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Consider the wave equation in one dimension:

$$\frac{\partial^2 u}{\partial t^2}-\frac{\partial^2 u}{\partial x^2}=0.$$

The most general solution of this can be written as $F(x-t)+G(x+t)$ for arbitrary functions $F, G$. It is commonly said that this is a consequence of the factorization

$$\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}=\left( \frac{\partial }{\partial t}-\frac{\partial}{\partial x}\right)\left( \frac{\partial }{\partial t}+\frac{\partial}{\partial x}\right).$$

Is this a general fact? If a differential operator $D$ factors into the product of two (commuting) operators $A, B$ then the kernel of $D$ can be expressed as the sum of the kernels of $A$ and $B$, perhaps?

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2 Answers 2

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If I understand the question correctly, the answer is no, because it is not even true for the square of an operator: solutions to $\frac{d^2}{dx^2}f(x)=0$ are not just sums of two of the constant solutions of $\frac{d}{dx}f(x)=0$.

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I'm sorry, I forgot to thank you. You clearly show that the problem is subtler than I thought. Of course we can say that every solution of $Au=0$ is also solution of $ABu=0$ but not the other way round. –  Giuseppe Negro Mar 2 '12 at 18:39

Clearly, $\mathrm{ker}(A)\cup\mathrm{ker}(B)\subset\mathrm{ker}(AB)$, and if the operators are linear, then $\mathrm{ker}(A)\oplus\mathrm{ker}(B)\subset \mathrm{ker}(AB)$. But take $f(x,t)=x$. Then, given $A=\partial_{t}-\partial_{x}$ and $B=\partial_{t}+\partial_{x}$, $f$ is in neither $\mathrm{ker}(A)$ nor in $\mathrm{ker}(B)$. But clearly, $f\in \mathrm{ker}(AB)$, so in general $\mathrm{ker}(A)\cup\mathrm{ker}(B)\subsetneq\mathrm{ker}(AB)$.

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The clear inclusion is that the sum $\text{ker}(A) + \text{ker}(B)$ lies in $\text{ker}(AB)$, not just the union, and the question is whether this inclusion is strict or not. –  Qiaochu Yuan Mar 1 '12 at 2:26

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