Operator Holes and extensions of sectorial operators and dual pairs of   contractions
  A description of the set of $m$-sectorial extensions of a dual pair $\{A_1,A_2\}$ of nonnegative operators is obtained. Some classes of nonaccretive extensions of the dual pair $\{A_1,A_2\}$ are described too. Both problems are reduced to similar problems for a dual pair $\{T_1,T_2\}$ of nondensely defined symmetric contractions $T_j=(I-A_j)(I+A_j)^{-1},\ j\in\{1,2\}$. In turn these problems are reduced to the investigation of the corresponding operator "holes". A complete description of the set of all proper and improper extensions of a nonnegative operator is obtained too. 
Mark Malamud