monotone matrix functions of two variables abstract monotone operator functions were studied by koranyi who generalised lowner s theorem on monotone matrix functions of arbitrary high order n to two variables an alternate proof of koranyi s representation of monotone operator functions is provided

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

the noncommutative lowner theorem for matrix monotone functions over operator systems abstract given a function f a b r lowner s theorem states f is monotone when extended to self adjoint matrices via the functional calculus if and only if f extends to a self map of the complex upper half plane in recent years several generalizations of lowner s theorem have been proven in several variables we use the relaxed agler m c carthy 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

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

toeplitz corona and the douglas property for free functions abstract the well known douglas lemma says that for operators a b on hilbert space that a a b b 0 implies b a c for some contraction operator c the result carries over directly to classical operator valued toeplitz operators simply by replacement of operator by toeplitz operator throughout free functions generalize the notion of free polynomials and formal power series and trace back to the work of j taylor in the 1970s they are of current interest in part because of their connections with free probability and engineering systems theory in this article for given free functions a and b with noncommutative domain k defined by free polynomial inequalities we obtain a sufficient condition in terms of the difference a a b b for the existence of a free function c taking contractive values on k such that b a c the connection to recent work of agler and mccarthy and their free toeplitz corona theorem is expounded

free pick functions representations asymptotic behavior and matrix monotonicity in several noncommuting variables r nevanlinna showed that elements of the pick class the set of complex analytic functions taking the upper half plane into itself have certain integral representations which reflect their asymptotic behavior at infinity later lowner connected the pick class to matrix monotone functions we generalize the nevanlinna representation theorems and lowner 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

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

noncommutative ball maps in this paper we analyze problems involving matrix variables for which we use a noncommutative algebra setting to be more specific we use a class of functions called nc analytic functions defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions we call such situations dimension free these types of functions have recently been used in the study of dimension free linear system engineering problems in this paper we characterize nc analytic maps that send dimension free matrix balls to dimension free matrix balls and carry the boundary to the boundary such maps we call nc ball maps we find that up to normalization an nc ball map is the direct sum of the identity map with an nc analytic map of the ball into the ball that is nc ball maps are very simple in contrast to the classical result of d angelo on such analytic maps in c another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary but on these our results are limited we shall be interested in several types of noncommutative balls conventional ones but also balls defined by constraints called linear matrix inequalities lmi what we do here is a small piece of the bigger puzzle of understanding how lmis behave with respect to noncommutative change of variables

cauchy transforms arising from homomorphic conditional expectations parametrize noncommutative pick functions abstract 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 noncommutative pick functions

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

a non commutative julia inequality we prove a julia inequality for bounded non commutative functions on polynomial polyhedra we use this to deduce a julia inequality for holomorphicfunctions on classical domains in mathbb c d we look at differentiability at a boundary point for functions that have a certain regularity there

noncommutative reproducing kernel hilbert spaces abstract the theory of positive kernels and associated reproducing kernel hilbert spaces especially in the setting of holomorphic functions has been an important tool for the last several decades in a number of areas of complex analysis and operator theory an interesting generalization of holomorphic functions namely free noncommutative functions e g functions of square matrix arguments of arbitrary size satisfying additional natural compatibility conditions is now an active area of research with motivation and applications from a variety of areas e g noncommutative functional calculus free probability and optimization theory in linear systems engineering the purpose of this article is to develop a theory of positive kernels and associated reproducing kernel hilbert spaces for the setting of free noncommutative function theory

loewner s theorem in several variables abstract in this paper we establish a multivariable non commutative generalization of loewner s theorem characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex half plane into itself the non commutative several variable theorem proved here represents several variable operator monotone functions of positive operator tuples not a priori assumed to be analytic or continuous as a schur complement of a positive linear matrix pencil to achieve this we establish the abstract integral formula for these functions using only matrix convexity and lmis that represents operator monotone and operator concave free functions as a conditional expectation of a schur complement of a linear matrix pencil on a tensor product operator algebra the results can be applied to any of the various multivariable operator means that have been constructed in the last decades or so including the karcher mean to obtain an explicit closed formula for these operator means

solution of the multivariate fokker planck equation by using a maximum path entropy principle this paper proposes an approach via the maximum entropy principle in order to determine the nonstationary solutions of the fokker planck equation with time varying coefficients the constraints are not the state moments as usual but their dynamic equations the maximum entropy principle herein utilized is a slight extension of jaynes principle which involves the path entropy of the stochastic process

rational cayley inner herglotz agler functions positive kernel decompositions and transfer function realizations abstract the bessmertny class consists of rational matrix valued functions of d complex variables representable as the schur complement of a block of a linear pencil a z z 1 a 1 z d a d whose coefficients a k are positive semidefinite matrices we show that it coincides with the subclass of rational functions in the herglotz agler class over the right poly halfplane which are homogeneous of degree one and which are cayley inner the latter means that such a function is holomorphic on the right poly halfplane and takes skew hermitian matrix values on i r d or equivalently is the double cayley transform over the variables and over the matrix values of an inner function on the unit polydisk using agler knese s characterization of rational inner schur agler functions on the polydisk extended now to the matrix valued case and applying appropriate cayley transformations we obtain characterizations of matrix valued rational cayley inner herglotz agler functions both in the setting of the polydisk and of the right poly halfplane in terms of transfer function realizations and in terms of positive kernel decompositions in particular we extend bessmertny s representation to rational cayley inner herglotz agler functions on the right poly halfplane where a linear pencil a z is now in the form a z a 0 z 1 a 1 z d a d with a 0 skew hermitian and the other coefficients a k positive semidefinite matrices

a tangential interpolation problem on the distinguished boundary of the polydisk for the schur agler class abstract we study a tangential interpolation problem with an arbitrary set of interpolating points on the distinguished boundary of the unit polydisk for schur agler class the description of all solutions is parametrized in terms of a linear fractional transformation

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry