Describe all three-dimensional vector spaces \(V\) of complex-valued \(C^\infty\) functions on \(\mathbb R\) that are invariant under differentiation.
Solution
Let \(D:V\to V\) denote differentiation.
1. For every Jordan block of \(D\) of size \(m\) and eigenvalue \(\lambda\),the corresponding invariant subspace is\[ e^{\lambda x}\operatorname{span}_{\CC}\{1,x,\ldots,x^{m-1}\}. \]
Choose a Jordan chain \(f_0,\ldots,f_{m-1}\) satisfying \[ (D-\lambda)f_0=0, \qquad (D-\lambda)f_j=f_{j-1}\quad(1\leq j<m). \] Write \(f_j=e^{\lambda x}g_j\). Then \[ (D-\lambda)f_j=e^{\lambda x}g_j', \] so \(g_0'=0\) and \(g_j'=g_{j-1}\). Inductively, \(g_j\) is a polynomial of degree at most \(j\), with nonzero coefficient of \(x^j\) because the Jordan chain is linearly independent. Hence \[ \operatorname{span}\{f_0,\ldots,f_{m-1}\} = e^{\lambda x}\operatorname{span}\{1,x,\ldots,x^{m-1}\}. \]
2. Every finite-dimensional differentiation-invariant space ofcomplex-valued smooth functions is a direct sum\[ \bigoplus_{j=1}^r e^{\lambda_j x} \operatorname{span}_{\CC}\{1,x,\ldots,x^{m_j-1}\}, \]where the \(\lambda_j\) are distinct and \(m_j\geq1\).
Over \(\CC\), the linear operator \(D\) has a Jordan decomposition. Group its Jordan blocks by eigenvalue. For a fixed eigenvalue \(\lambda\), there cannot be two Jordan blocks: each block contains a nonzero eigenvector, while every smooth solution of \[ f'=\lambda f \] is a scalar multiple of \(e^{\lambda x}\), so the \(\lambda\)-eigenspace of \(D\) on the ambient function space is one-dimensional. Thus there is at most one block for each eigenvalue. Step 1 identifies the subspace belonging to each block, and the Jordan decomposition gives their direct sum.
3. If \(\dim V=3\), then \(V\) is exactly one of the following forms:\[ \boxed{ \begin{aligned} &\operatorname{span}_{\CC}\{e^{\lambda x},e^{\mu x},e^{\nu x}\}, &&\lambda,\mu,\nu\text{ pairwise distinct};\\ &\operatorname{span}_{\CC}\{e^{\lambda x},xe^{\lambda x},e^{\mu x}\}, &&\lambda\neq\mu;\\ &\operatorname{span}_{\CC}\{e^{\lambda x},xe^{\lambda x},x^2e^{\lambda x}\}, &&\lambda\in\CC. \end{aligned}} \]
By step 2, the positive integers \(m_j\) sum to \(3\). The only partitions of \(3\) are \(1+1+1\), \(2+1\), and \(3\). Substituting these three possibilities into the decomposition of step 2 gives exactly the displayed list.
4. Every space in step 3 is three-dimensional and invariant underdifferentiation.
For each \(\lambda\) and \(m\), \[ D\left(e^{\lambda x}x^k\right) = \lambda e^{\lambda x}x^k +k e^{\lambda x}x^{k-1}, \] so each listed span is invariant. The displayed generators are linearly independent: within one exponential block this follows from independence of \(1,x,\ldots\), and blocks with distinct exponents are generalized eigenspaces of \(D\) for distinct eigenvalues. Their dimensions therefore sum to \(3\).
5. Q.E.D.
Step 3 shows that every possible \(V\) is on the list, and step 4 verifies that every space on the list has the required properties.