Boundary values in spaces spanned by rational functions and the index of invariant subspaces

Let μ$\mu$ be a positive compactly supported measure in the complex plane C$\mathbb {C}$ , and for each p,1⩽p<∞$p,1\leqslant p<\infty$ , let Hp(μ)$H^p(\mu )$ be the closed subspace of Lp(μ)$L^p(\mu )$ spanned by the polynomials. In 1991, Thomson gave a complete description of its structure, expressing Hp(μ)$H^p(\mu )$ as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non‐trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to Rp(μ)$R^p(\mu )$ , the closed subspace of Lp(μ)$L^p(\mu )$ spanned by the rational functions having no poles on the support of μ$\mu$ , by first establishing the existence of boundary values in these spaces. Our results all depend on the semiadditivity of analytic capacity, and ultimately on some form of the F. and M. Riesz theorem.