Eigenvalues of the differentiation operator I have a linear operator $T_1$ which acts on the vector space of polynomials in this way:
$$T_1(p(x))=p'(x).$$
How can I find its eigenvalues and how can I know whether it is diagonalizable or not?

Take the derivative of $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ (with $a_n\neq 0$), and set it equal to $\lambda a_nx^n+\cdots+\lambda a_0$.  Look particularly at the equality of the coefficients of $x^n$ to determine what $\lambda$ must be.  Once you know what the eigenvalues are, consider which possible diagonalized linear transformations have that eigenvalue set, and whether such linear transformations can be similar to differentiation.
