Stampfli has shown that for a given T ∈ B ( H ) T\, \in \,B\left ( H \right ) there exists a K ∈ C ( H ) K\, \in \,C\left ( H \right ) so that σ ( T + K ) = σ w ( T ) \sigma \left ( {T\, + \, K} \right )\,= \,{\sigma _w}\left ( T \right ) . An analogous result holds for the essential numerical range W e ( T ) {W_e}\left ( T \right ) . A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator T ∈ B ( H ) T\, \in \,B\left ( H \right ) if σ ( T + K ) = σ w ( T ) \sigma \left ( {T\, + \, K} \right )\, = \,{\sigma _w}\left ( T \right ) and W ( T + K ) ¯ = W e ( T ) \overline {W\left ( {T \, + \, K} \right )} \, = \,{W_e}\left ( T \right ) . Theorem. For each block diagonal operator T, there exists a compact operator K which preserves the Weyl spectrum and essential numerical range of T. The perturbed operator T + K T \, + \, K is not, in general, block diagonal. An example is given of a block diagonal operator T for which there can be no block diagonal perturbation which preserves the Weyl spectrum and essential numerical range of T.