Moving one family of commuting self-adjoint operators to another without losing commutativity on the way - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:03:08Z http://mathoverflow.net/feeds/question/51345 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51345/moving-one-family-of-commuting-self-adjoint-operators-to-another-without-losing-c Moving one family of commuting self-adjoint operators to another without losing commutativity on the way fedja 2011-01-06T21:39:22Z 2013-01-25T11:22:00Z <p>This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving his recent results could be replaced by the mean value theorem.</p> <p>Suppose that $A_1,\dots,A_n$ and $B_1,\dots,B_n$ are two commuting families of self-adjoint operators in a Hilbert space $H$ (that is all $A$'s commute, all $B$'s commute, but $A$'s may not commute with $B$'s). Assume that $\|A_k-B_k\|\le 1$ for all $k$. Is it true that there exists a one-parameter family $C_k(t)$ of self-adjoint commuting (for each fixed $t$) operators such that $C_k(0)=A_k$, $C_k(1)=B_k$ and $\int_0^1\left\|\frac d{dt}C_k(t)\right\|dt\le M(n)$ where $M(n)$ is a constant depending on $n$ only? In other words, is the set of commuting $n$-tuples of self-adjoint operators a "chord-arc set"?</p> http://mathoverflow.net/questions/51345/moving-one-family-of-commuting-self-adjoint-operators-to-another-without-losing-c/118615#118615 Answer by Rami for Moving one family of commuting self-adjoint operators to another without losing commutativity on the way Rami 2013-01-11T11:17:39Z 2013-01-11T11:17:39Z <p>Here is a "scratch of a proof". It might be completely wrong since I though about it in 1am.</p> <ol> <li><p>We can attach to the family $A_i$ protectors $P_\lambda$ where $\lambda \in \mathbb R^n$.</p></li> <li><p>For $J=(j_1,...,j_n)\in \mathbb Z^n$ Let $V_J=Im P_J \cap \bigcap Ker P_{j_1,..., j_i-1,...j_n}$.</p></li> <li><p>We have an $A_i$ invariant decomposition $V=\bigoplus V_J$ and $||(A_i-j_i)|_{V_J}|| \leq 1$. </p></li> <li><p>we can assume that $(A_i)|_{V_J}=j_i$ (by connecting it by strait line).</p></li> <li><p>We do the same for $B_i$.</p></li> <li><p>Now the problem should be similar to the f.d. case. This step I did not think through, but I hope it will be OK.</p></li> </ol> <p>Good luck</p>