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Multivalued Fredholm Type Operators with Abstract Generalised Inverses

✍ Scribed by T. Alvarez; R.W. Cross; Diane Wilcox

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
122 KB
Volume
261
Category
Article
ISSN
0022-247X

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✦ Synopsis

Multivalued semi-Fredholm type operators with topologically complemented kernels and ranges are investigated in normed linear spaces. It is shown that regularisers (generalised inverses) can be constructed for the classes given, and that the operators considered can be characterised in terms of their regularisers. Continuity of the inverses is discussed, and dual properties of adjoints are given. Β© 2001 Academic Press

1. Introduction

The existence of continuous generalised inverses for unbounded multivalued semi-Fredholm type operators in normed linear spaces is considered. We give characterisation theorems for the classes of multivalued Ξ±-and Ξ²-Atkinson operators, which are introduced formally below.

The concept of a generalised inverse may be traced back to early contributions by Fredholm, Hurwitz, Hilbert, and others, in the study of integral equations, which evolved into the rich theory on Fredholm type operators 403

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## Abstract A symbol calculus for the smallest Banach subalgebra π’œ~[__SO,PC__]~ of the Banach algebra ℬ︁(__L^n^~p~__(ℝ)) of all bounded linear operators on the Lebesgue spaces __L^n^~p~__(ℝ) (1 < __p__ < ∞, __n__ β‰₯ 1) which contains all the convolution type operators __W~a,b~__ = __a__β„±^βˆ’1^__b__β„± w