Let $T$ be a linear operator such that minimal polynomial of $T$ is the product of first degree polynomial then $T$ can be written as $D+N$ where $D$ is a diagonalizable linear operator and $N$ is nilpotent linear operator.
Now, this is something that is after the primary decomposition theorem. I am really confused understand if there is some link between the two.
My understanding is that $T$ w.r.t. some basis can be written as a triangulable linear operator and so it can be written as a sum of nilpotent operator and a diagonalizable linear operator.is this ok?
Well, if the minimal polynomial of $T$ is the product of first degree polynomials, then $T$ is diagonalizable. This is an immediate consequence of the Primary Decomposition Theorem.
Therefore
$$T = T + 0$$ where $T$ is diagonalizable and $0$ obviously nilpotent.