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

✍ Scribed by Mark Malamud

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 373 KB
• Volume: 279
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510382

✦ Synopsis

Abstract

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)