2,745 research outputs found
An inductive Julia-Caratheodory theorem for Pick functions in two variables
We study the asymptotic behavior of Pick functions, analytic functions which
take the upper half plane to itself. We show that if a two variable Pick
function has real residues to order at infinity and the imaginary
part of the remainder between and this expansion is of order then
has real residues to order and directional residues to order
Furthermore, has real residues to order if and only if the -th
derivative is given by a polynomial, thus obtaining a two variable analogue of
a higher order Julia-Carath\'eodory type theorem
The inverse function theorem and the resolution of the Jacobian conjecture in free analysis
We establish an invertibility criterion for free polynomials and free
functions evaluated on some tuples of matrices. We show that if the derivative
is nonsingular on some domain closed with respect to direct sums and
similarity, the function must be invertible. Thus, as a corollary, we establish
the Jacobian conjecture in this context. Furthermore, our result holds for
commutative polynomials evaluated on tuples of commuting matrices
Positivstellensatz\"e for noncommutative rational expressions
We derive some Positivstellensatz\"e for noncommutative rational expressions
from the Positivstellensatz\"e for noncommutative polynomials. Specifically, we
show that if a noncommutative rational expression is positive on a polynomially
convex set, then there is an algebraic certificate witnessing that fact. As in
the case of noncommutative polynomials, our results are nicer when we
additionally assume positivity on a convex set-- that is, we obtain a so-called
"perfect Positivstellensatz" on convex sets.Comment: 6 page
The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary
The edge-of-the-wedge theorem in several complex variables gives the analytic
continuation of functions defined on the poly upper half plane and the poly
lower half plane, the set of points in with all coordinates in
the upper and lower half planes respectively, through a set in real space,
The geometry of the set in the real space can force the
function to analytically continue within the boundary itself, which is
qualified in our wedge-of-the-edge theorem. For example, if a function extends
to the union of two cubes in which are positively oriented, with
some small overlap, the functions must analytically continue to a neighborhood
of that overlap of a fixed size not depending of the size of the overlap.Comment: 12 page
On the continuation of locally operator monotone functions
We generalize the phenomenon of continuation from complex anal- ysis to
locally operator monotone functions. Along the lines of the egde-of- the-wedge
theorem, we prove continuations exist dependent only on geometric features of
the domain and, namely, independent of the function values. We prove a
generalization of the Julia inequality for a class of functions containing
locally operator monotone functions, Pick functions
The noncommutative L\"owner theorem for matrix monotone functions over operator systems
Given a function L\"owner's theorem states
is monotone when extended to self-adjoint matrices via the functional
calculus, if and only if extends to a self-map of the complex upper half
plane. In recent years, several generalizations of L\"owner's theorem have been
proven in several variables. We use the relaxed Agler, McCarthy and Young
theorem on locally matrix monotone functions in several commuting variables to
generalize results in the noncommutative case. Specifically, we show that a
real free function defined over an operator system must analytically continue
to a noncommutative upper half plane as map into another noncommutative upper
half plane.Comment: 7 page
Committee spaces and the random column-row property
A committee space is a Hilbert space of power series, perhaps in several or
noncommuting variables, such that Such a space satisfies the true column-row property when
ever the map transposing a column multiplier to a row multiplier is
contractive. We describe a model for random multipliers and show that such
random multipliers satisfy the true column-row property. We also show that the
column-row property holds asymptotically for large random Nevanlinna-Pick
interpolation problems where the nodes are chosen identically and
independently. These results suggest that finding a violation of the true
column-row property for the Drury-Arveson space via na\"ive random search is
unlikely.Comment: 10 page
Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not
Nevanlinna showed that Cauchy transforms of probability measures parametrize
all functions from the upper half plane into itself satisfying a certain
asymptotic condition at infinity. We show that the correspondence fails in
general for the unbounded case for somewhat trivial reasons; however, we show
that in a setting of "homomorphic" operator valued free probability that Cauchy
transforms of homomorphic conditional expectations parametrize free Pick
functions.Comment: 12 page
Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables
We extend the study of the Pick class, the set of complex analytic functions
taking the upper half plane into itself, to the noncommutative setting. R.
Nevanlinna showed that elements of the Pick class have certain integral
representations which reflect their asymptotic behavior at infinity. Loewner
connected the Pick class to matrix monotone functions. We generalize the
Nevanlinna representation theorems and Loewner's theorem on matrix monotone
functions to the free Pick class, the collection of functions that map tuples
of matrices with positive imaginary part into the matrices with positive
imaginary part which obey the free functional calculus.Comment: 53 page
Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carath\'eodory theory
Let be a complex analytic function. The Julia
quotient is given by the ratio between the distance of to the boundary
of and the distance of to the boundary of A classical
Julia-Carath\'eodory type theorem states that if there is a sequence tending to
in the boundary of along which the Julia quotient is bounded, then
the function can be extended to such that is nontangentially
continuous and differentiable at and is in the boundary of
We develop an extended theory when and are taken to be
the upper half plane which corresponds to amortized boundedness of the Julia
quotient on sets of controlled tangential approach, so-called -Stolz
regions, and higher order regularity, including but not limited to higher order
differentiability, which we measure using -regularity. Applications are
given, including perturbation theory and moment problems.Comment: 26 page
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