Lie Algebras of Bounded Operators

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In proofs from the theory of finite-dimensional Lie algebras, the Jordan canonical structure of linear maps on finite-dimensional vector spaces plays a crucial role. However, classical results suggest the use of infinite-dimensional vector spaces as well. For instance, the classical Lie Theorem states that all finite-dimensional irreducible representations of solvable Lie algebras are one-dimensional, indicating that these algebras cannot be classified or distinguished from one another. This highlights the necessity for infinite-dimensional vector spaces. Yet, the structure of linear maps in such spaces is not well understood, as concepts like the Jordan canonical structure of matrices do not apply. Fortunately, a significant class of linear maps on vector spaces of arbitrary dimension exhibits similarities to matrices: the bounded linear operators on complex Banach spaces. Certain bounded operators, including Dunford spectral, Foia§ decomposable, scalar generalized, and Colojoara spectral generalized operators, possess a form of Jordan decomposition theorem. This work aims to present the key results achieved through the study of bounded operators in relation to Lie algebras.