The spectrum of orthogonal sums of subnormal pairs

This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.