Essential spectrum of a product of self-adjoint operators which have positive essential spectrum

Question

Given an infinite-dimensional Hilbert space $H$, a bounded linear self-adjoint operator $A:H\to H$ and a bounded linear invertible operator $B:H\to H$.
If both $A$ and $B$ have positive essential spectrum, does the product $BA$ have positive essential spectrum too? Also, in such case, is there a relation between the essential spectrum of $A$, $B$ and $BA$?
I note that in addition $A$ and $B$ does not commute and the definition of essential spectrum that I use is the Wolf or the Weyl one. What I do is the following: $$ \sigma_{ess}(BA)\setminus\{0\}=\sigma_{ess}((B^{\frac 1 2}B^{\frac 1 2})A)\setminus\{0\}=\sigma_{ess}(B^{\frac 1 2}(B^{\frac 1 2}A))\setminus\{0\}= \sigma_{ess}((B^{\frac 1 2}A)B^{\frac 1 2}))\setminus\{0\} $$ where $B^{\frac 1 2}$ is the square root of $B$.
Then, the operator $B^{\frac 1 2}A B^{\frac 1 2}$ is self-adjoint. But I did not know how to verify if $B^{\frac 1 2}A B^{\frac 1 2}$ has positive essential spectrum.