Rational dilation and constrained algebras

Abstract

If a set is a spectral set for an operator T, is it necessarily a complete spectral set? That is, if the spectrum of T is contained in , and von Neumann's inequality holds for T and rational functions with poles o of , does it still hold for all such matrix valued rational functions? Equivalently, if is a spectral set for T, does T have a dilation to a normal operator with spectrum in the boundary of ? This is true if is the disk or the annulus, but has been shown to fail in many other cases. There are also multivariable versions of this problem. For example, it is known that rational dilation holds for the bidisk, though it has been recently shown to fail for a distinguished variety in the bidisk called the Neil parabola. The Neil parabola is naturally associated to a constrained subalgebra of the disk algebra, as are many other distinguished varieties. We show that the rational dilation fails on certain distinguished varieties of the polydisk DN associated to the constrained subalgebra AB := C + B(z)A(D). Here A(D) is the algebra of functions that are analytic on the open unit disk D and continuous on the closure of D, and B(z) is a nite Blaschke product of degree N 2. To this end we identify and study the set of test functions B for H1 B := C+B(z)H1(D). Among others, we show that B is minimal (in a sense that there is no proper closed subset of B is su ces).