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

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

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

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

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

characteristic functions and joint invariant subspaces let t t1 tn be an n tuple of operators on a hilbert space such that t is a completely non coisometric row contraction we establish the existence of a one to one correspondence between the joint invariant subspaces under t1 tn and the regular factorizations of the characteristic function t associated with t in particular we prove that there is a non trivial joint invariant subspace under the operators t1 tn if and only if there is a non trivial regular factorization of t we also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function and prove the existence of joint invariant subspaces for certain classes of n tuples of operators we obtain criteria for joint similarity of n tuples of operators to cuntz row isometries in particular we prove that a completely non coisometric row contraction t is jointly similar to a cuntz row isometry if and only if the characteristic function of t is an invertible multi analytic operator

regular dilation on graph products of n abstract we extended the definition of regular dilation to graph products of n which is an important class of quasi lattice ordered semigroups two important results in dilation theory are unified under our result namely brehmer s regular dilation on n k and frazho bunce popescu s dilation of row contractions we further show that a representation of a graph product has an isometric nica covariant dilation if and only if it is regular a special case of our result was considered by popescu and we studied the connection with popescu s work

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

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

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

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

wavelet representations and fock space on positive matrices abstract we show that every biorthogonal wavelet determines a representation by operators on hilbert space satisfying simple identities which captures the established relationship between orthogonal wavelets and cuntz algebra representations in that special case each of these representations is shown to have tractable finite dimensional co invariant doubly cyclic subspaces further motivated by these representations we introduce a general fock space hilbert space construction which yields creation operators containing the cuntz toeplitz isometries as a special case

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

on fixed points of self maps of the free ball abstract in this paper we study the structure of the fixed point sets of noncommutative self maps of the free ball we show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace we provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete pick spaces

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

joint similarity and dilations for noncontractive sequences of operators a characteristic function t is defined in terms of multianalytic operators on fock spaces for any noncontractive sequence t t1 td d n or d of operators on a hilbert space h it is shown that if t is bounded then it is unitarily equivalent to a compression of an orthogonal projection on krein spaces this leads to a generalization of a theorem of davis and foias to multivariable setting more precisely it is proved that if t has bounded characteristic function then it is jointly similar to a contractive sequence of operators i e there exists a similarity s b h such that the operator defined by the row matrix st1s 1st2s 1 stds 1 is a contraction motivated by the similarity problem a multivariable dilation theory on fock spaces with indefinite metric is developed for noncontractive d tuples of operators wold type decompositions for sequences of bounded isometries on krein spaces and fourier representations for d orthogonal shifts are obtained and used to study the geometry of the canonical minimal isometric dilation associated with a sequence t of operators on a hilbert space

the isomorphism problem for some universal operator algebras abstract this paper addresses the isomorphism problem for the universal non self adjoint operator algebras generated by a row contraction subject to homogeneous polynomial relations we find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables and that this happens if and only if the associated subproduct systems are isomorphic the proof makes use of the complex analytic structure of the character space together with some recent results on subproduct systems restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry these commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties we prove a projective nullstellensatz connecting closed ideals and their zero sets under some technical assumptions we find that two such algebras are isomorphic as algebras if and only if they are similar and we obtain a clear geometrical picture of when this happens this result is obtained with tools from algebraic geometry reproducing kernel hilbert spaces and some new complex geometric rigidity results of independent interest the c envelopes of these algebras are also determined the banach algebraic and the algebraic classification results are shown to hold for the wot closures of these algebras as well

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

finitely correlated representations of product systems of c correspondences over nk abstract we study isometric representations of product systems of correspondences over the semigroup n k which are minimal dilations of finite dimensional fully coisometric representations we show the existence of a unique minimal cyclic coinvariant subspace for all such representations the compression of the representation to this subspace is shown to be a complete unitary invariant for a certain class of graph algebras the nonself adjoint wot closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace this class includes free semigroup algebras this result extends to a class of higher rank graph algebras which includes higher rank graphs with a single vertex

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

on joint spectral radius of commuting operators in hilbert spaces abstract our aim in this paper is to give a new formula of the joint spectral radius of commuting d tuples of operators on a complex hilbert space also we show that r t t for a commuting d tuple of operators t where r t and t denote respectively the joint spectral radius and the joint numerical radius of t this generalizes the well known relation between the spectral and the numerical radii of hilbert space operators which is proved in 10

a bipartite separable ball and its applications in this paper based on a matrix norm we first present a ball of separable un normalized states around the identity matrix for the bipartite quantum system which is larger than the separable ball in frobenius norm then the proposed ball is used to get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes but also a separable ball centered at the identity matrix for the multipartite quantum system

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

on the topological stable rank of non selfadjoint operator algebras we provide a negative solution to a question of m rieffel who asked if the right and left topological stable ranks of a banach algebra must always agree our example is found amongst a class of nest algebras we show that for many other nest algebras both the left and right topological stable ranks are infinite we extend this latter result to popescu s non commutative disc algebras and to free semigroup algebras as well

some remarks on vanishing theorems for holomorphic vector bundles this theorem generalizes kodaira s vanishing theorem 13 and has found a number of geometric applications 2 19 griffiths proof uses method ii above doing curvature calculations for line bundles then the object of this note is to reduce theorem g directly to the kodaira vanishing theorem answering a question of viehweg during the complex analysis meeting in oberwolfach 1982 thus avoiding any curvature calculation this proof has the further advantage that one obtains better results let us state as a sample

absolutely continuous representations and a kaplansky density theorem for free semigroup algebras we introduce notions of absolutely continuous functionals and representations on the non commutative disk algebra an absolutely continuous functionals are used to help identify the type l part of the free semigroup algebra associated to a extendible representation a extendible representation of an is regular if the absolutely continuous part coincides with the type l part all known examples are regular absolutely continuous functionals are intimately related to maps which intertwine a given extendible representation with the left regular representation a simple application of these ideas extends reflexivity and hyper reflexivity results moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of an is weak dense in the unit ball of the associated free semigroup algebra if and only if is regular we provide some explicit constructions related to the density theorem for specific representations a notion of singular functionals is also defined and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts

structure of free semigroupoid algebras abstract a free semigroupoid algebra is the wot closure of the algebra generated by a tck family of a graph we obtain a structure theory for these algebras analogous to that of free semigroup algebra we clarify the role of absolute continuity and wandering vectors these results are applied to obtain a lebesgue von neumann wold decomposition of tck families along with reflexivity a kaplansky density theorem and classification for free semigroupoid algebras several classes of examples are discussed and developed including self adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence

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

on tuples of commuting operators in positive semidefinite inner product spaces abstract this paper is concerned with the study of certain properties of operator tuples on a complex hilbert space h when a semi inner product induced by a positive operator a on h is considered in particular we show that r a t a t for every commuting operator tuple t t 1 t d such that each t k admits an a adjoint operator where r a t and a t denote respectively the a joint spectral radius and the a joint numerical radius of t this study allows to establish that r a t a t t a for every a normal commuting tuple of operators t where t a is denoted to be the a joint operator seminorm of t in addition the a joint spectral radius of an a m isometric tuple of commuting operators is studied

algebras of noncommutative functions on subvarieties of the noncommutative ball the bounded and completely bounded isomorphism problem abstract given a noncommutative nc variety v in the nc unit ball b d we consider the algebra h v of bounded nc holomorphic functions on v we investigate the problem of when two algebras h v and h w are isomorphic we prove that these algebras are weak continuously isomorphic if and only if there is an nc biholomorphism g w v between the similarity envelopes that is bi lipschitz with respect to the free pseudo hyperbolic metric moreover such an isomorphism always has the form f f g where g is an nc biholomorphism these results also shed some new light on automorphisms of the noncommutative analytic toeplitz algebras h b d studied by davidson pitts and by popescu in particular we find that aut h b d is a proper subgroup of aut b d when d and the varieties are homogeneous we remove the weak continuity assumption showing that two such algebras are boundedly isomorphic if and only if there is a bi lipschitz nc biholomorphism between the similarity envelopes of the nc varieties we provide two proofs in the noncommutative setting our main tool is the noncommutative spectral radius about which we prove several new results in the free commutative case we use a new free commutative nullstellensatz that allows us to bootstrap techniques from the fully commutative case

nest representations of directed graph algebras this paper is a comprehensive study of the nest representations for the free semigroupoid algebra mathfrak l g of a countable directed graph g as well as its norm closed counterpart the tensor algebra mathcal t g we prove that the finite dimensional nest representations separate the points in mathfrak l g and a fortiori in mathcal t g the irreducible finite dimensional representations separate the points in mathfrak l g if and only if g is transitive in components which is equivalent to being semisimple also the upper triangular nest representations separate points if and only if for every vertex x in mathcal t g supporting a cycle x also supports at least one loop edge we also study faithful nest representations we prove that mathfrak l g or mathcal t g admits a faithful irreducible representation if and only if g is strongly transitive as a directed graph more generally we obtain a condition on g which is equivalent to the existence of a faithful nest representation we also give a condition that determines the existence of a faithful nest representation for a maximal type mathbb n nest

dilating covariant representations of the non commutative disc algebras let be an isometric automorphism of the non commutative disc algebra an for n 2 we show that every contractive covariant representation of an dilates to a unitary covariant representation of on hence the c envelope of the semicrossed product an z is on z

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

applications of the wold decomposition to the study of row contractions associated with directed graphs based on a wold decomposition for families of partial isometries and projections of cuntz krieger toeplitz type we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks we prove a szego type factorization theorem for ckt families which leads to information on the structure of the unit ball in free semigroupoid algebras and show that joint similarity implies joint unitary equivalence for such families for each graph we prove a generalization of von neumann s inequality which applies to row contractions of operators on hilbert space which are related to the graph in a natural way this yields a functional calculus determined by quiver algebras and free semigroupoid algebras we establish a generalization of coburn s theorem for the c algebra of a ckt family and prove a universality theorem for c algebras generated by these families in both cases the c algebras generated by quiver algebras play the universal role

stable difference schemes for parabolic systems a numerical radius approach ii in this paper a numerical radius approach is taken in order to further discuss stability conditions for a well known family of finite difference schemes for the initial value problem associated with the petrowski well posed multispace dimensional parabolic system dst frac partial mathbf u mathbf x t partial t dst sum 1 leq p leq q leq s a pq dst frac partial 2 mathbf u mathbf x t partial x p partial x q sum 1 leq p leq s b p frac partial mathbf u mathbf x t partial x p c mathbf u mathbf x t noindent where a pq b p and c are constant matrices a pq being hermitian

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

lifting fixed points of completely positive mappings abstract let 1 n and let t j j 1 n b h be a row contraction on some hilbert space h let f t be the space of all x b h such that j 1 n t j x t j x we show that if non zero this space is completely isometric to the commutant of the cuntz part of the minimal isometric dilation of t j j 1 n

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

on stability of staggered schemes this paper investigates the theoretical stability bound of a lagrangian staggered scheme used to solve hydrodynamics equations we present the two dimensional 2d wave equation as a possible model for this study and by using the numerical radius of the amplification matrix we prove that the family of schemes defined with two time centering parameters is limited by a nonclassical stability bound limit defined with an analytical curve we further show that 2d numerical experiments agree with this theoretical result

on the numerical radius and its applications abstract we give a brief account of the numerical radius of a linear bounded operator on a hilbert space and some of its better known properties both finite and infinite dimensional aspects are discussed as well as applications to stability theory of finite difference approximations for hyperbolic initial value problems

joint numerical ranges of operators in semi hilbertian spaces abstract in this paper we aim to investigate the concept of numerical range and maximal numerical range relative to a positive operator of a d tuple of bounded linear operators on a hilbert space some properties and applications of these sets are studied mainly it is proved that they are convex for d 1 this generalizes the well known toeplitz hausdorff theorem 24 16 and stampfli s result 23 moreover under additional hypotheses we show that these sets are convex for d 2

noncommutative berezin transforms and multivariable operator model theory in this paper we initiate the study of a class dpm h of noncommutative domains of n tuples of bounded linear operators on a hilbert space h where m 2 n 2 and p is a positive regular polynomial in n noncommutative indeterminates these domains are defined by certain positivity conditions on p i e dpm h x x1 xn 1 p k x x 0 for 1 k m each such a domain has a universal model w1 wn of weighted shifts acting on the full fock space f2 hn with n generators the study of dpm h is close related to the study of the weighted shifts w1 wn their joint invariant subspaces and the representations of the algebras they generate the domain algebra an dpm the hardy algebra fn dpm and the c algebra c w1 wn a good part of this paper deals with these issues the main tool which we introduce here is a noncommutative berezin type transform associated with each n tuple of operators in dpm h the study of this transform and its boundary behavior leads to fatou type results functional calculi and a model theory for n tuples of operators in dpm h these results extend to noncommutative varieties vp qm h dpm h generated by classes q of noncommutative polynomials when m 2 n 2 p z1 zn and q 0 the elements of the corresponding variety vp qm h can be seen as multivariable noncommutative analogues of agler s m hypercontractions our results apply in particular when q consists of the noncommutative polynomials zizj zjzi i j 1 n in this case the model space is a symmetric weighted fock space fs2 dpm which is identified with a reproducing kernel hilbert space of holomorphic functions on a reinhardt domain in cn and the universal model is the n tuple m 1 m n of multipliers by the coordinate functions in this particular case we obtain a model theory for commuting n tuples of operators in dpm h recovering several results already existent in the literature

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

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties