Operator holes and extensions of sectorial operators and dual pairs of contractions

We find a criterion of existence and uniqueness of an m ‐sectorial extension of a dual pair { A 1 , A 2 } of nonnegative operators. A description of the set of all such extensions of a dual pair { A 1 , A 2 } is obtained too. A complete description of the set of all proper and improper m ‐sectorial extensions of a nonnegative operator is also obtained. All the problems are reduced to similar problems for a dual pair { T 1 , T 2 } of non‐densely defined symmetric contractions T j = ( I – A j )( I + A j ) –1 , j ∈ {1, 2}. In turn these problems are reduced to the investigation of the corresponding operator “holes”, intersections of two operator balls. Basically, complexity of the problem depends upon that whether the left/right radii of the operator ball(s) coincide or not. A parametrization of an operator hole with equal left and right radii is obtained. Solutions to the above problems are based on such a parametrization. Some classes of non‐contractive extensions of the dual pair { T 1 , T 2 } are described too. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)