= Spectral theory
{wiki}
<Spectral theory> studies a <linear operator> through the scalars at which it fails to be invertible in the required sense. For a bounded operator $T$ on a complex <Banach space>, its <spectrum> consists of $\lambda$ for which $T-\lambda I$ has no bounded inverse defined on the whole space.
In finite dimensions this is the set of <eigenvalues>. In infinite dimensions there can be more: multiplication by $x$ on $L^2([0,1])$ has <spectrum> $[0,1]$ but no <eigenvectors>. An eigenfunction at $\lambda$ would have to be supported on the single point $x=\lambda$, a null set. <Functions> concentrated near that point nevertheless have arbitrarily small $\|(T-\lambda I)f\|/\|f\|$, ruling out a bounded inverse.
Back to article page
