MHB Do Commuting Linear Integral Operators Share Eigenfunctions and Eigenvalues?

[sarrah1]

I have two linear integral operators

$Ky=\int_{a}^{b} \,k(x,s)y(s)ds$

$Ly=\int_{a}^{b} \,l(x,s)y(s)ds$

their kernels commute

Do they have same eigenfunctions like matrices and for instance in this case their product is the product of their eigenvalues. I am poorly read in operator theory but well read in matrices. Does the inverse of one has inverse eigenvalues
thanks
Sarrah

 

[Jhoneine]

A:The answer to your question is no.Let $K$ and $L$ be the operators you describe in your question, and let $\lambda$ be an eigenvalue of $K$ with eigenvector $v$. Then, by definition, $Kv = \lambda v$. However, it does not follow that $Lv = \lambda v$; in particular, it is not necessarily true that $Lv$ will have the same eigenvalue as $Kv$. It is possible that $Lv$ has the same eigenvalue as $Kv$, but there is no general guarantee that this is the case.In fact, in many cases it is possible to construct explicit examples of two linear integral operators $K$ and $L$ for which their kernels commute, but for which $K$ and $L$ do not have the same eigenfunctions and/or eigenvalues. For instance, let $k(x,s) = s^2$ and $l(x,s) = x^2$. It is straightforward to check that the kernels of $K$ and $L$ commute, but $K$ and $L$ do not have the same eigenfunctions (nor the same eigenvalues).