A Spectral Theory for Certain Operators on a Direct Sum of Hilbert Spaces

Let ℌ p = v ⊕ ... ⊕ ℌ be the direct sum of the Hilbert space ℌ with itself p times. Linear maps y = Ax in ℌ p have the form y i = Σ a ij x i where a ij are linear maps in ℌ and there are many natural problems concerned with the discovery of those properties, enjoyed by the operators a ij , that are shared by the operator A. Here we shall discuss, in particular, two such problems; the existence of a resolution of the identity and the existence of an operational calculus. These problems are, of course, closely related and, as is well known, an operator having the former property will have an operational calculus defined on the algebra of bounded Borel functions on its spectrum; but there may be quite a satisfactory operational calculus for an operator which has no resolution of the identity. We consider here only the case where the operators a ij are commuting normal operators in ℌ. This is just another way of saying that we assume all of the elements in the matrix representation of A = (a ij ) to belong to a commutative B*-subalgebra A of the B*-algebra B(ℌ) of bounded linear operators in ℌ. The algebra e p of such operators A is then a non-commutative (in case p > 1) B*-subalgebra of B(ℌ p ) and a consideration of the most elementary case, where p = 2 and the dimension of ℌ is 1, shows that the algebra A p contains non-normal operators. Do these non-normal operators in A p have resolutions of the identity? Unfortunately they need not, but it is easy to state a procedure for determining which ones do have such a spectral reduction and to see therefore that many operators which are not even similar to a normal operator do indeed have resolutions of the identity.