arXiv subjects

Thankarajan Prasad

Publications and source records attributed to Thankarajan Prasad.

7 recordsLinked to original sources

arXiv2026

Slice Regular Capacity and Rational Approximation

Approximation theory in the complex plane is a well established area of mathematical research, with extensive contributions documented in the literature, notably through the classical results of Weierstrass, Runge, Mergelyan, and Vitushkin. Approximation theory in quaternionic settings, particularly rational approximation on compact sets, which studies when slice regular functions can be uniformly approximated by rational functions, is one of the recent areas of interest in quaternionic rational approximation theory (See [6,13]). In this paper, we discuss Vitushkin approximation theorem in the quaternionic setting by using the quaternionic analogue of analytic capacity and also we establish Davies modified Vitushkin scheme in quaternionic setting. We also discuss quaternionic analogue of $γ_{+}$- capacity and Cauchy capacity and the remarkable results of Tolsa [35] and Younsi [45] related to these capacities in complex plane setting.

math.CV↗

arXiv2026

Model Theory for Operators Related to Square Roots of Normal Operators

In this paper we prove that a square root of a cyclic normal operator is unitarily equivalent to a block multiplication operator on a vector-valued Lebesgue space with a $2$--normal symbol. In addition, we show that a cyclic operator which admits a cyclic $2$--normal extension is unitarily equivalent to a block multiplication operator on a corresponding vector-valued Hardy space with $2$--normal symbol. We consider the existence of bounded point evaluations in the vector-valued Hardy space setting; as an application, we prove that an operator admitting a cyclic $2$--normal extension has nontrivial invariant subspaces. We also study uniqueness for minimal $n$--normal extensions of operators that have cyclic $2$--normal extensions.

math.FA↗

arXiv2026

Classes of operators related to subnormal operators

In this paper we attempt to lay the foundations for a theory encompassing some natural extensions of the class of subnormal operators, namely the $n$--subnormal operators and the sub-$n$--normal operators. We discuss inclusion relations among the above mentioned classes and other related classes, e.g., $n$--quasinormal and quasi-$n$--normal operators. We show that sub-$n$--normality is stronger than $n$--subnormality, and produce a concrete example of a $3$--subnormal operator which is not sub-$2$--normal. In \cite{CU1}, R.E. Curto, S.H. Lee and J. Yoon proved that if an operator $T$ is subnormal, left-invertible, and such that $T^n$ is quasinormal for some $n \le 2$, then $T$ is quasinormal. in \cite{JS}, P.Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility. In this paper we consider suitable analogs of this result for the case of operators in the above-mentioned classes. In particular, we prove that the weight sequence of an $n$--quasinormal unilateral weighted shift must be periodic with period at most $n$.

math.FA↗

arXiv2026

Hyponormal block Toeplitz operators with finite rank self-commutators

In this paper, we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. \ Recall that an operator $T_φ$ is hyponormal and $[T_φ^{*}, T_φ]$ is a finite rank operator if and only if there exists a finite Blaschke product $b$ in $\mathcal{E}(φ)$, where $$ \mathcal{E}(φ) := \{k \in H^\infty(\mathbb{T}): \left\|k\right\|_\infty \le 1 \textrm{ and } φ-k\cdot \barφ \in H^\infty(\mathbb{T})\}. $$ An analogous set $\mathcal{E}(Φ)$ can be defined for a matrix-valued symbol $Φ$. \ In the block Toeplitz operator case, we first establish that if a symbol $Φ$ is in $L^\infty(\mathbb{T}, M_n)$ and if $\mathcal{E}(Φ)$ contains a constant unitary matrix $U$, then $T_Φ$ is normal. \ We then obtain a suitable converse, under a mild assumption on the symbol. \ Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang, and W.Y. Lee. \ Concretely, assume that $Φ\in H^{\infty}(\mathbb{T}, M_n)$ is such that $Φ^{\ast}$ is of bounded type and $T_Φ$ is hyponormal. \ Then $[T_Φ^{\ast}, T_Φ]$ is a finite rank operator if and only if there exists a finite Blaschke-Potapov product in $\mathcal{E}(\widetildeΦ)$, where $\widetildeΦ:=\breveΦ^*$ and $\breveΦ(e^{iθ}):=Φ(e^{-iθ})$.

math.FA↗

arXiv2026

Subnormal block Toeplitz operators

In this paper we consider the subnormality of block Toeplitz operators $T_Φ$, where $Φ$ is an $n\times n$ matrix-valued function on the unit circle $\mathbb T$ of the form $$ Φ=QΦ^* \quad \hbox{($Q$ is a finite Blaschke--Potapov product).} $$ This is related to a matrix-valued version of Halmos's Problem 5 and Nakazi-Takahashi Theorem. We ask whether $T_Φ$ is either normal or analytic if $T_Φ$ is subnormal, where $Φ$ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when $Φ^*$ is not of bounded type.

math.FA↗

arXiv2025

Cesàro Operators on Rooted Directed Trees

In this paper, we introduce and investigate the notion of the Cesáro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ When $\mathscr T$ is the rooted tree with no branching vertex, then $C_{\mathscr T}$ is unitarily equivalent to the classical Cesáro operator $C_{0}$ on the sequence space $\ell^2(\mathbb N).$ We prove that for every narrow rooted directed tree $\mathscr T$, $C_{\mathscr T}$ is bounded, with norm bounded above by twice the width of $\mathscr T.$ When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming $\mathscr T$ is leafless and narrow, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree without any branching vertex. In particular, this demonstrates that the verbatim analogue of Kriete-Trutt theorem fails in the context of Cesáro operators on rooted directed trees. Nonetheless, under the same hypotheses, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator.

math.FA↗

arXiv2025

Weyl type theorems for hypercyclic, supercyclic, and Toeplitz operators

In this paper, we study property $(UW_E)$ for hypercyclic and supercyclic operators. The stability of variants of Weyl type theorems under compact perturbations for Toeplitz operators on the Bergman space is also studied. We also provide some examples of Toeplitz operators satisfying Weyl type theorems on the Bergman space and the harmonic Bergman space.

math.FA↗