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LINEAR FRACTIONAL RELATIONS IN BANACH SPACES: INTERIOR POINTS IN THE DOMAIN AND ANALOGUES OF THE LIOUVILLE THEOREM
Published online by Cambridge University Press: 09 August 2007
Abstract
In this paper we study linear fractional relations defined in the following way. Let i,
'i, i = 1,2, be Banach spaces. We denote the space of bounded linear operators by
. Let T ε
(
1 ⊕
2,
'1 ⊕
'2). To each such operator there corresponds a 2 × 2 operator matrix of the form
(*)
where Tij ε
(
j,
'i. For each such T we define a set-valued map GT from
(
1,
2) into the set of closed affine subspaces of
(
'1,
'2) by
The map GT is called a linear fractional relation.
The paper is devoted to the following two problems.
• Characterization of operator matrices of the form (*) for which the set GT(K) is non-empty for each K in some open ball of the space
(
1,
2).
• Characterizations of quadruples (
1,
2,
'1,
'2) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem “a bounded entire function is constant”.
- Type
- Research Article
- Information
- Copyright
- Copyright © Glasgow Mathematical Journal Trust 2007