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

models for n tuples of noncommuting operators abstract this paper proves versions of the rota model theorem the de branges rovnyak model theorem and the coisometric extension theorem for n tuples of not necessarily commuting operators this generalizes the work of a e frazho j funct anal 48 1982 1 11 for pairs of operators the methods involve applying the single operator results to matrices of operators

invariant subspaces and hyper reflexivity for free semigroup algebras in this paper we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras this enables us to prove that they are reflexive and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces such algebras are called hyper reflexive this property is very strong but it has been established in only a very few cases moreover the prototypes of this class of algebras are the natural candidate for a non commutative analytic toeplitz algebra on n variables the case we make for this analogy is very compelling in particular in this paper the key to the invariant subspace analysis is a good analogue of the beurling theorem for invariant subspaces of the unilateral shift this leads to a notion of inner outer factorization in these algebras in a sequel to this paper 13 we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in cn a free semigroup algebra is the weak operator topology closed algebra generated by a set s1 sn of isometries with pairwise orthogonal ranges these conditions are described algebraically by s i sj iji for 1 i j n f

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

nonlocality asymmetry and distinguishing bipartite states entanglement is a useful resource because some global operations cannot be locally implemented using classical communication we prove a number of results about what is and what is not locally possible we focus on orthogonal states which can always be globally distinguished we establish the necessary and sufficient conditions for a general set of 2 ifmmode times else texttimes fi 2 quantum states to be locally distinguishable and for a general set of 2 ifmmode times else texttimes fi n quantum states to be distinguished given an initial measurement of the qubit these results reveal a fundamental asymmetry to nonlocality which is the origin of nonlocality without entanglement and we present a very simple proof of this phenomenon

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

characterizing locally indistinguishable orthogonal product states bennett physical review a vol 59 no 2 p 1070 1999 identified a set of orthogonal product states in the hilbert space bbc 3 otimes bbc 3 such that reliably distinguishing those states requires nonlocal quantum operations while more examples have been found for this counterintuitive ldquononlocality without entanglementrdquo phenomenon a complete and computationally verifiable characterization for all such sets of states remains unknown in this paper we give such a characterization for both bbc 3 otimes bbc 3 and bbc 2 otimes bbc 2 otimes bbc 2 as a consequence we show that in both spaces there is no additional set of a fundamentally different structure than those of the known instances

an intrinsic beam model based on a helicoidal approximation part i formulation the focus of the present work is directed towards the one dimensional non linear analysis of space curved and twisted beams undergoing large displacements and finite rotations according to cosserat s model a beam is intended here as a continuum generated by the rigid motion of a cross section along a curve the novelty of the proposed methodology is implied in the assumption that the reference line of the beam both in the undeformed and in the deformed configurations is a helicoid in space this formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields in fact within the framework of helicoidal geometry a powerful description of space curved beam kinematics with large displacements and finite rotations can be developed furthermore the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non linear analysis the formulation discussed in this work adheres to a mixed variational principle the independent fields being the generalized displacements and the sectional stress resultants in a companion paper we address the linearization of the governing weak form the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations and the development of a helicoidal model for rigid body dynamics using finite elements in time

much ado about shear correction factors in timoshenko beam theory abstract many shear correction factors have appeared since the inception of timoshenko beam theory in 1921 while rational bases for them have been offered there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application herein three dimensional static and dynamic information and results for a beam of general both symmetric and non symmetric cross section are brought to bear on these issues only homogeneous isotropic beams are considered semi analytical finite element safe computer codes provide static and dynamic response data for our purposes greater clarification of issues relating to the bases for shear correction factors can be seen also comparisons of numerical results with timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary bernoulli euler theory an issue concerning principal shear axes arose in the definition of shear correction factors for non symmetric cross sections in this method expressions for the shear energies of two transverse forces applied on the cross section by beam and three dimensional elasticity theories are equated to determine the shear correction factors this led to the necessity for principal shear axes we will argue against this concept and show that when two forces are applied simultaneously to a cross section it leads to an inconsistency only one force should be used at a time and two sets of calculations are needed to establish the shear correction factors for a non symmetrical cross section

standard noncommuting and commuting dilations of commuting tuples we introduce a notion called maximal commuting piece for tuples of hilbert space operators given a commuting tuple of operators forming a row contraction there are two commonly used dilations in multivariable operator theory first there is the minimal isometric dilation consisting of isometries with orthogonal ranges and hence it is a noncommuting tuple there is also a commuting dilation related with a standard commuting tuple on boson fock space we show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation we use this result to classify all representations of the cuntz algebra o n coming from dilations of commuting tuples

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

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

when do local operations and classical communication suffice for two qubit state discrimination in this paper we consider the conditions under which a given ensemble of two qubit states can be optimally distinguished by local operations and classical communication locc we begin by completing the perfect distinguishability problem of two qubit ensembles both for separable operations and locc by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state then for the well known task of minimum error discrimination it is shown that almost all two qubit ensembles consisting of three pure states cannot be optimally discriminated using locc this is surprising considering that any two pure states can be distinguished optimally by locc special attention is given to ensembles that lack entanglement and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by locc thus providing new examples of the phenomenon known as non locality without entanglement we next consider an example of n parties who each share the same state but who are ignorant of its identity the state is drawn from the rotationally invariant trine ensemble and we establish a tight connection between the n copy ensemble and shor s lifted single copy ensemble for any finite n we prove that optimal identification of the states cannot be achieved by locc however as n locc can indeed discriminate the states optimally this is the first result of its kind finally we turn to the task of unambiguous discrimination and derive new lower bounds on the locc inconclusive probability for symmetric states when applied to the double trine ensemble this leads to a rather different distinguishability character than when the minimum error probability is considered

noncommutative hyperbolic geometry on the unit ball of b h n in this paper we introduce a hyperbolic poincare bergman type distance 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 it is proved that is invariant under the action of the free holomorphic automorphism group of b h n 1 i e x y x y x y b h n 1 for all aut b h n 1 moreover we show that the topology and the usual operator norm topology coincide on b h n 1 while the open ball b h n 1 is not a complete metric space with respect to the operator norm topology we prove that b h n 1 is a complete metric space with respect to the hyperbolic metric we obtain an explicit formula for in terms of the reconstruction operator rx x1 r1 xn rn x x1 xn b h n 1 associated with the right creation operators r1 rn on the full fock space with n generators in the particular case when h c we show that the hyperbolic distance coincides with the poincare bergman distance on the open unit ball bn z z1 zn cn z 2 1 we obtain a schwarz pick lemma for free holomorphic functions on b h n 1 with respect to the hyperbolic metric i e if f f1 fm is a contractive f 1 free holomorphic function then f x f y x y x y b h n 1 as consequences we show that the caratheodory and the kobayashi distances with respect to coincide with on b h n 1 the results of this paper are presented in the more general context of harnack parts of the closed ball b h n 1 which are noncommutative analogues of the gleason parts of the gelfand spectrum of a function algebra

schur class multipliers on the arveson space de branges rovnyak reproducing kernel spaces and commutative transfer function realizations an interesting and recently much studied generalization of the classical schur class is the class of contractive operator valued multipliers s for the reproducing kernel hilbert space h kd on the unit ball bd cd where kd is the positive kernel kd 1 1 on bd the reproducing kernel space h ks associated with the positive kernel ks i s s kd is a natural multivariable generalization of the classical de branges rovnyak canonical model space a special feature appearing in the multivariable case is that the space h ks in general may not be invariant under the adjoints m j of the multiplication operators m j f jf on h kd we show that invariance of h ks under m j for each j 1 d is equivalent to the existence of a realization for s of the form s d c i 1a1 dad 1 1b1 dbd such that connecting operator u a1b1 adbdcd has adjoint u which is isometric on a certain natural subspace u is weakly coisometric and has the additional property that the state operators a1 ad pairwise commute in this case one can take the state space to be the functional model space h ks and the state operators a1 ad to be given by aj m j h ks a de branges rovnyak functional model realization we show that this special situation always occurs for the case of inner functions s where the associated multiplication operator ms is a partial isometry and that inner multipliers are characterized by the existence of such a realization such that the state operators a1 ad satisfy an additional stability property

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

local hypothesis testing between a pure bipartite state and the white noise state in this paper we treat a local discrimination problem in the framework of asymmetric hypothesis testing we choose a known bipartite pure state psi rangle as an alternative hypothesis and the completely mixed state as a null hypothesis as a result we analytically derive an optimal type 2 error and an optimal positive operator valued measure povm for one way local operations and classical communication locc povm and separable povm for two way locc povm we study a family of simple three step locc protocols and show that the best protocol in this family has strictly better performance than any one way locc protocol in low dimensional systems when there may exist differences between two way locc povm and one way locc povm

on the analysis of prismatic beams using first order warping functions abstract a one dimensional theory for isotropic homogeneous beams with a general cross section is developed that utilizes both deformations in the cross section plane and warping out of the cross section plane where these deformation functions are known from saint venant s elasticity solutions for bending and flexure of a prismatic beam first order warping functions in contrast to existing technical beam theories which assume that local cross section deformations and some stresses can be neglected the present theory makes no assumption the finite element model is developed using the resulting differential equations along with a partially weak form formulation full three dimensional constitutive equations are used solutions are found using the finite element method numerical results which are presented for a wide variety of loadings and boundary conditions show that the calculated stresses and displacements are in excellent agreement with existing elasticity solutions

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

hyperbolic geometry on noncommutative polyballs in this paper we study the noncommutative balls c x1 xn b h n x1 xn 1 0 where is the joint operator radius for n tuples of bounded linear operators on a hilbert space in particular 1 is the operator norm 2 is the joint numerical radius and is the joint spectral radius we introduce a harnack type equivalence relation on c 0 and use it to define a hyperbolic distance on the harnack parts equiv alence classes of c we prove that the open ball c 0 is the harnack part containing 0 and obtain a concrete formula for the hyperbolic distance in terms of the reconstruction operator associated with the right creation operators on the full fock space with n gener ators moreover we show that the topology and the usual operator norm topology coincide on c 1 while the open ball c 1 is not a complete metric space with respect to the operator norm topology we prove that it is a complete metric space with respect to the hyper bolic metric in the particular case when 1 and h c the hyperbolic metric coincides with the poincare bergman distance on the open unit ball of c n

curvature invariant for hilbert modules over free semigroup algebras abstract we introduce a notion of relative curvature resp euler characteristic for finite rank contractive hilbert modules over c f n the complex free semigroup algebra generated by the free semigroup f n on n generators asymptotic formulas and basic properties for both the curvature and the euler characteristic are established in particular it is shown that the standard relative curvature invariant resp euler characteristic of a hilbert module h is a nonnegative number less than or equal to the rank of h and it depends only on the properties of the completely positive map t x ni 1 tixt i where t1 tn is the row contraction of not necessarily commuting operators uniquely determined by the c f n module structure of h moreover we prove that for every t 0 there is a hilbert module h such that curv h h t the module structure defined by the left creation operators on the full fock space f2 hn on n generators occupies the position of the rank one free module in the algebraic theory we obtain a complete description of the closed submodules resp quotients of the free hilbert module f2 hn and calculate their curvature invariant it is shown that the curvature is a complete invariant for the finite rank submodules of the free hilbert module f2 hn k where k is a finite dimensional hilbert space a noncommutative version of the gauss bonnet chern theorem from riemannian geometry is obtained for graded hilbert modules over c f n in particular it is proved that the curvature and euler characteristic coincide for certain classes of pure hilbert modules our investigation is based on noncommutative poisson transforms noncommutative dilation theory and harmonic analysis on fock spaces

operators affiliated to the free shift on the free hardy space abstract the smirnov class for the classical hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators this definition extends naturally to the commutative and non commutative multi variable settings of the drury arveson space and the full fock space over c d identifying the fock space with the free multi variable hardy space of non commutative or free holomorphic functions in a non commutative open unit ball in several matrix variables we prove that any closed densely defined operator affiliated to the right free multiplier algebra of the full fock space acts as right multiplication by a non commutative function in the right free smirnov class and analogously replacing right with left

realizing a 2 d positive operator valued measure by local operations and classical communication we study the realization of a 2 d positive operator valued measure povm by local operations and classical communication locc we derive that any povm on a 2 d hilbert space in which each party s space is finite dimensional can be realized by one way locc this implies that any optimal discrimination of quantum states spanning such a 2 d hilbert space is possible by one way locc regardless of the optimality criterion used and how entangled the states are

gleason s problem and homogeneous interpolation in hardy and dirichlet type spaces of the ball we solve gleason s problem in the reproducing kernel hilbert spaces with reproducing kernels 1 1 1nzjwj r for real r 0 and their counterparts for r 0 and study the homogeneous interpolation problem in these spaces

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

models for noncommuting operators abstract this paper develops a model theory for a pair of noncommuting operators using backward shift operators on a fock space rota s theorem is generalized i e it is shown that any two bounded operators on a hilbert space are simultaneously similar to part of a pair of backward shift operators on a fock space these shift operators and the fock space framework are also used to develop a dilation theory for two noncommuting operators

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

saint venant solutions for prismatic anisotropic beams abstract an analytical model is presented for determining the displacement and stress distributions of the saint venant extension bending torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy the determination of the complete displacement field requires solving a coupled two dimensional boundary value problem for the local in plane deformations and warping out of the section plane the principle of minimum potential energy is applied to a discretized representation of the cross section ritz method to calculate solutions to this problem the behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions where definitions are presented for the shear center center of twist torsion constant and a new geometric parameter the line of extension bending centers two sets of numerical results are presented to illustrate how section geometry beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam

interpolatability distinguishes locc from separable von neumann measurements local operations with classical communication locc and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement separable operations are strictly more powerful than locc but no simple explanation of this phenomenon is known we show that in the case of von neumann measurements the ability to interpolate measurements is an operational principle that sets apart locc and separable operations

limitations on separable measurements by convex optimization we prove limitations on locc and separable measurements in bipartite state discrimination problems using techniques from convex optimization specific results that we prove include an exact formula for the optimal probability of correctly discriminating any set of either three or four bell states via locc or separable measurements when the parties are given an ancillary partially entangled pair of qubits an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements and an optimal bound on the success probability for any locc or separable measurement for the recently proposed state discrimination problem of yu duan and ying

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

spectral theory of multiplication operators on hardy sobolev spaces abstract for a pointwise multiplier of the hardy sobolev space h 2 on the open unit ball b n in c n we study spectral properties of the multiplication operator m h 2 h 2 in particular we compute the spectrum and essential spectrum of m and develop the fredholm theory for these operators

clark theory in the drury arveson space abstract we extend the basic elements of clark s theory of rank one perturbations of backward shifts to row contractive operators associated to de branges rovnyak type spaces h b contrastively contained in the drury arveson space on the unit ball in c d the aleksandrov clark measures on the circle are replaced by a family of states on a certain noncommutative operator system and the backward shift is replaced by a canonical solution to the gleason problem in h b in addition we introduce the notion of a quasi extreme multiplier of the drury arveson space and use it to characterize those h b spaces that are invariant under multiplication by the coordinate functions

local distinguishability of any three quantum states we prove that any three linearly independent pure quantum states can always be locally distinguished with nonzero probability regardless of their dimension entanglement or multipartite structure almost always all three states can be unambiguously identified the only exceptional case where one state is locally knowable but the other two are not is found among multi qubit states

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

implicit inverse function theorems for free noncommutative functions abstract we prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices we apply these results to study dependence of the solution of the initial value problem for odes in noncommutative spaces on the initial data and to extremal problems with noncommutative constraints

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

distinguishability of quantum states by separable operations in this paper we study the distinguishability of multipartite quantum states by separable operations we first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations an analytical version of this condition is derived for the case of d 1 pure states where d is the total dimension of the state space under consideration a number of interesting consequences of this result are then carefully investigated remarkably we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication before our work only a class of 3 otimes 3 nonlocal separable operations was known bennett phys rev a 59 1070 1999 we also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states i e has an orthogonal schmidt number not less than three thus generalize the recent work about indistinguishable bipartite subspaces watrous phys rev lett 95 080505 2005 notably we obtain an explicit construction of indistinguishable subspaces of dimension 7 or 6 by considering a composite quantum system consisting of two qutrits resp three qubits which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8

hardy algebras w correspondences and interpolation theory given a von neumann algebra m and a w correspondence e over m we construct an algebra h e that we call the hardy algebra of e when m open image in new window e h e is the classical hardy space h open image in new window of bounded analytic functions on the unit disc when m open image in new window and e open image in new window h e is the free semigroup algebra studied by popescu davidson and pitts and many others we show that given any faithful normal representation of m on a hilbert space h there is a natural correspondence e over the commutant m called the dual of e and that h e can be realized in terms of b h valued functions on the open unit ball open image in new window e in the space of adjoints of elements in e we prove analogues of the nevanlinna pick theorem in this setting and discover other aspects of the value distribution theory for elements in h e we also analyze the boundary behavior of elements in h e and obtain generalizations of the sz nagy foias functional calculus and the functional calculus of popescu for c n c row contractions the correspondence e has a dual that is naturally isomorphic to e and the commutants of certain so called induced representations of h e can be viewed as induced representations of h e for these induced representations a double commutant theorem is proved

fixed point theorems for noncommutative functions abstract we establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices the conclusions are stronger if such a mapping also respects matrix similarities i e is a noncommutative function as a special case we prove the corresponding contractive mapping theorem which can be viewed as a new version of the banach fixed point theorem this result is then applied to prove the existence and uniqueness of a solution of the initial value problem for odes in noncommutative spaces as a by product of the ideas developed in this paper we establish a noncommutative version of the principle of nested closed sets

interpolation problems in several variables abstract we obtain a noncommutative nudel man type interpolation theorem in several variables and a commutative analogue forh b n the hardy space of bounded analytic functions in the open ball of c n using this result we obtain an operatorial analogue in several variables of the rosenblum rovnyak restriction theorem as well as an extension to the noncommutative analytic toeplitz algebraf we provide new extensions of the caratheodory fejer and nevanlinna pick interpolation theorems forf

complete nevanlinna pick kernels abstract we give a new treatment of quiggin s and mccullough s characterization of complete nevanlinna pick kernels we show that a kernel has the matrix valued nevanlinna pick property if and only if it has the vector valued nevanlinna pick property we give a representation of all complete nevanlinna pick kernels and show that they are all restrictions of a universal complete nevanlinna pick kernel

distinguishing bipartitite orthogonal states using locc best and worst cases two types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications we examine sets of states that can be perfectly distinguished in particular showing that any three orthogonal maximally entangled states in c3 c3 form such a set in cases where orthogonal states cannot be distinguished we obtain upper bounds for the probability of error using locc taken over all sets of k orthogonal states in cn cm in the process of proving these bounds we identify some sets of orthogonal states for which perfect distinguishability is not possible

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

transfer function realization for multipliers of the arveson space abstract an interesting and recently much studied generalization of the classical schur class is the class of contractive operator valued multipliers for the reproducing kernel hilbert space h k d on the unit ball b d c d where k d is the positive kernel k d 1 1 on b d we study this space from the point of view of realization theory and functional models of de branges rovnyak type we highlight features which depart from the classical univariate case coisometric realizations have only partial uniqueness properties the nonuniqueness can be described explicitly and this description assumes a particularly concrete form in the functional model context

general behavior and shear center location of prismatic anisotropic beams via power series abstract the behavior of a tip loaded anisotropic cantilever beam with an arbitrary cross section is studied using saint venant s semi inverse method along with a power series solution for the local in plane and out of plane deformation warping functions the power series coefficients are determined by solving a set of variationally derived linear algebraic equations using the resulting three dimensional displacement solutions the shear deformation associated with applied tip loads is investigated as well as the shear center location two different definitions of the shear center are presented for anisotropic beams by extending existing approaches developed for isotropic sections both of the extended definitions reveal the linear dependency of the shear center location on beam length numerical results are presented for three different cross sections ellipse triangle naca 0012 airfoil and two different materials al 6061 t6 off angle high strength graphite epoxy fibers

testing for a pure state with local operations and classical communication we examine the problem of using local operations and classical communication locc to distinguish a known pure state from an unknown possibly mixed state bounding the error probability from above and below we study the asymptotic rate of detecting multiple copies of the pure state and show that if the overlap of the two states is great enough then they can be distinguished asymptotically as well with locc as with global measurements otherwise the maximal schmidt coefficient of the pure state is sufficient to determine the asymptotic error rate

on bohr s inequality bohr s inequality says that if is a bounded analytic function on the closed unit disc then for 0 leq r 1 3 and that 1 3 is sharp in this paper we give an operator theoretic proof of bohr s inequality that is based on von neumann s inequality since our proof is operator theoretic our methods extend to several complex variables and to non commutative situations we obtain bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel hilbert spaces on various higher dimensional domains for the non commutative disc algebra an and for the reduced respectively full group c algebra of the free group on n generators we also include an application to banach algebras we prove that every banach algebra has an equivalent norm in which it satisfies a non unital version of von neumann s inequality 2000 mathematical subject classification 47a20 47a56

characteristic function for polynomially contractive commuting tuples abstract in this note we develop the theory of characteristic function as an invariant for n tuples of operators the operator tuple has a certain contractivity condition put on it this condition and the class of domains in c n that we consider are intimately related a typical example of such a domain is the open euclidean unit ball given a polynomial p in c z 1 z 2 z n whose constant term is zero all the coefficients are nonnegative and the coefficients of the linear terms are nonzero one can naturally associate a reinhardt domain with it which we call the p ball definition 1 1 using the reproducing kernel hilbert space h p c associated with this reinhardt domain in c n s pott constructed the dilation for a polynomially contractive commuting tuple definition 1 2 s pott standard models under polynomial positivity conditions j operator theory 41 1999 365 389 mr 2000j 47019 given any polynomially contractive commuting tuple t we define its characteristic function t which is a multiplier we construct a functional model using the characteristic function exploiting the model we show that the characteristic function is a complete unitary invariant when the tuple is pure the characteristic function gives newer and simpler proofs of a couple of known results one of them is the invariance of the curvature invariant and the other is a beurling theorem for the canonical operator tuple on h p c it is natural to study the boundary behaviour of t in the case when the domain is the euclidean unit ball we do that and here essential differences with the single operator situation are brought out

invariant subspaces and operator model theory on noncommutative varieties let mathcal q subset mathbb c left be an arbitrary set of polynomials in noncommutative indeterminates such that q 0 0 for all q in mathcal q the noncommutative variety begin aligned mathcal v f mathcal q m mathcal h left x x 1 ldots x n in mathbf d f m mathcal h q x 0 text for all q in mathcal q right end aligned where mathbf d f m mathcal h is a noncommutative regular domain in b mathcal h n and b mathcal h is the algebra of bounded linear operators on a hilbert space mathcal h admits a universal model b m b 1 m ldots b n m such that q b m 0 q in mathcal q acting on a model space which is a subspace of the full fock space with n generators in this paper we obtain a beurling type characterization of the joint invariant subspaces under the operators b 1 m ldots b n m in terms of partially isometric multi analytic operators acting on model spaces more generaly a beurling lax halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under b 1 m otimes i mathcal e ldots b n m otimes i mathcal e and to characterize when they are generating for the corresponding invariant subspaces similar results are obtained for any pure n tuple x 1 ldots x n in the noncommutative variety mathcal v f mathcal q m mathcal h we characterize the elements in the noncommutative variety mathcal v f mathcal q m mathcal h which admit characteristic functions develop an operator model theory for the completely non coisometric elements and prove that the characteristic function is a complete unitary invariant for this class of elements this extends the classical sz nagy foias functional model for completely non unitary contractions based on characteristic functions our results apply in particular when mathcal q consists of the noncommutative polynomials z iz j z jz i i j 1 ldots n in this case the model space is a symmetric weighted fock space which is identified with a reproducing kernel hilbert space of holomorphic functions on a reinhardt domain in mathbb c n and the universal model is the n tuple m lambda 1 ldots m lambda n of multipliers by the coordinate functions

multivariable bohr inequalities operator valued multivariable bohr type inequalities are obtained for i a class of noncommutative holomorphic functions on the open unit ball of generalizing the analytic functions on the open unit disc ii the noncommutative disc algebra and the noncommutative analytic toeplitz algebra iii a class of noncommutative selfadjoint harmonic functions on the open unit ball of generalizing the real valued harmonic functions on the open unit disc iv the cuntz toeplitz algebra the reduced resp full group algebra resp of the free group with generators v a class of analytic functions on the open unit ball of the classical bohr inequality is shown to be a consequence of fejer s inequality for the coefficients of positive trigonometric polynomials and haager up de la harpe inequality for nilpotent operators moreover we provide an inequality which for analytic polynomials on the open unit disc is sharper than bohr s inequality

non commutative clark measures for the free and abelian toeplitz algebras abstract we construct a non commutative aleksandrov clark measure for any element in the operator valued free schur class the closed unit ball of the free toeplitz algebra of vector valued full fock space over c d here the free analytic toeplitz algebra is the unital weak operator topology wot closed algebra generated by the component operators of the free shift the row isometry of left creation operators this defines a bijection between the free operator valued schur class and completely positive maps non commutative ac measures on the operator system of the free disk algebra the norm closed algebra generated by the free shift identifying drury arveson space with symmetric fock space we determine the relationship between the non commutative ac measures for elements of the operator valued commutative schur class the closed unit ball of the wot closed toeplitz algebra generated by the arveson shift and the ac measures of their free liftings to the free schur class

on the chernoff distance for asymptotic locc discrimination of bipartite quantum states motivated by the recent discovery of a quantum chernoff theorem for asymptotic state discrimination we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication locc in the limit of many copies while for two pure states a result of walgate et al shows that locc is just as powerful as global measurements data hiding states divincenzo et al show that locality can impose severe restrictions on the distinguishability of even orthogonal states here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states extremal d d werner states by a linear programming approach surprisingly the single copy optimal measurement remains optimal for n copies in the sense that the best strategy is measuring each copy separately followed by a simple classical decision rule we also put a lower bound on the bias with which states can be distinguished by separable operations

an operator valued poisson kernel abstract versions of the cauchy and poisson formulas in the unit ball of c n are defined by means of appropriate operator valued kernels connections with some problems of dilation theory are then made

non commutative markov chains and multi analytic operators journal of mathematical analysis and applications volume 364 issue 1 1 april 2010 pages 275 288

on a bitangential interpolation problem for contractive valued functions on the unit ball we solve the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the schur class consisting of matrix valued analytic functions on the ball the interpolation conditions are formulated via a generalized functional calculus with operator argument thereby generalizing in a compact way the simple first order interpolation conditions considered for this class of functions in earlier work the description of all solutions is given via a redheffer transform whose entries are given explicitly in terms of the interpolation data

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

operator structures and quantum one way locc conditions we conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one way local operations and classical communications locc we show how operator structures such as operator systems operator algebras and hilbert c modules naturally arise in this setting we make use of these structures to derive new results and bounds in the study of one way locc and we use the approach to uncover new derivations of some previously established results

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

a sufficient and necessary condition for 2n 1 orthogonal states to be locally distinguishable in a c2 cn system we investigate the local distinguishability of a set of orthogonal states on c2 cn by mathematical induction we prove that a set of 2n 1 orthogonal states on c2 cn can be locally discriminated by local operation and classical communication if and only if at most one of them is entangled that is the sum of schmidt rank of these states should be no greater than the total hilbert space dimension 2n furthermore we give some discussion about locally distinguishing any number of orthogonal states on c2 cn and explain why this schmidt rank criterion cannot be generalized to locally discriminate any number of orthogonal states