Proceedings of the American Mathematical Society

In this paper we investigate situations in which the space C π ( X ) {C_\pi }(X) of continuous, real-valued functions on X X is a Borel subset of the product space R X {{\mathbf {R}}^X} . We show that for completely regular, nondiscrete spaces, C π ( X ) {C_\pi }(X) cannot be a G δ {G_\delta } , an F σ {F_\sigma } , or a G δ σ {G_{\delta \sigma }} subset of R X {{\mathbf {R}}^X} , but it can be an F σ δ {F_{\sigma \delta }} or G δ σ δ {G_{\delta \sigma \delta }} subset.

LetR⟨x⟩\mathbb R\langle x \rangledenote the ring of polynomials inggfreely noncommuting variablesx=(x1,…,xg)x=(x_1,\dots ,x_g). There is a natural involution∗*onR⟨x⟩\mathbb R\langle x \rangledetermined byxj∗=xjx_j^*=x_jand(pq)∗=q∗p∗(pq)^*=q^* p^*, and a free polynomialp∈R⟨x⟩p\in \mathbb R\langle x \rangleis symmetric if it is invariant under this involution. IfX=(X1,…,Xg)X=(X_1,\dots ,X_g)is aggtuple of symmetricn×nn\times nmatrices, then the evaluationp(X)p(X)is naturally defined and furtherp∗(X)=p(X)∗p^*(X)=p(X)^*. In particular, ifppis symmetric, thenp(X)∗=p(X)p(X)^*=p(X). The main result of this article says ifppis symmetric,p(0)=0p(0)=0and for eachnnand each symmetric positive definiten×nn\times nmatrixAAthe set{X:A−p(X)≻0}\{X:A-p(X)\succ 0\}is convex, thenpphas degree at most two and is itself convex, or−p-pis a hermitian sum of squares.

We provide a short proof of a theorem of W. B. Arveson in operator theory. The conclusion of this theorem is the same as that of the Corona Theorem but the hypotheses are operator theoretic. Our proof yields an exact value for the constant involved. We also comment on this theorem as a new approximation problem.

A new model for generalized Nevanlinna functions Q ∈ N κ Q\in \mathbf {N}_\kappa will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of Q Q . The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.