Publications and Preprints
A stronger form of Yamamoto's theorem II - Spectral operators
(with Dr. Soumyashant Nayak).
(with Dr. Soumyashant Nayak).
On the Jordan-Chevalley-Dunford decomposition of operators in type \(I\) Murray-von Neumann algebras
(with Dr. Soumyashant Nayak).
(with Dr. Soumyashant Nayak).
My research lies at the intersection of functional analysis and operator algebras, with a focus on structural and spectral properties of bounded linear operators on Hilbert spaces. I am interested in exploring key equivalence relations such as similarity, quasi-similarity, unitary equivalence, and using these to address approximation problems in operator algebras.
“A Stronger Form of Yamamoto’s Theorem II — Spectral Operators” is a follow-up to Nayak’s earlier work titled “A Stronger Form of Yamamoto’s Theorem on Singular Values,” which extended Yamamoto’s classical singular-value theorem by proving that the normalized power sequence (NPS), \(\{∣A^n∣^\frac{1}{n}\}_{n \in \mathbb N}\), of a matrix \(A\) converges in norm and identifying its positive-semidefinite limit in terms of certain projections associated with the diagonalizable part in the Jordan-Chevalley decomposition of \(A\). The subsequent paper further generalizes this result to the setting of spectral operators on complex Hilbert spaces and identifies the limit of the NPS through the idempotent-valued spectral resolution of the spectral operator. The result bridges finite-dimensional matrix theory and infinite-dimensional operator theory, complementing Haagerup-Schultz convergence results in type \(II_1\) factors.
“On the Jordan-Chevalley-Dunford decomposition of operators in type \(I\) Murray-von Neumann algebras”, shows at first that, for \(n \ge 3\), the mapping on \(M_n(\mathbb C)\) which sends a matrix to its diagonalizable part in its Jordan-Chevalley decomposition, is norm-unbounded on any neighbourhood of the zero matrix. Then leveraging the unboundedness of the Jordan-Chevalley decomposition in \(M_n(\mathbb C)\), establishes a canonical Jordan-Chevalley-Dunford decomposition for densely-defined closed operators affiliated with a type \(I\) finite von Neumann algebra, by meaningfully identifying a scalar-type part and a quasinilpotent part; mirroring the Jordan-Chevalley decomposition for matrices, and the Dunford decomposition for spectral operators. It also proves the convergence of normalized power sequence of such operators by making an essential use of the corresponding result for spectral operators. For a (bounded) operator in a type \(I\) finite von Neumann algebra, the aforesaid scalar-type and quasinilpotent parts in the Jordan-Chevalley-Dunford decomposition need not be bounded, and the functorial nature of this decomposition indicates that considering unbounded affiliated operators is both necessary and natural in the quest for a Jordan-Chevalley-Dunford decomposition for bounded operators in type $II_1$ von Neumann algebras.