Spectra of constructs of a system of operators

This paper describes the spectrum and the upper and lower Fredholm spectra of (n + m)-tuples (F(A₁),., F(A_n), G(B₁),., G(B_m)) of operators, where (A_i) and (B_j) are systems of operators in two Hilbert spaces H₁ and H₂, and F and G are certain linear operators defined on L(Hi). Using spectral mapping theorems the spectra of operators constructed by the action of a polynomial on a system (F(A₁),., F(A_n), G(B₁),., G(B_m)) is obtained. In particular, the spectra of the elementary operator and tensor products of operators is determined.