Publications and Preprints

Broadly, my research interest lies in the theory of operator algebras. It is well established that focusing on Hilbert space operators often provides insights into the study of structure theory of von Neumann algebras. 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”, extends Yamamoto’s asymptotic singular-value theorem from finite matrices to spectral operators on a Hilbert space, proving that the normalized power sequence, \(∣A_n∣^\frac{1}{n}\), of a spectral operator \(A\), converges in norm and identifying its limit through the idempotent-valued spectral resolution of \(A\). 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 using the unboundedness of the Jordan-Chevalley decomposition in \(M_n(\mathbb C)\), and convergence of the normalized power sequence for spectral operators in \(\mathscr B(\mathscr H)\), it gives 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. It also proves the convergence of normalized power sequence such operators. By observing that 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, the functorial nature of Murray-von Neumann algebras indicates that considering unbounded affiliated operators is both necessary and natural in the quest for a Jordan-Chevalley-Dunford decomposition for bounded operators in finite von Neumann algebras.