Distinguished algebraic varieties in
have been the focus of much research in recent years for good reasons. This note gives
a different perspective.
We find a new characterization of an algebraic variety
which is distinguished with respect to the bidisc. It is in terms of the joint
spectrum of a pair of commuting linear matrix pencils.
There is a known characterization of
due to a seminal work of Agler and McCarthy. We show that Agler–McCarthy
characterization can be obtained from the new one and vice versa.
En route, we develop a new realization formula for operator-valued
contractive analytic functions on the unit disc.
There is a one-to-one correspondence between operator-valued contractive
holomorphic functions and
canonical model triples. This pertains to the
new realization formula mentioned above.
Pal and Shalit gave a characterization of an algebraic variety, which is
distinguished with respect to the symmetrized bidisc, in terms of a matrix
of numerical radius no larger than
.
We refine their result by making the class of matrices strictly smaller.
In a generalization in the direction of more than two variables, we
characterize all one-dimensional algebraic varieties which are distinguished
with respect to the polydisc.
At the root of our work is the Berger–Coburn–Lebow theorem characterizing a
commuting tuple of isometries.
