arXiv subjects
Publications and source records attributed to Supratim Jana.
We obtain a complete characterization of the class of conjugation operators $C$ on $H^2$ that map shift-invariant subspaces to shift-invariant subspaces. Alongside, we also obtain the existence and non-existence of conjugations that send shift-invariant subspaces to coinvariant subspaces.
We introduce and systematically study a class of operators that arise naturally due to the Beurling decomposition of the Hardy space $H^2=K_θ\oplus θH^2$. While the compressions of classical Toeplitz and Hankel operators to the Beurling subspace $θH^2$ and the model space $K_θ$ account for the diagonal components of the decomposition, the corresponding off-diagonal operators have remained largely unexplored. Motivated by this, we introduce and analyze a new class of operators, termed \emph{restricted Toeplitz} and \emph{restricted Hankel operators}, acting between Beurling subspace $ηH^2$ and model space $K_θ$. Within this framework, we obtain necessary and sufficient conditions for the vanishing, finite-rank, and compactness properties of these operators. We further establish algebraic characterizations in the spirit of Brown-Halmos \cite{BH} and Sarason \cite{SAR, DES}, showing that these operators can be identified through certain operator equations involving compressed shifts. As an application, we introduce the notions of small and big truncated Toeplitz operators, and provide criteria for when they vanish, have finite rank, or are compact.
We introduce the notion of the Dual Truncated Hankel Operator (DTHO) and provide several operator equation characterizations using the dual compressed shift operator. These characterizations are similar to classical results concerning Hankel operators and align with recent findings related to Truncated Hankel Operators (THO) \cite{GM}. Additionally, our work addresses comprehensive solutions to various operator equations encountered in studying THO and the classical Hankel operator. We have also established some fundamental operator-theoretic properties of DTHO that apply to general symbols and symbols under specific conditions.
We present the notion of the nearly dual compressed shift-invariant subspaces of the orthogonal complement of the model space and obtain their structure using Hitt's algorithm \cite{DH}.
In the classical Hardy space $H^2(\mathbb{D})$, it is well-known that the kernel of the Hankel operator is invariant under the action of shift operator S and sometimes nearly invariant under the action of backward shift operator $S^{*}$. It appears in this paper that kernels of finite rank perturbations of Hankel operators are almost shift invariant as well as nearly $S^*$- invariant with finite defect. This allows us to obtain a structure of the kernel in several important cases by applying a recent theorem due to Chalendar, Gallardo, and Partington.