I gave a talk last week in the Geometry, Analysis and Physics seminar with the title “The limit operator symbol”. This was an attempt to distill some of the ideas from my series of posts on the Lindner-Seidel and Spakula-Willett papers, especially post IV of the series. In particular, I wanted to explain the crucial move from having a series of inequalities *witnessed* to having a similar series of inequalities *centrally witnessed*. As Nigel put it during the seminar, we are attempting to describe a “witness (re)location program”: our witnesses are scattered all over \(\Gamma\), and we are attempting to move them all to the “courthouse”, that is, to a neighborhood of the identity, at the same time.

Here is the key fact. We are working on a \(\Gamma\) that has property A.

**Lemma** Let \(C,R,\epsilon>0\) be given. There is an increasing sequence \(r_1,r_2,\ldots\) that has the following property: Suppose that \(A\) is an operator in the translation algebra with propagation \(\le R\) and norm \(\le C\). Suppose that \(\nu(A)\le b\). Then for each \(N\in{\mathbb N}\) there exists a translate \(B_N=A^{\gamma_N}\) of \(A\) with the property that the inequality \( \nu(B_N)\le b \) is \( (2^{-n}\epsilon,r_n) \) – centrally witnessed for all \(n=1,\ldots, N\).

*Proof.* Observe that, using property A, there is an increasing sequence \(s_1(C,\epsilon),s_2(C,\epsilon),\ldots\) having the property that for any operator \(A\) (with propagation \(\le m\) and norm \(\le C\)), any true inequality on the lower norm is \( (2^{-k}\epsilon,s_k(C,\epsilon)) \)-witnessed. (We choose not to indicate the dependence on \(m\) as we will not need to change this in the course of the induction.) See post III for discussion of this. Also, we assume without loss of generality that \(\epsilon<1\).

We proceed now by induction on \(N\). The case \(N=1\) is immediate: We can take \(r_1(C,\epsilon)=s_1(C,\epsilon)\); there is only one inequality to witness and an appropriate translation will produce a central witness.

Suppose by induction that the result has been proved for \(N-1\). We take \(r_N(C,\epsilon) = 2s_N(C,\epsilon)+r_{N-1}(C+4,\epsilon/2)\).

Consider the inequality \(\nu(A) \le b \). It is true, and therefore it is \( (2^{-N},s_N(C,\epsilon))\)-witnessed. Choose a translation \(\gamma\) so that the translated inequality \(\nu(A^\gamma) \le b \) is centrally witnessed with the same constants: that is to say, there is a unit vector \(\eta\) supported in \( B(e;s_N(C,\epsilon))\) such that \( \|A^\gamma\eta\| \le b 2^{-N}\epsilon\). Let \(P\) denote the orthogonal projection onto the functions supported within \(B=B(e;s_N)\).

Let \(X\) denote the operator

\[ X = PA^\gamma P + c(I-P), \]

where the constant \(c\) is chosen strictly greater than \(b+2\cdot 2^{-N}+2 \) and less than \(C+4\). This operator has the same propagation bound as \(A\) and has norm at most \(C+4\). Moreover, it satisfies the inequality \(\nu(X) \le b + \cdot 2^{-N}\epsilon \), as is shown by the vector \(\xi\). By the inductive assumption, then, there is some \(\delta\in\Gamma\) such that, for each \(k=1,\ldots,{N-1}\), the ball \(B(\delta; r_k(C+4,\epsilon/2)) \) contains a \(2^{-k-1}\epsilon\) witness to the above inequality.

Now observe that some part of the support of all the witnesses described above must lie within the original ball \(B\). (Otherwise, we would have \(X\xi = c\xi\), which is not compatible with the inequality we are supposed to be witnessing.) Moreover, an elementary estimate shows that if we truncate the witnesses to \(B\) and renormalize (that is, replace \(\xi\) by \( \|P\xi\|^{-1} P\xi \) ), then what we obtain are still witnesses to the same inequality. Imagine that this is done. The new (truncated and renormalized) \(\xi\)’s are now \( (2^{-k-1}\epsilon,r_k(C+4,\epsilon/2)) \)-witnesses to \(\nu(A^\gamma) \le b + 2^{-N}\epsilon\), and they are all supported in balls \( B(\delta;r_k) \) as well as in \(B\).

From the fact that \(B\cap B(\delta;r_{N-1}\neq\emptyset \) we find that \(|\delta|<s_N +r_{N-1}\). Now translate everything by \(\delta^{-1}\), moving \(\delta\) to the identity. Then, inductively, the inequality \(\nu(A^{\gamma\delta})\le b \) is \( (2^{-k},r_k(C+4,\epsilon/2)) \)-centrally witnessed for \(k=1,\ldots,N-1\). Moreover, it is \((2^{-N},r_N(C,\epsilon))\) centrally witnessed as well, by the \(\delta\)-translate of \(\eta\), which is supported in the ball of radius \( |\delta|+2s_N(C,\epsilon) \le r_N(C,\epsilon) \). This completes the proof.