Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multivariable operator-valued Nevanlinna-Pick interpolation: a survey   The theory of Nevanlinna-Pick and Carath\'eodory-Fej\'er interpolation for
matrix- and operator-valued Schur class functions on the unit disk is now well
established. Recent work has produced extensions of the theory to a variety of
multivariable settings, including the ball and the polydisk (both commutative
and noncommutative versions), as well as a time-varying analogue. Largely
independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S.
Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock
space built from a $W^*$-correspondence E over a $W^{*}$-algebra $\cA$ and a
*-representation $\sigma$ of $\cA$. In this review we provide an exposition of
the Muhly-Solel interpolation theory accessible to operator theorists, and
explain more fully the connections with the already existing interpolation
literature. The abstract point evaluation first introduced by Muhly-Solel leads
to a tensor-product type functional calculus in the main examples. A second
kind of point-evaluation for the $W^*$-correspondence Hardy algebra, also
introduced by Muhly and Solel, is here further investigated, and a
Nevanlinna-Pick theorem in this setting is proved. It turns out that, when
specified for examples, this alternative point-evaluation leads to an
operator-argument functional calculus and corresponding Nevanlinna-Pick
interpolation. We also discuss briefly several Nevanlinna-Pick interpolation
results for Schur classes that do not fit into the Muhly-Solel
$W^*$-correspondence formalism.

19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph