Abstract
When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that there exists some operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 if and only if there exists some left invertible operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 . A necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ Inv ( K , H ) is given, where Inv ( K , H ) denotes the set of all the invertible operators of B ( K , H ) . In addition, we give a necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for all C ∈ Inv ( K , H ) .
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