Let $T$ be an operator acting on a Banach space $X$. We show that between extensions of $T$ to some Banach space $Y\supset X$ which do not increase the defect spectrum (or the spectrum) it is possible to find an extension with the minimal possible defect spectrum.
 B. Bollobás: To what extent can the spectrum of an operator be diminished under an extension, in: Linear and Complex Analysis Problem Book, V. P. Havin, S. V. Hruščev and N. K. Nikol’skij (eds.), Lecture Notes in Math. 1043, Springer, Berlin 1984, 210.
 V. Müller: On the joint essential spectrum of commuting operators
. Acta Sci. Math. Szeged 57 (1993), 199–205. MR 1243277