On Some Decompositions of Operators in Hilbert Spaces

Abstract

In this project, we study the Cartesian, polar, and direct sum decompositions of operators in Hilbert spaces to explore the properties of their components. We start by decomposing an operator T into its real and imaginary parts (T = A+iB), analyzing fundamental properties of this decomposition. Next, we investigate the polar decomposition, where operators are represented as a product of a unitary operator and a positive operator. Lastly, we investigate the direct sum decomposition, combining different operator components to compare and contrast their individual properties.