Reference request: Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.

Question

I am looking for a proof of the following result.

Let $T$ be a self-adjoint operator, and assume that $V$ is compact and self-adjoint. Then $\sigma_{ess}(T + V ) = \sigma_{ess}(T)$.

Here the context is in a separable Hilbert space and all the operators are bounded. I believe this theorem can be proven with Weyl's criterion but I haven't been able to find the proof online.

Here $\sigma_{ess} = \{t \in \mathbb{R}: \operatorname{dim}\operatorname{range} E(t-\epsilon,t+\epsilon) = \infty \quad \forall \epsilon > 0 \}$ where $E$ denotes the spectral family associated to $T$.