On operators with infinitely many spectral components

For a unital complex Banach algebra A and n∈ N, let Cₙ( A) be the set of all elements whose spectrum has at least n connected components, and let C_∞( A)=n∈ N Cₙ( A). We prove that C_∞( A)≠ if and only if A contains a sequence of non-zero pairwise orthogonal idempotents, and that the number of components of a spectrum is always finite, ₀ or c. This allows us to extend the spectral depth d_σ( A) to a cardinal-valued invariant with values in N∪\₀, c\. We show that d_σ( A)= c precisely when A contains a dyadic family of idempotents indexed by the full binary tree. In that case, some element has spectrum homeomorphic to the Cantor space 2N. The supremum defining d_σ( A) is attained when the spectral classes of A are dense and when A is commutative, but not in general. We construct a unital complex Banach algebra with d_σ( A)=₀ and C_∞( A)=. If $Δ$ is the character space of a commutative A and π₀(Δ) is the space of all components of $Δ$, then d_σ( A)=(π₀(Δ)) when π₀(Δ) is finite, d_σ( A)=₀ when π₀(Δ) is infinite and scattered, and d_σ( A)= c when π₀(Δ) is not scattered. We also prove that the spectral depth is additive with respect to finite direct sums and that the spectral classes are invariant under injective holomorphic functions.