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Let $D$ be a $3\times 3 $ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain.
Denote by $(L^2(\Omega))^3$ the set of square integrable functions over $\Omega$ and by $A$ a bounded self adjoint-linear operator that acts from $(L^2(\Omega))^3$ to itself.
The question : is there a relation between the essential spectrum of the operator $DA$ and the other one of $A$ ? I note that in my situation, the operators $A$ and $D$ does not commute.
Does we have $$ \sigma_{ess}(DA)\subset W(D)\,\sigma_{ess}(A) ?\; (W(D) \mbox{ denotes the numerical range of }D) $$ I note that the possible definition of essential spectrum related to my work is the Wolf or the Weyl one.
Edit : I found that the operator $A$ is self-adjoint, I found also its essential spectrum. More precisely, using an orthogonal decomposition of $(L^2(\Omega))^3$ in the following form

$$ (L^2(\Omega))^3=E_1+E_2+E_3 \;(E_i,i=1,2,3 \mbox{ are some subspaces}) $$ I found that the restriction of $A$ on $E_1$ (resp. $E_2$) is the identity operator (resp. the null operator) and that its restriction on $E_3$ is spectrally equivalent to the operator $\frac 1 2 I$.

Remark: if $\alpha$ is an eigenvalue of $A$ of infinite multiplicity, then the set $$ \{\alpha\beta\,|\,\beta \mbox{ is an eigenvalue }of D\} $$ is contained in the essential spectrum of $DA$.
Could we found any result from the multiplicative perturbation theory of self-adjoint positive operator? Thanks.

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