Abstract
When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator matrix acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = A C 0 B . In this paper, for given A and B, the sets ⋂ C ∈ B l ( K , H ) σ l ( M C ) , ⋂ C ∈ Inv ( K , H ) σ l ( M C ) and ⋃ C ∈ Inv ( K , H ) σ l ( M C ) are determined, where σ l ( T ) , B l ( K , H ) and Inv ( K , H ) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H .
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