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Shanola S. Sequeira

Publications and source records attributed to Shanola S. Sequeira.

8 recordsLinked to original sources
arXiv2026

Quasinormal and hyponormal operators in the $β(H)$-class

Following \cite{BR} we study hyponormal operators in the strictly larger class $β(H)$ of operators whose restriction to every non-zero reducing subspace is norm attaining. The Weyl-type characterization of positive operators via the essential spectrum used in the absolutely norm attaining case is replaced by a well-ordering condition on the eigenvalues (Theorem~\ref{thm:positive}). We prove a transfinite, block-diagonal representation $T=\bigoplus_iλ_iS_i$ (with isometries $S_i$) for the $β(H)$-operators whose $|T|$-eigenspaces are reducing, equivalently the quasinormal operators with $|T|\inβ(H)$, and we show by examples that hyponormality alone does not give such a decomposition. For normal operators we obtain a complete representation, and for hyponormal $β(H)$-operators we show that normality is equivalent to the spectrum having planar Lebesgue measure zero. An analogous area-zero criterion for normality is proved for hyponormal operators attaining the minimum modulus on every non-zero reducing subspace.

math.FA↗
arXiv2026

Lévy measures for Dirichlet-type spaces on the unit bidisc

For the Dirichlet-type space $\mathcal D(ρ^{(1)},ρ^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $ρ^{(1)}$ and $ρ^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the Lévy measure $ν_{(\mathscr M_z,1)}$ associated with the completely alternating multisequence $\left\{\|z^α\|_{\mathcal D(ρ^{(1)},ρ^{(2)})}^2 \right\}_{α\in\mathbb Z_+^2}$ admits the explicit representation \begin{equation*} dν_{(\mathscr M_z,1)}(x)=\frac{1}{1-x_1}d((S_{*}ρ^{(1)})\timesδ_1)(x)+\frac{1}{1-x_2}d(δ_1\times (S_{*}ρ^{(2)}))(x) \end{equation*} on $[0,1]^2\backslash\{(1,1)\},$ where $S_{*}ρ^{(i)},$ $i=1,2$, denotes the pushforward measure of $ρ^{(i)}$ by the map $S:\overline{\mathbb D}\rightarrow [0,1]$ defined by $S(z)=|z|^2,$ $z\in \overline{\mathbb D},$ and $δ_1$ denotes the Dirac measure at 1.

math.FA↗
arXiv2026

Sum of self-commutators of commuting operators

It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal $d$-tuples on a complex Hilbert space $\mathcal H$, namely, commuting $d$-tuples ${\bf T}=(T_1, \ldots, T_d)$ satisfying $\sum_{j=1}^d [T^*_j, T_j] \geqslant 0$. We show that every sum-hyponormal $d$-tuple of compact operators decomposes as the direct sum of a normal $d$-tuple and a quasinilpotent sum-hyponormal $d$-tuple. As a consequence, every sum-hyponormal $d$-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal $d$-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal $d$-tuples with commuting imaginary parts. As a consequence, we prove that ${\bf T}$ is normal whenever the planar Lebesgue measure of $s(σ({\bf T}))$ is zero, where $σ({\bf T})$ denotes the Taylor spectrum of ${\bf T}$ and ${s}$ is the complex-linear polynomial with alternating coefficients $1$ and $i$.

math.FA↗
arXiv2023

Representation of Compact Operators between Banach spaces

In this article, we give a representation for compact operators acting between reflexive Banach spaces, which generalizes the representation given by Edmunds et al. for compact operators between reflexive Banach spaces with strictly convex duals. Further, we give a representation for operators on Banach spaces that are comparable to compact normal operators on Hilbert spaces and illustrate our result with an example.

math.FA↗
arXiv2023

Conditions implying the normality of $\ast$-paranormal operators in the closure of $\mathcal{AN}$-operators

In this article, we first prove the existence of an invariant subspace for a norm-attaining $\ast$-paranormal operator. Then give a representation for $\ast$-paranormal operators in the closure of absolutely norm-attaining operators and further study a few sufficient conditions for the normality of such operators. Finally, we discuss Toeplitz and Hankel $\ast$-paranormal operators in the closure of absolutely norm-attaining operators on the Hardy space.

math.FA↗
arXiv2022

Representation and normality of Hyponormal operators in the closure of $\mathcal{AN}$-operators

Let $H_1$, $H_2$ be complex Hilbert spaces. A bounded linear operator $T : H_1 \to H_2$ is said to be norm attaining if there exists a unit vector $x \in H_1$ such that $\|Tx\| = \|T\|$. If $T|_{M} : M \to H_2$ is norm attaining for every closed subspace $M$ of $H_1$, then we say that $T$ is an absolutely norm attaining ($\mathcal{AN}$-operator). If the norm of the operator is replaced by the minimum modulus $m(T) = \inf\{\|Tx\| : x \in H_1, \|x\| =1\}$, then $T$ is said to be a minimum attaining and an absolutely minimum attaining operator ($\mathcal{AM}$-operator), respectively. In this article, we give representations of quasinormal $\mathcal{AN}$, $\mathcal{AM}$-operators and the operators in the closure of these two classes. Later we extend these results to the class of hyponormal operators in the closure of $\mathcal{AN}$-operators and a further look at some sufficient conditions under which these operators become normal.

math.FA↗
arXiv2022

On the closure of Absolutely Norm attaining Operators

Let $H_1$ and $H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a bounded linear operator. We say $T$ to be norm attaining, if there exists $x\in H_1$ with $\|x\|=1$ such that $\|Tx\|=\|T\|$. If for every closed subspace $M$ of $H_1$, the restriction $T|_{M}:M\rightarrow H_2$ is norm attaining then, $T$ is called absolutely norm attaining operator or $\mathcal{AN}$-operator. If we replace the norm of the operator by the minimum modulus $m(T)=\inf{\{\|Tx\|:x\in H_1,\; \|x\|=1}\}$, then $T$ is called the minimum attaining and the absolutely minimum attaining operator (or $\mathcal{AM}$-operator) respectively. In this article, we discuss about the operator norm closure of the $\mathcal{AN}$-operators. We completely characterize operators in this closure and study several important properties. We mainly give the spectral characterization of the positive operators in this class and give the representation when the operator is normal. Later we also study the analogous properties for $\mathcal{AM}$-operators and prove that the closure of $\mathcal{AM}$-operators is same as that of the closure of $\mathcal{AN}$-operators. As a consequence, we prove similar results for operators in the norm closure of $\mathcal{AM}$-operators.

math.FA↗