Question

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.

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"?

Answer 1

Here is a "scratch of a proof". It might be completely wrong since I though about it in 1am.

1. We can attach to the family $A_i$ protectors $P_\lambda$ where $\lambda \in \mathbb R^n$.

2. For $J=(j_1,...,j_n)\in \mathbb Z^n$ Let $V_J=Im P_J \cap \bigcap Ker P_{j_1,..., j_i-1,...j_n}$.

3. We have an $A_i$ invariant decomposition $V=\bigoplus V_J$ and $||(A_i-j_i)|_{V_J}|| \leq 1$.

4. we can assume that $(A_i)|_{V_J}=j_i$ (by connecting it by strait line).

5. We do the same for $B_i$.

6. 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.

Good luck