Shekhawat, Renu2026-05-042026-02-25127p.http://hdl.handle.net/10263/7665This thesis has been completed under the supervision of Dr. Soumyashant NayakThe classical Jordan–Chevalley decomposition expresses a matrix A ∈ Mn(C) as a unique commuting sum A = D + N, where D is diagonalizable and N is nilpotent. Although this decomposition is algebraic in origin, it encodes significant spectral information and, as shown by Nayak, has an important analytic consequence: the convergence of the normalized power sequence {|A^n|^ 1/n }n∈N ; |A| := (A∗A)^1/2 . In this thesis we study Jordan-Chevalley–type decompositions in infinite-dimensional settings and their connection with the convergence behaviour of normalized power sequences. In particular, we discuss this phenomenon for Dunford’s spectral operators and compact operators on a complex Hilbert space, and further extend the theory to operators affiliated with finite type I von Neumann algebras.enJordan-Chevalley DecompositionJordan Canonical FormRoth's removal rulespectral radius formulaYamamoto's theoremDunford decompositionspectral operatorscompact operatorsnormalized power sequencesMurray von Neumann algebrasaffiliated operatorsThesis