Functional calculus of unitary matrices and commutator norms: reference request - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:02:10Z http://mathoverflow.net/feeds/question/108418 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108418/functional-calculus-of-unitary-matrices-and-commutator-norms-reference-request Functional calculus of unitary matrices and commutator norms: reference request Terry Loring 2012-09-29T19:07:18Z 2012-10-01T14:31:04Z <p>Suppose we have a normal matrix $A$ and a general matrix $B$. For a continuous function $f$ on the disk we can find upper-bounds on <code>$\Vert[f(A),B]\Vert$</code> in terms of <code>$\Vert[A,B]\Vert$</code>. The more we know about $B$ and $f$ the more hope we have of a tight estimate. I work here with the operator norm, and add the restrictions $\left\Vert A\right\Vert \leq1$ and $\left\Vert B\right\Vert \leq1$ for simplicity.</p> <p>If the spectrum of $A$ is in $[-1,1]$ then we are assuming $A$ is a Hermitian contraction. One method of attack uses the Fourier transform, where for nice $f$ we write <code>$f^{\prime}(x)=\int_{-\infty}^{\infty}g(t)e^{itx}\, dx$</code> and <code>$\left[f(A),B\right]=\int_{-\infty}^{\infty}\frac{g(t)}{it}\left[e^{itA},B\right]\, dx.$</code> From here we get quickly the estimate <code>$\left\Vert \left[f(A),B\right]\right\Vert \leq\left\Vert \widehat{f^{\prime}}\right\Vert _{1}\left\Vert \left[A,B\right]\right\Vert .$</code> Matt Hastings familiarized me with this trick. I suppose this is folklore. Uffe Haggerup thinks such a result and generalizations are in some old papers on derivations.</p> <p>My current interest is in another special case, assuming $A$ has spectrum in the unit circle. Let's replace $A$ by $U$, now a unitary matrix. This is easier, as we can watch what happens for a trig-polynomial. Assume $f$ on the circle is given by <code>$f(z)=\sum_{-N}^{N}a_{n}z^{n}$</code> and use the estimate <code>$\left\Vert \left[U^{n},B\right]\right\Vert \leq|n|\left\Vert \left[U,B\right]\right\Vert$</code> to find <code>$\left\Vert \left[f(U),B\right]\right\Vert \leq\sum_{-N}^{N}|n||a_{n}|\left\Vert \left[U,B\right]\right\Vert .$</code> If we let $q(x)=f(e^{ix})$ then we can use classical Fourier series notation and can use the replacement <code>$\left\Vert \widehat{q^{\prime}}\right\Vert _{1}=\sum_{-N}^{N}|n||a_{n}|.$</code></p> <p>This leads to a useful result.</p> <blockquote> <p>For nice periodic functions $q$ we define $f$ so that $q(x)=f(e^{ix})$. The formula <code>$\left\Vert \left[f(U),B\right]\right\Vert \leq\left\Vert \widehat{q^{\prime}}\right\Vert _{1}\left\Vert \left[U,B\right]\right\Vert$</code> should hold true for at least all unitary matrices and all matrices $B$ of norm at most one. Where is this written down? I am flexible as to the meaning of nice here. Thrice differentiable handles most cases I find interesting.</p> </blockquote>