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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

REFERENCES

1. Azizov, T. Ya. and Iokhvidov, I. S., Linear operators in spaces with indefinite metric and their applications (in Russian), Itogi Nauki Tekh., Ser. Mat. Anal. 17 (1979), 113–206; English translation: J. Soviet Math. 15 (1981), 438–490.Google Scholar
2. Conway, J. B., A course in operator theory, Graduate Studies in Mathematics, Vol. 21 (American Mathematical Society, Providence, R.I., 2000).Google Scholar
3. Iokhvidov, I. S., Banach spaces with a J-metric. J-nonnegative operators (in Russian) Dokl. Akad. Nauk SSSR 169 (1966) 259–261; English translation: Soviet Math. Dokl. 7 (1966), 896–899.Google Scholar
4. Iokhvidov, I. S., J-nondilating operators in a Banach space (in Russian), Dokl. Akad. Nauk SSSR 169 (1966), 519–522; English translation: Soviet Math. Dokl. 7 (1966), 962–965.Google Scholar
5. James, R. C., Quasicomplements, J. Approximation Theory 6 (1972), 147–160.Google Scholar
6. Johnson, W. B., On quasi-complements, Pacific J. Math. 48 (1973), 113–118.Google Scholar
7. Khatskevich, V. A., Fixed points of generalized linear-fractional transformations (in Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 1130–1141; English translation: Math. USSR–Izvestiya 9 (1975), 1069–1080.Google Scholar
8. Khatskevich, V. A., Ostrovskii, M. I., and Shulman, V. S., Linear fractional relations for Hilbert space operators, Math. Nachrichten 279 (2006), 875–890.Google Scholar
9. Khatskevich, V. A., Ostrovskii, M. I., and Shulman, V. S., An analogue of the Liouville theorem for linear fractional relations in Banach spaces, Bull. Australian Math. Soc. 73 (2006), 89–105.CrossRefGoogle Scholar
10. Khatskevich, V. A., Senderov, V. A., and Shulman, V. S., On operator matrices generating linear fractional maps of operator balls, in Complex analysis and dynamical systems, Contemp. Math. 364 (2004), 93–102.Google Scholar
11. Khatskevich, V. A. and Shulman, V. S., Operator fractional-linear transformations: convexity and compactness of image; applications, Studia Math. 116 (1995), 189–195.CrossRefGoogle Scholar
12. Krein, M. G., A new application of the fixed-point principle in the theory of operators in a space with an indefinite metric, (in Russian), Dokl. Akad. Nauk SSSR 154 (1964), 1023–1026; English translation: Soviet Math. Doklady 5 (1964), 224–228.Google Scholar
13. Krein, M. G. and Smuljan, Ju. L., On linear-fractional transformations with operator coefficients (in Russian), Matematicheskie Issledovaniya (Kishinev) 2 (1967), no. 3, 64–96; English translation: Amer. Math. Soc. Transl., Ser. 2 103 (1974), 125–152.Google Scholar
14. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, I-Sequence spaces. (Springer–Verlag, 1977).CrossRefGoogle Scholar
15. Ostrovskii, M. I. and Fonf, V. P., Operators with dense range and extensions of minimal systems (in Russian), Izvestiya vuzov. Matem. (1990), 45–47; English translation: Soviet Math. (Iz. VUZ) 34 (1990), 53–56.Google Scholar
16. Pitt, H. R., A note on bilinear forms, J. London Math. Soc. 11 (1936), 174–180.Google Scholar
17. Plichko, A. N., Selection of subspaces with special properties in a Banach space and some properties of quasicomplements (in Russian), Funktsional. Anal. i Prilozhen 15 (1981), 82–83; English translation: Funct. Anal. Appl. 15 (1981), 67–68.CrossRefGoogle Scholar