865 research outputs found
Quasi-Convex Free Polynomials
Let \Rx denote the ring of polynomials in freely non-commuting
variables . There is a natural involution * on \Rx
determined by and and a free polynomial p\in\Rx
is symmetric if it is invariant under this involution. If is
a tuple of symmetric matrices, then the evaluation is
naturally defined and further . In particular, if is
symmetric, then . The main result of this article says if is
symmetric, and for each and each symmetric positive definite
matrix the set is convex, then has
degree at most two and is itself convex, or is a hermitian sum of squares
Boundary Representations for Operator Algebras
All operator algebras have (not necessarily irreducible) boundary
representations. A unital operator algebra has enough such boundary
representations to generate its C*-envelope.Comment: 7 pages. Includes instructions for processing in pdfLaTe
Test Functions, Kernels, Realizations and Interpolation
Jim Agler revolutionized the area of Pick interpolation with his realization
theorem for what is now called the Agler-Schur class for the unit ball in
. We discuss an extension of these results to algebras of
functions arising from test functions and the dual notion of a family of
reproducing kernels, as well as the related interpolation theorem. When working
with test functions, one ideally wants to use as small a collection as
possible. Nevertheless, in some situations infinite sets of test functions are
unavoidable. When this is the case, certain topological considerations come to
the fore. We illustrate this with examples, including the multiplier algebra of
an annulus and the infinite polydisk.Comment: 22 page
The failure of rational dilation on a triply connected domain
For R a bounded triply connected domain with boundary consisting of disjoint
Jordan loops there exists an operator T on a complex Hilbert space H so that
the closure of R is a spectral set for T, but T does not dilate to a normal
operator with spectrum in B, the boundary of R. There is considerable overlap
with the construction of an example on such a domain recently obtained by
Agler, Harland and Rafael using numerical computations and work of Agler and
Harland.Comment: 43 page
Free convex sets defined by rational expressions have LMI representations
Suppose p is a symmetric matrix whose entries are polynomials in freely
noncommutating variables and p(0) is positive definite. Let D(p) denote the
component of zero of the set of those g-tuples X of symmetric matrices (of the
same size) such that p(X) is positive definite. By a previous result of the
authors, if D(p) is convex and bounded, then D(p) can be described as the set
of all solutions to a linear matrix inequality (LMI). This article extends that
result from matrices of polynomials to matrices of rational functions in free
variables.
As a refinement of a theorem of Kaliuzhnyi-Verbovetskyi and Vinnikov, it is
also shown that a minimal symmetric descriptor realization r for a symmetric
free matrix-valued rational function R in g freely noncommuting variables
precisely encodes the singularities of the rational function. This
singularities result is an important ingredient in the proof of the LMI
representation theorem stated above
Szego and Widom Theorems for the Neil Algebra
Versions of well known function theoretic operator theory results of Szego
and Widom are established for the Neil algebra. The Neil algebra is the
subalgebra of the algebra of bounded analytic functions on the unit disc
consisting of those functions whose derivative vanishes at the origin.Comment: 11 pages, Version
Agler interpolation families of kernels
An abstract Pick interpolation theorem for a family of positive semi-definite
kernels on a set is formulated. The result complements those in \cite{Ag}
and \cite{AMbook} and will subsequently be applied to Pick interpolation on
distinguished varieties \cite{JKM}.Comment: 14 page
Non-commutative varieties with curvature having bounded signature
The signature(s) of the curvature of the zero set V of a free
(non-commutative) polynomial is defined as the number of positive and negative
eigenvalues of the non-commutative second fundamental form on V determined by
p. With some natural hypotheses, the degree of p is bounded in terms of the
signature. In particular, if one of the signatures is zero, then the degree of
p is at most two
Semidefinite programming in matrix unknowns which are dimension free
One of the main applications of semidefinite programming lies in linear
systems and control theory. Many problems in this subject, certainly the
textbook classics, have matrices as variables, and the formulas naturally
contain non-commutative polynomials in matrices. These polynomials depend only
on the system layout and do not change with the size of the matrices involved,
hence such problems are called "dimension-free". Analyzing dimension-free
problems has led to the development recently of a non-commutative (nc) real
algebraic geometry (RAG) which, when combined with convexity, produces
dimension-free Semidefinite Programming. This article surveys what is known
about convexity in the non-commutative setting and nc SDP and includes a brief
survey of nc RAG. Typically, the qualitative properties of the non-commutative
case are much cleaner than those of their scalar counterparts - variables in
R^g. Indeed we describe how relaxation of scalar variables by matrix variables
in several natural situations results in a beautiful structure.Comment: 25 pages; surve
Matrix Convex Hulls of Free Semialgebraic Sets
This article resides in the realm of the noncommutative (free) analog of real
algebraic geometry - the study of polynomial inequalities and equations over
the real numbers - with a focus on matrix convex sets and their projections
. A free semialgebraic set which is convex as well as bounded and open
can be represented as the solution set of a Linear Matrix Inequality (LMI), a
result which suggests that convex free semialgebraic sets are rare. Further,
Tarski's transfer principle fails in the free setting: The projection of a free
convex semialgebraic set need not be free semialgebraic. Both of these results,
and the importance of convex approximations in the optimization community,
provide impetus and motivation for the study of the free (matrix) convex hull
of free semialgebraic sets.
This article presents the construction of a sequence of LMI domains
in increasingly many variables whose projections are
successively finer outer approximations of the matrix convex hull of a free
semialgebraic set . It is based on free analogs of
moments and Hankel matrices. Such an approximation scheme is possibly the best
that can be done in general. Indeed, natural noncommutative transcriptions of
formulas for certain well known classical (commutative) convex hulls does not
produce the convex hulls in the free case. This failure is illustrated on one
of the simplest free nonconvex .
A basic question is which free sets are the projection of a free
semialgebraic set ? Techniques and results of this paper bear upon this
question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a
Mathematica notebook) can be found at
http://www.math.auckland.ac.nz/~igorklep/publ.htm
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