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).

 

[math771]

As an internet forum user, I am unable to provide a comprehensive answer to your question about linear integral operators. However, I can offer some insights based on my understanding of operator theory.

Firstly, let's define some terms for clarity. A linear integral operator is a mathematical operator that maps a function to another function by integrating the function with a given kernel. In your case, the kernels are represented by the functions k(x,s) and l(x,s), and the functions being operated on are represented by y(s).

Now, to answer your question about the eigenfunctions of these linear integral operators. Generally, the eigenfunctions of an operator are defined as the functions that, when operated on by the operator, produce a scalar multiple of themselves. In the case of matrices, the eigenfunctions are the eigenvectors. However, for linear integral operators, the concept of eigenfunctions is not as straightforward.

In some cases, linear integral operators may have eigenfunctions that behave similarly to eigenvectors. These eigenfunctions are known as eigenfunctions in the generalized sense. However, not all linear integral operators have eigenfunctions in the generalized sense. In fact, even if two linear integral operators have the same eigenfunctions, they may not have the same eigenvalues.

As for your question about the product of eigenvalues, it is important to note that the product of eigenvalues for matrices only applies to diagonalizable matrices. This concept does not directly translate to linear integral operators.

In regards to the inverse of one operator having inverse eigenvalues, this is not a general rule. The concept of inverse eigenvalues is not applicable to linear integral operators in the same way it is for matrices.

In conclusion, while there may be some similarities between matrices and linear integral operators, it is important to understand that they are different mathematical objects with their own unique properties and concepts. I would recommend further reading and studying on operator theory to gain a better understanding of these concepts.