and Bhawna Gupta
Associate professor, University of Delhi, Delhi, India., University of Delhi, Delhi, India.
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 $k^{th}$-order slant Toeplitz operator with symbol $\phi $ and $ S_{H_\xi}^k $ is $k^{th}$-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.
Keywords : $k^{th}$-order slant Toeplitz operator, $k^{th}$-order slant Hankel operator, $k^{th}$-order (slant Toeplitz + slant Hankel) operator, Fredholm operator.
MSC numbers : 47B35, 47B30.
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