A Dialogue Between Two Lifting Theorems

The relation between the lifting theorems due to Nagy-Foias and Cotlar-Sadosky is discussed. PRESENTATION. The Nagy-Foias commutant lifting theorem is a basic result in Operator Theory and its applications to interpolation problems. Its scope is shown in a fundamental book due to Foias and Frazho where we can read that “the work on the general framework of the commutant lifting theorem continued to grow mainly in Romania, the U.S.A. and Venezuela.” [FF, p. viii] Now, the “Southamerican” contribution to the subject stems from the purpose of understanding the relations between the Nagy-Foias theorem and the Cotlar-Sadosky theorem on “weakly positive” matrices of measures. The aim of this note is to recall some aspects of a “dialogue” between those two theorems that ends by showing that they can be seen as alternative ways of describing the same facts: see below, theorems (4) and (7). ∗Universidad de La República, Montevideo, Uruguay. 30 Asociación Matemática Venezolana THE COTLAR-SADOSKY THEOREM. We shall use the following notation en(t) = e, n ∈ Z and t ∈ R, P is the space of trigonometric polynomials, i.e. of finite sums ∑ anen, with n ∈ Z and an ∈ C, P+ = { ∑ anen ∈ P : an = 0 if n < 0}, P− = { ∑ anen ∈ P : an = 0 if n ≥ 0}; T denotes the unit circle on the complex plane C, C(T) is the Banach space of complex continuous functions on T and M(T) its dual, i.e., the space of complex Radon measures on T; for any p ≥ 1, H = {f ∈ L ≡ L(T) : f̂(n) = 0 if n < 0}, where f̂ is the Fourier transform of f . If μ = {μjk}j,k=1,2 is a matrix with entries in M(T) and f = (f1, f2) ∈ C(T) × C(T), we set