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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

M. I. OSTROVSKII
M. I. OSTROVSKII*
Affiliation:
Department of Mathematics and Computer Science, St. John's University, 8000 Utopia Parkway, Queens, NY 11439, USA e-mail: ostrovsm@stjohns.edu
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Abstract

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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 T ij ε ( j , 'i . For each such T we define a set-valued map G T from ( 1, 2) into the set of closed affine subspaces of ('1, '2) by

The map G T 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 G T(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”.

Keywords

47A5646B2047B50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 257 - 268
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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