We study the problem \[ ( ∗ ) T m x m = ∑ n = 1 k λ n V m n x m , 0 ≠ x m ∈ H m , m = 1 , … , k , (\ast )\qquad {T_m}{x_m} = \sum \limits _{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k, \] where T m {T_m} and V m n {V_{mn}} are selfadjoint linear operators on separable Hilbert spaces H m {H_m} , with T m {T_m} positive, T m − 1 T_m^{ - 1} compact and V m n {V_{mn}} bounded. We assume “left definiteness” which involves positivity of certain linear combinations of cofactors in the determinant with ( m , n ) (m,\,n) th entry ( x m , V m n x m ) ({x_m},\,{V_{mn}}{x_m}) . We establish a spectral theory for ( ∗ ) (\ast ) that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators B n = Δ n − 1 Δ 0 {B_n} = \Delta _n^{ - 1}{\Delta _0} , where the Δ n {\Delta _n} are determinantal operators on ⊗ m = 1 k H m \otimes _{m = 1}^k{H_m} . This complements an established approach to the alternative “right definite” problem (where Δ 0 {\Delta _0} is positive) via the operators Γ n = Δ 0 − 1 Δ n {\Gamma _n} = \Delta _0^{ - 1}{\Delta _n} .