Let A be an nxn matrix of operators acting on a finite sum of Banach spaces. Gershgorin's disc theorem extends to this setting; the discs centered at the diagonal entries of the matrix now being replaced by "discs" surrounding the spectrum of each diagonal entry operator. The union of these "discs" is the Gershgorin set for A. Taking advantage that similar operators have the same spectrum, it is useful to consider the intersection of all the Gersgorin's sets corresponding to "similar" operators to A. These "minimal Gershgorin's sets" have been studied by Varga and others in the scalar and partitioned finite matrices cases. More refined results are obtained when the spaces are Hilbert spaces and when the diagonal entries operators are normal. The boundary of the minimal Gershgorin's set is particularly interesting to look at.