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.