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

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

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

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

quasilinear elliptic problems in r n involving oscillatory nonlinearities in this paper we study an elliptic problem in r n which involves the p laplacian p n 2 and the nonlinear term has an oscillatory behavior by means of a direct variational approach we establish the existence of infinitely many homoclinic solutions whose w 1 p r n norms tend to zero to infinity respectively whenever the nonlinearity oscillates at zero at infinity respectively the solutions have invariance properties with respect to certain subgroups of the orthogonal group o n

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

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

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

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

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

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

a nonsmooth principle of symmetric criticality and variational hemivariational inequalities in this paper we prove the principle of symmetric criticality for motreanu panagiotopoulos type functionals i e for convex proper lower semicontinuous functionals which are perturbed by a locally lipschitz function by means of this principle a variational hemivariational inequality is studied on certain type of unbounded strips

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

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

subcritical perturbations of a singular quasilinear elliptic equation involving the critical hardy sobolev exponent abstract in this work we improve some known results for a singular operator and also for a wide class of lower order terms by proving a multiplicity result the proof is made by applying the generalized mountain pass theorem due to ambrosetti and rabinowitz to do this we show that the minimax levels are in a convenient range by combining a special class of approximating functions due to gazzola and ruf with the concentrating functions of the best sobolev constant

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

existence and multiplicity of weak solutions for a singular quasilinear elliptic equation abstract in this paper we consider the singular quasilinear elliptic equation 0 1 div x a p d u p 2 d u v x u p 2 u h x u q 2 u h x u r 2 u x r n 0 where 0 a n p p 1 p n 1 q r p n p n p the function v x is continuous and nonnegative the functions h x h x l b r 1 b with 0 r 0 r using variational methods we obtain a number of results on existence and multiplicity of solutions by considering all possible orderings of exponents p q r

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

ground state solutions for singular p laplacian equation in rn abstract in this work we study the ground state solution for the class of singular quasilinear elliptic problem of the form div x a p u p 2 u h x u m 2 u h x u q 2 u x r n where 1 p n a n p p and h r h r 0 we prove the compactly embedding results of the weighted z d a r 1 p r n l h r m r n space of radially symmetric functions and then obtain the existence of nontrivial ground state solution

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

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

weak solutions of quasilinear problems with nonlinear boundary condition the growing attention for the study of the p laplacian operator p in the last few decades is motivated by the fact that it arises in various applications for instance in fluid mechanics the shear stress and the velocity gradient pu of certain uids obey a relation of the form x a x pu x where pu u p 2 u here p 1 is an arbitrary real number and the case p 2 respectively pi 2 p 2 corresponds to a newtonian respectively pseudoplastic dilatant uid the resulting equations of motion then involve div a pu which reduces to a pu a div pu provided that a is constant the p laplacian also appears in the study of torsional creep elastic for p 2 plastic as p see 7 ow through porous media p 2 see 12 or glacial sliding p 1 3 see 9 let rn be an unbounded domain with possible noncompact smooth boundary and n is the unit outward normal on we consider the nonlinear elliptic boundary value problem div a x u p 2 u 1 x 1 u p 2u 1 x 2 u q 2u in

a supersymmetry approach to almost diagonal random matrices we develop a supersymmetric field theoretical description of the gaussian ensemble of the almost diagonal hermitian random matrices the matrices have independent random entries hi j with parametrically small off diagonal elements we derive a regular virial expansion of correlation functions in the number of interacting supermatrices associated with different sites in the real space and demonstrate that the perturbation theory constructed in this way is controlled by a small parameter the general form of the integral expression for the mth virial coefficient governed by the interaction of m supermatrices is presented and calculated explicitly in the cases of 2 and 3 matrix interaction the suggested technique allows us to calculate both the spectral correlations and the correlations of the eigenfunctions taken at different energies and in different space points

multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity some multiplicity results are presented for the eigenvalue problem div x 2a u x 2b f u x 2c g u in u 0o n

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

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665