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

Abstract

AbstractWe 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 Tj = (I – Aj )(I + Aj )–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)