arXiv subjects

Ashish Kujur

Publications and source records attributed to Ashish Kujur.

2 recordsLinked to original sources

arXiv2025

Higher order weighted Dirichlet type spaces with poly-superharmonic weights and Dirichlet type operators of finite order

We study higher-order weighted Dirichlet-type spaces on the unit disc associated with a class of poly-superharmonic weights. A higher-order Littlewood Paley formula is established enabling the computation of higher-order weighted Dirichlet integrals and allowing us to relate iterates of the Laplacian of the weight to higher-order defect operators of the shift operator on these spaces. This leads to the introduction of Dirichlet-type operators of finite order, a class containing $m$-isometries as well as completely hyperexpansive and completely hypercontractive operators of finite order. We prove that every cyclic operator in this class admits a functional model as the shift on a suitable higher-order weighted Dirichlet-type space, thereby providing a unified extension of the model theories for cyclic completely hyperexpansive operators and cyclic $m$-isometries.

math.FA↗

arXiv2024

Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space

A well known result of Brown and Halmos shows that the Toeplitz operators induced by $L^{\infty}(\mathbb T)$ symbols on the Hardy space of the unit disc $\mathbb D$ are characterized by the operator identity $T_{\bar{z}}AT_z=A,$ where $T_z, T_{\bar{z}}$ are the Toeplitz operators induced by the function $z$ and $\bar{z}$ on the unit circle $\mathbb T$ respectively. In this paper we introduce and study a class of Toeplitz operators on the Dirichlet space $\mathcal{D} _0$ induced by a symbol class $\mathcal T(\mathcal D _0)= \overline{H^{\infty}_0(\mathbb D)} + \mathcal M(\mathcal D_0 ),$ where $H^{\infty}_0(\mathbb D)$ denotes the set of all bounded analytic function on $\mathbb D$ vanishing at $0$ and $\mathcal M(\mathcal D _0)$ denotes the multiplier algebra of the Dirichlet space $\mathcal D_0.$ We find that the Toeplitz operators on the Dirichlet space $\mathcal D$ induced by the symbol class $\mathcal T(\mathcal D _0)$ is completely characterized by the operator identity $T_{\bar{z}}AT_z=A.$

math.FA↗