the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

second order cone relaxations for binary quadratic polynomial programs several types of relaxations for binary quadratic polynomial programs can be obtained using linear second order cone or semidefinite techniques in this paper we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming using our framework we re derive previous relaxation schemes and provide new ones in particular we present three relaxations for binary quadratic polynomial programs the first two relaxations based on second order cone and semidefinite programming represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems from a practical point of view due to the computational cost semidefinite based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances to improve the computational efficiency for solving such problems we propose a third relaxation based purely on second order cone programming computational tests on different classes of nonconvex binary quadratic polynomial problems including quadratic knapsack problems show that the second order cone based relaxation outperforms the semidefinite based relaxations that are proposed in the literature in terms of computational efficiency and it is comparable in terms of bounds

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

convection diffusion equation with absorption and non decaying initial data in this paper we give the precise description of the large time behavior of the solution u of the cauchy problem tu u a u u in rn 0 u x 0 x 0in rn by using the ordinary differential equation and a linear parabolic equation here n 1 a rn 1 1 0 and is a bounded continuous function such that lp rn for some 1 p furthermore we study the large time behavior of the hot spots for the solution u

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

fluctuations of casimir forces on finite objects i spheres and hemispheres the mean square forces that result from the zero point fluctuations of quantized fields are calculated when acting on spheres and hemispheres of variable sizes for the maxwell field the boundary conditions of a perfectly conducting surface are imposed the scalar field is investigated for neumann and dirichlet boundary conditions the force is averaged over a finite time t small and large objects are distinguished on the scale of ct the results for the sphere and the hemisphere are compared with those for a piston that is embedded in an infinite plane a small hemisphere and a small piston are found to have fluctuations of the same order of magnitude while on a small sphere the fluctuations are by two orders of magnitude smaller because of correlations of fluctuations on the two sides of the sphere large spheres are shown to fit into the picture of large objects being composed of many patches each with the fluctuations impinging as on a large piston

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

fluctuations of casimir forces on finite objects ii flat circular disk for pt i see ibid vol 25 p 3015 37 1992 the zero point fluctuations of a scalar field with neumann boundary conditions are investigated on a flat circular disk the helmholtz equation is separated in oblate spheroidal coordinates the mean square force acting perpendicularly on the disk is calculated by picturing the disk as an oblate spheroid with zero eccentricity the force is averaged over a time t and the disk is taken small compared with ct the results show that the fluctuations on a disk are roughly equal to those on a small sphere for a one sided disk divergences arise from the sharp edge but the mean square force has the same power dependence on t as for a hemisphere

sparse sos relaxations for minimizing functions that are summations of small polynomials this paper discusses how to find the global minimum of functions that are summations of small polynomials small means involving a small number of variables some sparse sum of squares sos techniques are proposed we compare their computational complexity and lower bounds with prior sos relaxations under certain conditions we also discuss how to extract the global minimizers from these sparse relaxations the proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems lastly we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization

moment problems for multi sequences of operators abstract we give an explicit solution to the scalar moment problem on semi algebraic compact subsets of r n and apply this result to the study of some operator multi sequences

generic properties for semialgebraic programs in this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints in which every problem of the class is obtained by linear perturbations of the objective function while the core objective function and the constraint functions are kept fixed assume that the linear independence constraint qualification is satisfied at every point in the constraint set it is shown that almost all problems in the class are such that i the restriction of the objective function on the constraint set is coercive and regular at infinity ii there is a unique optimal solution lying on a unique active manifold at which the strict complementarity and second order sufficiency conditions the quadratic growth condition and the holder type global error bound hold and iii all minimizing sequences converge furthermore the active manifold is constant and the optimal solution and the optimal value function depend analytically under local perturbations of the

quasinormality of powers of commuting pairs of bounded operators abstract we study jointly quasinormal and spherically quasinormal pairs of commuting operators on hilbert space as well as their powers we first prove that up to a constant multiple the only jointly quasinormal 2 variable weighted shift is the helton howe shift second we show that a left invertible subnormal operator t whose square t 2 is quasinormal must be quasinormal third we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal n tuples fourth we show that if a 2 variable weighted shift w and its powers w 2 1 and w 1 2 are all spherically quasinormal then w may not necessarily be jointly quasinormal moreover it is possible for both w 2 1 and w 1 2 to be spherically quasinormal without w being spherically quasinormal finally we prove that for 2 variable weighted shifts the common fixed points of the toral and spherical aluthge transforms are jointly quasinormal

subcritical nonlinear heat equation abstract we study asymptotic behavior in time of small solutions to nonlinear heat equations in subcritical case we find a new family of self similar solutions which change a sign we show that solutions are stable in the neighborhood of these self similar solutions

minimizer extraction in polynomial optimization is robust in this article we present a robustness analysis of the extraction of optimizers in polynomial optimization optimizers can be extracted by solving moment problems using flatness and the gelfand n

asymptotics for nonlinear heat equations abstract we study the cauchy problem for the nonlinear heat equation u t u u 1 0 x r n t 0 u 0 x u 0 x x r n in the sub critical case of 0 2 n in the present paper we intend to give a more precise estimate for the remainder term in the asymptotic representation known from paper escobedo and kavian 1987 5 u t x t 1 w 0 x t o t 1 as t uniformly with respect to x r n where w 0 is a positive solution of equation w 2 w w 1 1 w which decays rapidly at infinity lim 2 w 0 0

semidefinite representations for finite varieties we consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities when the polynomial equations have a finite set of complex solutions we can reformulate this problem as a semidefinite programming problem our semidefinite representation involves combinatorial moment matrices which are matrices indexed by a basis of the quotient vector space x 1 xn i where i is the ideal generated by the polynomial equations in the problem moreover we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by lasserre semidefinite approximations can be constructed by considering truncated combinatorial moment matrices rank conditions are given in a grid case that ensure that the approximation solves the original problem to optimality

necessary global optimality conditions for nonlinear programming problems with polynomial constraints in this paper we develop necessary conditions for global optimality that apply to non linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization in particular we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints and to classes of fractional programming problems our necessary conditions become also sufficient for global optimality for polynomial programming problems our approach makes use of polynomial over estimators and a polynomial version of a theorem of the alternative which is a variant of the positivstellensatz in semi algebraic geometry we discuss numerical examples to illustrate the significance of our optimality conditions

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

parametric optimization and optimal control using algebraic geometry methods we present two algebraic methods to solve the parametric optimization problem that arises in non linear model predictive control we consider constrained discrete time polynomial systems and the corresponding constrained finite time optimal control problem the first method is based on cylindrical algebraic decomposition the second uses grobner bases and the eigenvalue method for solving systems of polynomial equations both methods aim at moving most of the computational burden associated with the optimization problem off line by pre computing certain algebraic objects then an on line algorithm uses this pre computed information to obtain the solution of the original optimization problem in real time fast and efficiently introductory material is provided as appropriate and the algorithms are accompanied by illustrative examples

algebraic degree of polynomial optimization consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials under some genericity assumptions we prove that the optimality conditions always hold on optimizers and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials we also give a general formula for the algebraic degree of the optimal coordinates the derivation of the algebraic degree is equivalent to counting the number of all complex critical points as special cases we obtain the algebraic degrees of quadratically constrained quadratic programming qcqp second order cone programming socp and p th order cone programming pocp in analogy to the algebraic degree of semidefinite programming j nie k ranestad and b sturmfels the algebraic degree of semidefinite programming math programm to appear

large time behaviour of the solutions of a semilinear parabolic equation in rn abstract the asymptotic behaviour as t tends to of the solution of u t u u u p 1 0 in r n r p 1 was studied it was proved that the behaviour depends strongly on the sign of n 2 n p and also on the rate of decay of the admissible initial data u 0 x as x tends to

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

a dynamic inequality generation scheme for polynomial programming hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs this approach rapidly becomes computationally expensive and is often tractable only for problems of small size in this paper we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs when used iteratively this scheme improves the bounds without incurring an exponential growth in the size of the relaxation as a result the proposed scheme is in principle scalable to large general polynomial programming problems when all the variables of the problem are non negative or when all the variables are binary the general algorithm is specialized to a more efficient algorithm in the case of binary polynomial programs we show special cases for which the proposed scheme converges to the global optimal solution we also present several examples illustrating the computational behavior of the scheme and provide comparisons with lasserre s approach and for the binary linear case with the lift and project method of balas ceria and cornuejols

on the complexity of testing attainment of the optimal value in nonlinear optimization we prove that unless hbox p hbox np there exists no polynomial time or even pseudo polynomial time algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low degree polynomials is attained if the degrees of these polynomials are fixed our results along with previously known frank wolfe type theorems imply that exactly one of two cases can occur either the optimal value is attained on every instance or it is strongly np hard to distinguish attainment from non attainment we also show that testing for some well known sufficient conditions for attainment of the optimal value such as coercivity of the objective function and closedness and boundedness of the feasible set is strongly np hard as a byproduct our proofs imply that testing the archimedean property of a quadratic module is strongly np hard a property that is of independent interest to the convergence of the lasserre hierarchy finally we give semidefinite programming sdp based sufficient conditions for attainment of the optimal value in particular a new characterization of coercive polynomials that lends itself to an sdp hierarchy

on square positive extensions and cubature formulas we consider linear functionals ld defined over pd p r x1 xn and study how to extend ld to a square positive l p r i e to a linear functional l with l p2 0 for all p p we use the connection of square positive functionals to real ideals to orthogonal polynomials and to moment matrices finally we report how such extension techniques are used to construct and analyse cubature formulas

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

positive definiteness of paired symmetric tensors and elasticity tensors abstract in this paper we consider higher order paired symmetric tensors and strongly paired symmetric tensors elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors a strongly paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics which plays an important role in nonlinear elasticity theory we mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional strongly paired symmetric tensors we first show that the concerned strongly paired symmetric tensor is positive definite if and only if its smallest m eigenvalue is positive second we propose several necessary and sufficient conditions under which the concerned strongly paired symmetric tensor is positive definite third we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional strongly paired symmetric tensor can be written as a sum of squares of polynomials and further propose several necessary and or sufficient conditions to judge whether the concerned strongly paired symmetric tensors are positive definite or not fourth by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest m eigenvalue of a fourth order three dimensional strongly paired symmetric tensor by which we can check positive definiteness of the concerned tensor the preliminary numerical results confirm our theoretical findings

semidefinite approximations for global unconstrained polynomial optimization we consider the problem of minimizing a polynomial function on mathbb r n known to be hard even for degree 4 polynomials therefore approximation algorithms are of interest lasserre siam j optim 11 2001 pp 796 817 and parrilo math program 96 2003 pp 293 320 have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations we propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming the method is applied to several examples

univariate parameterization for global optimization of mixed integer polynomial problems abstract this paper presents a new relaxation technique to globally optimize mixed integer polynomial programming problems that arise in many engineering and management contexts using a bilinear term as the basic building block the underlying idea involves the discretization of one of the variables up to a chosen accuracy level teles j p castro p m matos h a 2013 multiparametric disaggregation technique for global optimization of polynomial programming problems j glob optim 55 227 251 by means of a radix based numeric representation system coupled with a residual variable to effectively make its domain continuous binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed integer non linear problem into a larger one of the mixed integer linear programming type the new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms

biquadratic optimization over unit spheres and semidefinite programming relaxations this paper studies the so called biquadratic optimization over unit spheres min x in mathbb r n y in mathbb r m sum 1 leq i k leq n 1 leq j l leq m b ijkl x i y j x k y l subject to x 1 y 1 we show that this problem is np hard and there is no polynomial time algorithm returning a positive relative approximation bound then we present various approximation methods based on semidefinite programming sdp relaxations our theoretical results are as follows for general biquadratic forms we develop a frac 1 2 max m n 2 approximation algorithm under a slightly weaker approximation notion for biquadratic forms that are square free we give a relative approximation bound frac 1 nm when min n m is a constant we present two polynomial time approximation schemes ptass which are based on sum of squares sos relaxation hierarchy and grid sampling of the standard simplex for practical computational purposes we propose the first order sos relaxation a convex quadratic sdp relaxation and a simple minimum eigenvalue method and show their error bounds some illustrative numerical examples are also included

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

an exact jacobian sdp relaxation for polynomial optimization given polynomials f x gi x hj x we study how to minimize f x on the set s left x in mathbb r n h 1 x cdots h m 1 x 0 g 1 x geq 0 ldots g m 2 x geq 0 right let fmin be the minimum of f on s suppose s is nonsingular and fmin is achievable on s which are true generically this paper proposes a new type semidefinite programming sdp relaxation which is the first one for solving this problem exactly first we construct new polynomials varphi 1 ldots varphi r by using the jacobian of f hi gj such that the above problem is equivalent to begin gathered underset x in mathbb r n min f x hfill rm s t h i x 0 varphi j x 0 1 leq i leq m 1 1 leq j leq r hfill quad g 1 x nu 1 cdots g m 2 x nu m 2 geq 0 quad forall nu in 0 1 m 2 hfill end gathered second we prove that for all n big enough the standard n th order lasserre s sdp relaxation is exact for solving this equivalent problem that is its optimal value is equal to fmin some variations and examples are also shown

the hierarchy of local minimums in polynomial optimization this paper studies the hierarchy of local minimums of a polynomial in the vector space mathbb r n rn for this purpose we first compute h h minimums for which the first and second order necessary optimality conditions are satisfied to compute each h h minimum we construct a sequence of semidefinite relaxations based on optimality conditions we prove that each constructed sequence has finite convergence under some generic conditions a procedure for computing all local minimums is given when there are equality constraints we have similar results for computing the hierarchy of critical values and the hierarchy of local minimums

certifying convergence of lasserre s hierarchy via flat truncation consider the optimization problem of minimizing a polynomial function subject to polynomial constraints a typical approach for solving it globally is applying lasserre s hierarchy of semidefinite relaxations based on either putinar s or schmudgen s positivstellensatz a practical question in applications is how to certify its convergence and get minimizers in this paper we propose flat truncation as a certificate for this purpose assume the set of global minimizers is nonempty and finite our main results are 1 putinar type lasserre s hierarchy has finite convergence if and only if flat truncation holds under some generic assumptions the same conclusion holds for the schmudgen type one under weaker assumptions 2 flat truncation is asymptotically satisfied for putinar type lasserre s hierarchy if the archimedean condition holds the same conclusion holds for the schmudgen type one if the feasible set is compact 3 we show that flat truncation can be used as a certificate to check exactness of standard sos relaxations and jacobian sdp relaxations

minimizing polynomials via sum of squares over the gradient ideal a method is proposed for finding the global minimum of a multivariate polynomial via sum of squares sos relaxation over its gradient variety that variety consists of all points where the gradient is zero and it need not be finite a polynomial which is nonnegative on its gradient variety is shown to be sos modulo its gradient ideal provided the gradient ideal is radical or the polynomial is strictly positive on the real gradient variety this opens up the possibility of solving previously intractable polynomial optimization problems the related problem of constrained minimization is also considered and numerical examples are discussed experiments show that our method using the gradient variety outperforms prior sos methods

a frank wolfe type theorem for nondegenerate polynomial programs in this paper we study the existence of optimal solutions to a constrained polynomial optimization problem more precisely let f 0 f 0 and f 1 ldots f p mathbb r n rightarrow mathbb r f 1 f p r n r be convenient polynomial functions and let s x in mathbb r n f i x le 0 i 1 ldots p ne emptyset s x r n f i x 0 i 1 p under the assumption that the map f 0 f 1 ldots f p mathbb r n rightarrow mathbb r p 1 f 0 f 1 f p r n r p 1 is non degenerate at infinity we show that if f 0 f 0 is bounded from below on s s then f 0 f 0 attains its infimum on s s

solution of the general moment problem via a one parameter imbedding this paper presents a computational theory for the general scalar moment problem the formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics and in nonuniform multidimensional sampling given a finite set of moments the theory provides a test for the existence of a positive measure which is consistent with such data at the same time the theory also provides a characterization of all such consistent positive measures it should be noted that classical results e g in the theory of the trigonometric moment problem hamburger stieljes nevanlinna pick interpolation etc are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels the centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function spl psi which plays the role of a free parameter the differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure for each spl psi the fixed point determines a corresponding measure suitable choice of spl psi allows recovering any measure which is consistent with the data the fixed point of the differential equation corresponds to an extremum of an entropy like functional and the differential equation is constructed via an appropriate homotopy that follows changes in the lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments

embry truncated complex moment problem abstract let t be a cyclic subnormal operator on a hilbert space h with cyclic vector x 0 and let ij t i t j x 0 x 0 for any i j n 0 the bram halmos characterization for subnormality of t involved a moment matrix m n in a parallel approach we construct a moment matrix e n corresponding to embry s characterization for subnormality of t we discuss the relationship between m n and e n via the full moment problem next given a collection of complex numbers ij 0 i j 2 n i j n with 00 0 and ji ij we consider the truncated complex moment problem for this entails finding a positive borel measure supported in the complex plane c such that ij z i z j d z we show that this moment problem can be solved when e n 0 and e n admits a flat extension e n k where k 1 when n is odd and k 2 when n is even

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets