isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

complements to models for noncommuting operators abstract this paper develops a dilation theory for two noncommutative operators many of the results and techniques in dilation theory for one operator are extended to this setting this is done by extending the schaffer construction for one operator to the fock space setting a new wold decomposition for two isometrics with orthogonal range is obtained this wold decomposition and the schaffer construction is used to extend the sz nagy foias lifting theorem and the characteristic function theory to the noncommutative setting the characteristic function is operator valued and defined in a fock space

composition operators on noncommutative hardy spaces abstract in this paper we initiate the study of composition operators on the noncommutative hardy space h ball 2 which is the hilbert space of all free holomorphic functions of the form f x 1 x n k 0 k a x f n a 2 1 where the convergence is in the operator norm topology for all x 1 x n in the noncommutative operatorial ball b h n 1 and b h is the algebra of all bounded linear operators on a hilbert space h when the symbol is a free holomorphic self map of b h n 1 we show that the composition operator c f f f h ball 2 is bounded on h ball 2 several classical results about composition operators boundedness norm estimates spectral properties compactness similarity have free analogues in our noncommutative multivariable setting the most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of b h n the operator algebras generated by the left creation operators on the full fock space with n generators and the classical complex function theory in the unit ball of c n in a more general setting we establish basic properties concerning the composition operators acting on fock spaces associated with noncommutative varieties v p 0 h b h n 1 generated by sets p 0 of noncommutative polynomials in n indeterminates such that p 0 0 p p 0 in particular when p 0 consists of the commutators x i x j x j x i for i j 1 n we show that many of our results have commutative counterparts for composition operators on the symmetric fock space and consequently on spaces of analytic functions in the unit ball of c n

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

periodic solutions for coupled first order nonlinear differential systems of hamiltonian type soient f g 0 t r r r et p q 0 t r des fonctions continues donnees telles que f 0 x y f t x y g 0 x y g t x y x y er p o p t q o q t et 0 t p t dt s 0 t q t dt 0 on considere l existence des solutions periodiques pour les systemes differentiels non lineaires d ordre 1 couples de type hamiltonien x f t x y p t g t x y q t t d t x o x t g o y t 0

free holomorphic functions and interpolation in this paper we obtain a noncommutative multivariable analogue of the classical nevanlinna pick interpolation problem for analytic functions with positive real parts on the open unit disc given a function f lambda to mathbb c where lambda is an arbitrary subset of the open unit ball mathbb b n z in mathbb c n z 1 we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball b mathcal h n 1 such that rm re g geq 0 quad rm and quad g z f z quad z in lambda where b mathcal h is the algebra of all bounded linear operators on a hilbert space mathcal h the proof employs several results from noncommutative multivariable operator theory and a noncommutative cayley transform introduced and studied in the present paper acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions all the results of this paper are obtained in the more general setting of free holomorphic functions with operator valued coefficients as consequences we deduce some results concerning operator valued analytic interpolation on the unit ball mathbb b n

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

free holomorphic functions on the unit ball of b h n abstract in this paper we continue the study of free holomorphic functions on the noncommutative ball b h n 1 x 1 x n b h n x 1 x 1 x n x n 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h and n 1 2 or n several classical results from complex analysis have free analogues in our noncommutative setting we prove a maximum principle a naimark type representation theorem and a vitali convergence theorem for free holomorphic functions with operator valued coefficients we introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by schwarz and harnack the borel caratheodory theorem is extended to our noncommutative setting using a noncommutative generalization of schwarz s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball b h n 1 we obtain an analogue of julia s lemma for free holomorphic functions f b h n 1 b h m 1 we also obtain pick julia theorems for free holomorphic functions with operator valued coefficients we provide a noncommutative generalization of a classical inequality due to lindelof which turns out to be sharper then the noncommutative von neumann inequality finally we introduce a pseudohyperbolic metric on b h n 1 which is invariant under the action of the free holomorphic automorphism group of b h n 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on b n the open unit ball of c n in this setting we obtain a schwarz pick type lemma we also provide commutative versions of these results for operator valued multipliers of the drury arveson space

gronwall bellman type nonlinear delay integral inequalities on time scales abstract in this paper some gronwall bellman type nonlinear delay integral inequalities on time scales are established which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales our results generalize some of the main results in lipovan 2006 1 pachpatte 2000 2 ferreira and torres 2009 3 zhang and meng 2008 4 cheung and ren 2006 5 kim 2009 6 and some of our results unify continuous and discrete analysis in the literature

free functions with symmetry in 1936 margarete c wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables we show that wolf s theorem is a special case of a general theory of the ring of invariant free polynomials every ring of invariant free polynomials is isomorphic to a free polynomial ring furthermore we show that this isomorphism extends to the free functional calculus as a norm preserving isomorphism of function spaces on a domain known as the row ball we give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures specifically we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case

solvability of a boundary value problem with the nonlinearity satisfying a sign condition soit f 0 r r une fonction donnee satisfaisant des conditions de caratheodory et h 0 r une fonction donnee dans l 1 0 on obtient l existence d une solution au probleme aux valeurs limites u x u x f x u x h x x 0 u 0 u 0 pour h x telle que 0 h x sinxdx 0 et f x u telle que f x u u 0 pour u r et il y a une constante 0 telle que limsup u f x u u 3 uniformement pour x 0

noncommutative transforms and free pluriharmonic functions abstract in this paper we study free pluriharmonic functions on noncommutative balls b h n 0 and their boundary behavior these functions have the form f x 1 x n k 1 k b x a 0 i k 1 k a x a b c where the convergence of the series is in the operator norm topology for any x 1 x n b h n and b h denotes the algebra of all bounded linear operators on a hilbert space h the main tools used in this study are certain noncommutative transforms which are introduced in the present paper and which generalize the classical transforms of berezin poisson fantappie herglotz and cayley several classical results from complex analysis have free analogues in our noncommutative multivariable setting

some new delay integral inequalities and their applications the main objective of this paper is to establish some new delay integral inequalities which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations

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

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

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

a generalization of retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

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

integral inequalities for retarded volterra equations integral inequalities are very useful in the qualitative analysis of differential and integral equations starting with o lipovan a retarded gronwall like inequality and its applications j math anal appl 252 2000 389 401 several recent investigations see o lipovan a retarded integral inequality and its applications j math anal appl 285 2003 436 443 b g pachpatte explicit bounds on certain integral inequalities j math anal appl 267 2002 48 61 b g pachpatte on some retarded integral inequalities and applications j inequal pure appl math 3 2002 article 18 b g pachpatte on a certain retarded integral inequality and its applications j inequal pure appl math 5 2004 article 19 b g pachpatte on some new nonlinear retarded integral inequalities j inequal pure appl math 5 2004 article 80 were devoted to retarded integral inequalities in this paper we consider the case of retarded volterra integral equations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the asymptotic behaviour of the solutions

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

unbounded perturbations of forced second order ordinary differential equations at resonance we prove the existence of at least one solution for the differential equation x t m2x t g t x t e t with periodicity conditions x 0 x 2 x 0 x 2 0 where m 0 is an integer e is integrable and g satisfies caratheodory conditions our results are obtained for the case when there is resonance at the eigenvalue m2 of the linear second order differential equation t x t 0 r with x 0 x 2 x 0 x 2 0 the function g may be unbounded and touching of the eigenvalue m 1 2 resp m 1 2 if m 0 on a subset of positive measure is allowed our approach also works when periodicity conditions are replaced by dirichlet or neumann boundary conditions the proofs are based on topological degree mawhin s continuation theorem and leray schauder techniques

integral inequalities in two independent variables for retarded volterra equations in this paper we obtain some retarded volterra integral inequalities in two independent variables which can be used as tools in the theory of partial differential and integral equations with time delays

on retarded integral inequalities and their applications abstract the main results of j math anal appl 285 2003 436 443 are here generalized to the following retarded integral inequalities u m t c m m n m m n 0 t f s u n s w u s g s u n s d s and u m t c m m n m m n 0 t f s u n s w u s d s m m n 0 t g s u n s w u s d s where m n 0 are constants and t r 0 the results given here can be applied to the global existence of solutions to differential equations with time delay

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

on certain integral inequalities in two independent variables for retarded equations in this paper we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays applications are given to illustrate the usefulness of the inequalities

on retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

on some new integral inequalities and their applications the aim of the present paper is to establish some new integral inequalities in two independent variables which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain partial differential equations