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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.

Abstract. We give some perturbation theorems for multivalued linear operators in a Banach space. Two different approaches are suggested: the resolvent approach and the modified resolvent approach. The results allow us to handle degenerate abstract Cauchy problems (inclusions). A very wide application of obtained abstract results to initial boundary value problems for degenerate parabolic (elliptic-parabolic) equations with lower-order terms is studied. In particular, integro-differential equations have been considered too. Mathematics Subject Classification (2010

A subspace of a Banach space is called an operator range if it is the continuous linear image of a Banach space. Operator ranges and operator ideals with fixed range space are investigated. Properties of strictly singular, strictly cosingular, weakly sequentially precompact, and other classes of operators are derived. Perturbation theory and closed semi-Fredholm operators are discussed in the final section. 1980 Mathematics subject classification (Amer. Math. Soc.): Primary 47 A 05; secondary 47 A 53, 47 A 55, 47 D 30, 47 D 40, 46 B 25.