minimal strongly eventually positive realization for a class of externally positive systems the minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state space matrices in this paper the minimal strongly eventually positive realization which relaxes the nonnegativity constraint on the state space model and only requires that the state trajectory is nonnegative after a certain number of steps is proposed for an externally positive system it is shown that a discrete time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization a constructive proof is provided the continuous time counterpart is also addressed by transforming it into a corresponding discrete time case

a lower bound on the dimension of minimal positive realizations for discrete time systems abstract in some applications one is interested in having a state space realization with nonnegative matrices positive realization of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself moreover in most cases it is desirable to have a realization with minimal dimension unfortunately it is not known to date how to determine in general the minimum dimension of a positive realization and only lower and upper bounds to it are available this letter provides an upper bound on the dimension of a minimal positive realization for transfer functions with simple poles this is a considerable improvement on an earlier upper bound in which only transfer functions with real poles were considered

a non commutative positivstellensatz on isometries a symmetric non commutative polynomial p when evalu ated at a tuple of operators on a flnite dimensional real hilbert space h has a value which is a symmetric operator we show that any such polynomial which takes positive semideflnite values on the variety z of spherical isometries is represented as a sum of squares of polynomials plus a residual part vanishing on z here by spherical isometries we mean tuples a a 1 a2 an of operators on h such that a t a1 a tan i this observation improves prior theorems known only for strictly pos itive polynomials it is known that for commutative polynomials the result is false

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system theory

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

caratheodory interpolation on the non commutative polydisk the caratheodory problem in the n variable non commutative herglotz agler class and the caratheodory fejer problem in the n variable non commutative schur agler class are posed it is shown that the caratheodory resp caratheodory fejer problem has a solution if and only if the non commutative polynomial with given operator coefficients the data of the problem indexed by an admissible set takes operator values with positive semidefinite real part resp contractive operator values on n tuples of jointly nilpotent contractive n n matrices for all n n

an example of how positivity may force realizations of large dimension abstract a standard result of linear system theory states that an siso proper rational transfer function of degree n always has a realization of dimension n in some applications one is interested in having a realization with nonnegative entries and it is known that when the dominant poles display a specific pattern forcing nonnegativity leads to a system which is not jointly reachable and observable in this paper we show that the minimal dimension of a positive realization may be large even in the case of a single dominant pole more precisely we provide a family of transfer functions each of which is of degree n 3 such that for any integer n 3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than n

equivalent relaxations of optimal power flow several convex relaxations of the optimal power flow opf problem have recently been developed using both bus injection models and branch flow models in this paper we prove relations among three convex relaxations a semidefinite relaxation that computes a full matrix a chordal relaxation based on a chordal extension of the network graph and a second order cone relaxation that computes the smallest partial matrix we prove a bijection between the feasible sets of the opf in the bus injection model and the branch flow model establishing the equivalence of these two models and their second order cone relaxations our results imply that for radial networks all these relaxations are equivalent and one should always solve the second order cone relaxation for mesh networks the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second order cone relaxation therefore for mesh networks one should either solve the chordal relaxation or the socp relaxation trading off tightness and the required computational effort simulations are used to illustrate these results

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

the real positive definite completion problem for a simple cycle abstract we consider the question of whether a real partial positive definite matrix in which the specified off diagonal entries consist of a full n cycle has a positive definite completion this lies in contrast to the previously studied chordal case we give two solutions in one we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sufficient the second is more computational and allows presentation of all positive definite completions as well as answering the existence question

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

noncommutative plurisubharmonic polynomials part ii local assumptions abstract we say that a symmetric noncommutative nc polynomial is nc plurisubharmonic nc plush on an nc open set if it has an nc complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size in this paper we show that if an nc polynomial is nc plurisubharmonic on an nc open set then the polynomial is actually nc plurisubharmonic everywhere and has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic greene et al 2011 1 has shown that if p is nc plurisubharmonic everywhere then p has the form in eq 0 1 in other words 1 makes a global assumption while the current paper makes a local assumption but both reach the same conclusion we show that if p is nc plurisubharmonic on an nc open set local then p is in fact nc plurisubharmonic everywhere global and has the form expressed in eq 0 1 this paper requires a technique that is not used in 1 we use a gram like vector and matrix representation called the border vector and middle matrix for homogeneous degree 2 nc polynomials we then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix this rigid structure plus the theorems in 1 ultimately force the form in eq 0 1

large scale semidefinite programming via a saddle point mirror prox algorithm in this paper we first demonstrate that positive semidefiniteness of a large well structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked in a linear fashion to the matrix we derive also the dual counterpart of the outlined representation which expresses the possibility of positive semidefinite completion of a well structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix using the representations we then reformulate well structured large scale semidefinite problems into smooth convex concave saddle point problems which can be solved by a prox method developed in 6 with efficiency mathcal o epsilon 1 implementations and some numerical results for large scale lovasz capacity and maxcut problems are finally presented

some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties abstract we examine several modified versions of the heteroskedasticity consistent covariance matrix estimator of hinkley 1977 and white 1980 on the basis of sampling experiments which compare the performance of quasi t statistics we find that one estimator based on the jackknife performs better in small samples than the rest we also examine the finite sample properties of using modified critical values based on edgeworth approximations as proposed by rothenberg 1984 in addition we compare the power of several tests for heteroskedasticity and find that it may be wise to employ the jackknife heteroskedasticity consistent covariance matrix even in the absence of detected heteroskedasticity

optimal convergence rates bahadur representation and asymptotic normality of partitioning estimators abstract this paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives mean square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions the uniform rate explicitly accounts for the effect of moment assumptions which is useful in semiparametric inference a general asymptotic integrated mean square error approximation is obtained and used to derive an optimal plug in tuning parameter selector a uniform bahadur representation is developed for linear functionals of the estimator using this representation asymptotic normality is established along with consistency of a standard error estimator the finite sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

a positivstellensatz for non commutative polynomials a non commutative polynomial which is positive on a bounded semi algebraic set of operators has a weighted sum of squares representation this positivstellensatz parallels similar results in the commutative case a broader issue is to what extent does real semi algebraic geometry extend to non commutative polynomials our strict positivstellensatz is positive news on the opposite extreme from strict positivity would be a real nullstellensatz we give an example which shows that there is no non commutative real nullstellensatz along certain lines however we include a successful type of non commutative nullstellensatz proved by george bergman

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

geometry of maximum likelihood estimation in gaussian graphical models author s uhler caroline advisor s sturmfels bernd abstract algebraic statistics exploits the use of algebraic techniques to develop new paradigms and algorithms for data analysis the development of computational algebra software provides a powerful tool to analyze statistical models in part i of this thesis we use methods from computational algebra and algebraic geometry to study gaussian graphical models algebraic methods have proven to be useful for statistical theory and applications alike we describe a particular application to computational biology in part ii part i of this thesis investigates geometric aspects of maximum likelihood estimation in gaussian graphical models more generally we study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection in chapter 2 we examine these problems at the interface of statistics and optimization from the perspective of convex algebraic geometry and characterize the cone of all sufficient statistics for which the maximum likelihood estimator mle exists in chapter 3 we develop an algebraic elimination criterion which allows us to find exact lower bounds on the number of observations needed to ensure that the mle exists with probability one this is applied to bipartite graphs grids and colored graphs we also present the first instance of a graph for which the mle exists with probability one even when the number of observations equals the treewidth computational algebra software can be used to study graphs with a limited number of vertices and edges in chapter 4 we study the problem of existence of the mle from an asymptotic point of view by fixing a class of graphs and letting the number of vertices grow to infinity we prove that for very large cycles already two observations are sufficient for the existence of the mle with probability one part ii of this thesis describes an application of algebraic statistics to association studies rapid research progress in genotyping techniques have allowed large genome wide association studies existing methods often focus on determining associations between single loci and a specific phenotype however a particular phenotype is usually the result of complex relationships between multiple loci and the environment we develop a method for finding interacting genes i e epistasis using markov bases we test our method on simulated data and compare it to a two stage logistic regression method and to a fully bayesian method showing that we are able to detect the interacting loci when other methods fail to do so finally we apply our method to a genome wide dog data set and identify epistasis associated with canine hair length

noncommutative plurisubharmonic polynomials part i global assumptions abstract we consider symmetric polynomials p in the noncommutative nc free variables x 1 x 2 x g we define the nc complex hessian of p as the second directional derivative replacing x t by y q x x t h h t 2 p s t x t h y s k t s 0 y x t k h t we call an nc symmetric polynomial nc plurisubharmonic nc plush if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n n matrices for every size n i e q x x t h h t 0 for all x h r n n g for every n 1 in this paper we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables i e an nc symmetric polynomial p is nc plush if and only if it has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic in this paper we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the frobenius theorem a subsequent paper j m greene preprint 6 proves that if the nc complex hessian q of p takes positive semidefinite values on an nc open set then q takes positive semidefinite values on all tuples x h thus p has the form in eq 0 1 the proof in j m greene preprint 6 draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions

positive semidefinite quadratic forms on unitary matrices abstract let f x 1 x n i j 1 n ij x i x j a ij a ji r be a real quadratic form such that the trace of the hermitian matrix f v 1 v n i j 1 n ij v i v j is nonnegative for all unitary 2 n 2 n matrices v 1 v n we prove that f u 1 u n is positive semidefinite for all unitary matrices u 1 u n of arbitrary size m m

phase type distributions and representations some results and open problems for system theory in this paper we consider phase type distributions these distributions correspond to the random hitting time of an absorbing markov chain they are used for modelling various random times in particular those which appear in manufacturing systems as processing times times to failure repair times etc the markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation in this paper we give a system theory oriented introduction to phase type distributions we concentrate mainly on the representation problem which consists of finding a markov chain associated with some phase type distribution this is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting indeed this problem has strong connections with the positive realization problem in control theory the paper recalls known results gives s

the tracial moment problem and trace optimization of polynomials the main topic addressed in this paper is trace optimization of polynomials in noncommuting nc variables given an nc polynomial f what is the smallest trace f underline a can attain for a tuple of matrices underline a a relaxation using semidefinite programming sdp based on sums of hermitian squares and commutators is proposed while this relaxation is not always exact it gives effectively computable bounds on the optima to test for exactness the solution of the dual sdp is investigated if it satisfies a certain condition called flatness then the relaxation is exact in this case it is shown how to extract global trace optimizers with a procedure based on two ingredients the first is the solution to the truncated tracial moment problem and the other crucial component is the numerical implementation of the artin wedderburn theorem for matrix algebras due to murota kanno kojima kojima and maehara trace optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side two topics with many applications the most prominent being to linear systems engineering and quantum physics the optimization problems discussed here facilitate new possibilities for applications e g in operator algebras and statistical physics

minimal eventually positive realizations of externally positive systems it is a well known fact that externally positive linear systems may fail to have a minimal positive realization in order to investigate these cases we introduce the notion of minimal eventually positive realization for which the state update matrix becomes positive after a certain power eventually positive realizations capture the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive but only transiently as a consequence we show that in discrete time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of markov coefficients of the impulse response in continuous time instead if the sampling time is chosen sufficiently long a minimal eventually positive realization leads always to a sampled realization which is minimal and positive

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

bounds on the size of minimal nonnegative realizations for discrete time lti systems nonnegative linear time invariant lti state space models are characterized by the property that their state variables and outputs remain nonnegative whenever the initial state and the inputs are nonnegative necessary and sufficient conditions for the existence of a nonnegative state space realization of a given discrete time dt impulse response were obtained recently in this letter we look at minimal nonnegative realizations of dt impulse responses and develop bounds on the number of state variables that are necessary

the convex positivstellensatz in a free algebra abstract given a monic linear pencil l in g variables let p l p l n n n where p l n x s n g l x 0 and s n g is the set of g tuples of symmetric n n matrices because l is a monic linear pencil each p l n is convex with interior and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form p l the main result of this paper establishes a perfect noncommutative nichtnegativstellensatz on a convex semialgebraic set namely a noncommutative matrix valued polynomial p is positive semidefinite on p l if and only if it has a weighted sum of squares representation with optimal degree bounds p s s j finite f j l f j where s f j are matrices of noncommutative polynomials of degree no greater than deg p 2 this noncommutative result contrasts sharply with the commutative setting where there is no control on the degrees of s f j and assuming only p nonnegative as opposed to p strictly positive yields a clean positivstellensatz so seldom that such cases are noteworthy

the symmetric m matrix and symmetric inverse m matrix completion problems abstract the symmetric m matrix and symmetric m 0 matrix completion problems are solved and results of johnson and smith linear algebra appl 290 1999 193 are extended to solve the symmetric inverse m matrix completion problem 1 a pattern i e a list of positions in an n n matrix has symmetric m completion i e every partial symmetric m matrix specifying the pattern can be completed to a symmetric m matrix if and only if the principal subpattern r determined by its diagonal is permutation similar to a pattern that is block diagonal with each diagonal block complete or in graph theoretic terms if and only if each component of the graph of r is a clique 2 a pattern has symmetric m 0 completion if and only if the pattern is permutation similar to a pattern that is block diagonal with each diagonal block either complete or omitting all diagonal positions or in graph theoretic terms if and only if every principal subpattern corresponding to a component of the graph of the pattern either omits all diagonal positions or includes all positions 3 a pattern has symmetric inverse m completion if and only if its graph is block clique and no diagonal position is omitted that corresponds to a vertex in a graph block of order 2 the techniques used are also applied to matrix completion problems for other classes of symmetric matrices

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

on the reachable set for third order linear discrete time systems with positive control the case of complex eigenvalues abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has a complex conjugate pair of eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

heteroscedasticity robust inference in linear regression models with many covariates we consider inference in linear regression models that is robust to heteroscedasticity and the presence of many control variables when the number of control variables increases at the same rate as

frequency limited mmb h _ infty model reduction for positive systems in this paper the problem of frequency limited mmb h infty model reduction for positive linear time invariant systems is investigated specifically our goal is to find a stable positive reduced order model for a given positive system such that the mmb h infty norm of the error system is bounded over a frequency interval of interest a new condition in terms of matrix inequality is developed for characterizing the frequency limited mmb h infty performance then an equivalent parametrization of a positive reduced order model is derived based on which an iterative algorithm is constructed for optimizing the reduced order model the algorithm utilizes coarse reduced order models resulting from generalized balanced truncation as the initial value both continuous and discrete time systems are considered in the same framework numerical examples clearly show the effectiveness and advantages of the proposed model reduction method

minimal positive realizations of transfer functions with real poles a standard result of linear system theory states that a single input single output siso rational n th order transfer function always has a state space realization of the same order in some applications one is interested in having a realization with nonnegative entries i e a positive system and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order n in this technical note necessary and sufficient conditions for a third order transfer function with distinct real poles to have a third order positive realization are given these conditions are expressed in terms of lower bounds for the first three samples of the impulse response and therefore are very easy to check this result is an extension of a previous result for transfer functions with distinct real positive poles

heteroskedasticity robust standard errors for fixed effects panel data regression the conventional heteroskedasticity robust hr variance matrix estimator for cross sectional regression with or without a degrees of freedom adjustment applied to the fixed effects estimator for panel data with serially uncorrelated errors is inconsistent if the number of time periods t is fixed and greater than 2 as the number of entities n increases we provide a bias adjusted hr estimator that is consistent under any sequences n t in which n and or t increase to this estimator can be extended to handle serial correlation of fixed order

on the reachable set for third order linear discrete time systems with positive control abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has all real eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations moreover we provide necessary and sufficient conditions for finite time reachability of every reachable state and characterize the minimum number of steps needed to reach every state in terms of eigenvalue locations

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

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

null and positivstellensatze for rationally resolvable ideals abstract hilbert s nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in c x in the free algebra c x the vanishing set of a two sided ideal i is defined in a dimension free way using images in finite dimensional representations of c x i in this article nullstellensatze for a simple but important class of ideals in the free algebra called tentatively rationally resolvable here are presented an ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the x variables using noncommutative rational functions in the remaining variables whether such an ideal i satisfies the nullstellensatz is intimately related to embeddability of c x i into free skew fields these notions are also extended to free algebras with involution for instance it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two sided ideal i generated by 1 j x j x j this is then applied to free real algebraic geometry polynomials positive semidefinite on spherical isometries are sums of hermitian squares modulo i similar results are obtained for nc unitary groups

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples