Countability of eigenvalues of a linear operator - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:07:20Z http://mathoverflow.net/feeds/question/78220 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78220/countability-of-eigenvalues-of-a-linear-operator Countability of eigenvalues of a linear operator Kofi 2011-10-15T19:16:06Z 2011-10-15T19:30:00Z <p>Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues?</p> <p>Or put the other way around, if I want to ensure that a (not necessarily bounded) linear operator on a separable Hilbert space only has countably many eigenvalues, is closedness (or better said, closability) a sufficient condition?</p> <p>(By the term eigenvalue, I do not only mean a point in the spectrum of course, but one that actually fulfills $Tx = \lambda x$.)</p> http://mathoverflow.net/questions/78220/countability-of-eigenvalues-of-a-linear-operator/78224#78224 Answer by Matthew Daws for Countability of eigenvalues of a linear operator Matthew Daws 2011-10-15T19:28:39Z 2011-10-15T19:28:39Z <p>Let $T:\ell^2\rightarrow\ell^2$ be the backwards shift operator, $T(a_n) = (a_2,a_3,\cdots)$. This is a contraction. For any $\lambda\in\mathbb C$, consider the sequence given by $a_n = \lambda^n$. Thus $(a_{n+1}) = (\lambda^2,\lambda^3,\cdots) = \lambda(\lambda,\lambda^2,\cdots)$ and so, if $(a_n)\in\ell^2$, then $(a_n)$ is an eigenvector of $T$, for eigenvalue $\lambda$. Of course, $\sum_n |\lambda^n|^2 = \sum_n |\lambda^2|^n &lt;\infty$ if and only if $|\lambda|&lt;1$.</p> <p>So even a bounded operator can have a continuum of eigenvectors.</p> http://mathoverflow.net/questions/78220/countability-of-eigenvalues-of-a-linear-operator/78225#78225 Answer by Faisal for Countability of eigenvalues of a linear operator Faisal 2011-10-15T19:30:00Z 2011-10-15T19:30:00Z <p>Certainly not. In fact there are bounded operators with uncountably many eigenvalues. For example, the left shift $S^\ast$ defined on $\ell^2$ by $S^\ast(x_1,x_2,\ldots)=(x_2,x_3,\ldots)$ has point spectrum equal to the open unit disk.</p>