LINEAR FRACTIONAL RELATIONS IN BANACH SPACES: INTERIOR POINTS IN THE DOMAIN AND ANALOGUES OF THE LIOUVILLE THEOREM

OSTROVSKII, M. I.

Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 257-268

Résumé

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”.

Détail

DOI : 10.1017/S0017089507003539

Mots-clés : 47A56, 46B20, 47B50

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