Not sure if anyone is still interested in this, but here we go: The claim is false, very much so indeed. Let $A=S+S^*$ be the free Jacobi matrix on $\ell^2$; I write $S$ for the shift $(Sy)_n = y_{n+1}$. Then $\sigma_{ac}(A)=[-2,2]$. Let $B$ be multiplication by a bounded sequence $b_n\ge c$. Then $$ BAB = b(S+S^*)b = bb_+S + bb_-S^* $$ is another Jacobi matrix, and pretty much any choice of $b$ provides a counterexample as it's so easy to destroy ac spectrum of one-dimensional operators. For example, any $b$ that takes only two values and is not eventually periodic works: $\sigma_{ac}(BAB)=\emptyset$

See perhaps my paper http://annals.math.princeton.edu/2011/174-1/p04 for background on this.

This is false. Let $A=S+S^*$ be the free Jacobi matrix on $\ell^2$; I write $S$ for the shift $(Sy)_n = y_{n+1}$. Then $\sigma_{ac}(A)=[-2,2]$. Let $B$ be multiplication by a bounded sequence $b_n\ge c$. Then $$ BAB = b(S+S^*)b = bb_+S + bb_-S^* $$ is another Jacobi matrix, and pretty much any choice of $b$ provides a counterexample as it's so easy to destroy ac spectrum of one-dimensional operators. For example, any $b$ that takes only two values and is not eventually periodic works: $\sigma_{ac}(BAB)=\emptyset$

See perhaps my paper http://annals.math.princeton.edu/2011/174-1/p04 for background on this.