Commuting Linear Operators In Hilbert Spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:10:15Z http://mathoverflow.net/feeds/question/109339 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109339/commuting-linear-operators-in-hilbert-spaces Commuting Linear Operators In Hilbert Spaces Miguel 2012-10-10T23:05:28Z 2012-10-12T04:01:29Z <p>Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find all the linear operators that commute with $L$. </p> <p>Now suppose that $H$ is a Hilbert space and let $L:H\rightarrow H$ be a continuous linear operator. There is some method to determine all the continuous linear operatores that commute with $L$?</p> http://mathoverflow.net/questions/109339/commuting-linear-operators-in-hilbert-spaces/109349#109349 Answer by Nik Weaver for Commuting Linear Operators In Hilbert Spaces Nik Weaver 2012-10-11T02:30:31Z 2012-10-11T02:30:31Z <p>Well ... yes if $L$ is normal (meaning $LL^* = L^*L$; in particular, if $L$ is self-adjoint). Assuming that $H$ is separable, we have a structure theorem which says that $H$ is isomorphic to the $L^2$ sections of a bundle over $[-\|L\|, \|L\|]$ whose fibers are Hilbert spaces, in such a way that $L$ goes to multiplication by $x$. The operators that commute with $L$ are then morally just the operators which preserve each fiber, though one has to be a little careful with measurability issues when making this precise.</p> <p>If $L$ is not normal then at least you can say any weak operator limit of polynomials in $L$ commutes with $L$. But I don't know if you can say much more than that in general.</p> http://mathoverflow.net/questions/109339/commuting-linear-operators-in-hilbert-spaces/109360#109360 Answer by András Bátkai for Commuting Linear Operators In Hilbert Spaces András Bátkai 2012-10-11T07:30:05Z 2012-10-11T07:30:05Z <p>I am far from being an expert, but there is a list of results for special cases in a book by <a href="http://books.google.hu/books?id=ru3eM5LZgYEC&amp;lpg=PP1&amp;hl=de&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">Radjavi and Rosenthal</a>, especially in Chapter 9. </p>