Mathematics is often divided into Analysis and Algebra. I want to know under which area Operator Theory lies. I have studied functional analysis where we studied operators on infinite dimensional spaces. I have come across terms like operator algebra and got confused whether it is a part of algebra or analysis. I want to study operator theory so I would like to know which is required more - Algebra or Analysis?


"Division" is a poor choice, as it would suggest that analysis and algebra are two disparate areas of mathematics when in reality they are closely intertwined. (Also, whoever did the division apparently didn't care much for geometry.) This is perhaps most apparent in the area of functional analysis, into which operator theory is most commonly considered to fall under.

Functional analysis, as you may know, is the study of vector spaces endowed with a topology, the so-called topological vector spaces (TVS), and operators on topological vector spaces that respect these two combined structures: namely, linear operators that are continuous with respect to the underlying topologies. In operator theory, one focuses on these operators, their properties, and collections of operators with possibly additional algebraic structure, such as $C^*$-algebras and von Neumann algebras.

To properly study these objects one needs to be equally comfortable with algebra, analysis, and topology. The algebraic, analytic, and topological properties of structures defined using TVS-s and operators on TVS-s are closely intertwined. For example, an important theorem of operator theory is the von Neumann bicommutant theorem, which relates a purely topological concept (the weak and strong operator closures of a von Neumann algebra) and a purely algebraic concept (the bicommutant of a von Neumann algebra).

That being said, functional analysis is a big field. There are areas of functional analysis that really emphasize skill with analysis as opposed to algebra; you see this more toward the side of functional analysis that interacts with PDEs. Operator theory is sort of on the other side of functional analysis, more abstract and less concerned with quantitative estimates; here some skill with algebra becomes more important. At the research level operator theory can become extremely algebraic (as far as I can tell). But you won't get anywhere in any part of functional analysis without a decent understanding of algebra, analysis, and topology all at the same time.


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