Spectra of constructs of a system of operators (original) (raw)

1984, Proceedings of the American Mathematical Society

This paper describes the spectrum and the upper and lower Fredholm spectra of ( n + m ) (n + m) -tuples ( F ( A 1 ) , ⋯ , F ( A n ) , G ( B 1 ) , ⋯ , G ( B m ) ) (F({A_1}), \cdots ,F({A_n}),G({B_1}), \cdots ,G({B_m})) of operators, where ( A i ) ({A_i}) and ( B j ) ({B_j}) are systems of operators in two Hilbert spaces H 1 {\mathcal {H}_1} and H 2 {\mathcal {H}_2} , and F F and G G are certain linear operators defined on L ( H i ) \mathcal {L}({\mathcal {H}_i}) . Using spectral mapping theorems the spectra of operators constructed by the action of a polynomial on a system ( F ( A 1 ) , ⋯ , F ( A n ) , G ( B 1 ) , ⋯ , G ( B m ) ) (F({A_1}), \cdots ,F({A_n}),G({B_1}), \cdots ,G({B_m})) is obtained. In particular, the spectra of the elementary operator and tensor products of operators is determined.