An inner function is a bounded analytic function on the unit disc whose radial boundary values have modulus one almost everywhere. Beurling's theorem makes them the exact generators of the shift-invariant subspaces of the Hardy space H2, and every such function factors uniquely into a Blaschke product carrying its zeros times a singular inner factor carried by a measure on the circle. This concept covers that factorization, the model spaces K_theta = H2 minus theta H2 it produces, and the operators those spaces support: restricted shifts and the Sz.-Nagy-Foias functional model for contractions, truncated Toeplitz and Hankel operators, universal operators in the sense of Rota and Caradus and their link to the invariant subspace problem, Clark measures and rank-one unitary perturbations, the Crofoot transform, Frostman shifts, and the numerical ranges and Poncelet-type geometry of compressed shifts.
An inner function is an analytic self-map theta of the open unit disc whose boundary values have modulus one almost everywhere. Every non-zero Hardy space function factors uniquely, up to a unimodula…