Norm of an integral operator

[sarrah1]

Hello

A simple question.
I have a linear integral operator (self-adjoint)

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

where $k$ is the kernel. Can I say that its norm (I believe in $L^2$) equals the spectral radius of $K?$

Thanks!
Sarah

 

[Jhoneine]

No, the norm of an integral operator is not necessarily equal to its spectral radius. The norm of an integral operator is given by the supremum of its operator norm, which is defined as $\sup_{x\neq 0}\frac{||Kx||}{||x||}$. The spectral radius on the other hand is the largest eigenvalue of the operator.

 

[math771]

Hi Sarah,

That's a great question! Yes, you are correct in saying that the norm of $K$ in $L^2$ is equal to its spectral radius. This is known as the "spectral radius formula" and it holds for any bounded linear operator on a Hilbert space. In fact, the spectral radius formula is a powerful tool in the study of linear operators and their spectra.

Hope this helps!
[Your username]