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

krein adler transformations for shape invariant potentials and pseudo virtual states for 11 examples of one dimensional quantum mechanics with shape invariant potentials the darboux crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to krein adler transformations deleting multiple eigenstates with shifted parameters these are based upon infinitely many polynomial wronskian identities of classical orthogonal polynomials i e the hermite laguerre and jacobi polynomials which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape invariant potentials

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

the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

geometric mapping properties of semipositive matrices abstract semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices nevertheless the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices it is shown that for a semipositive matrix a there exist a proper polyhedral cone k 1 of nonnegative vectors and a polyhedral cone k 2 of nonnegative vectors such that a k 1 k 2 the set of all nonnegative vectors mapped by a to the nonnegative orthant is a proper polyhedral cone as a consequence a belongs to a proper polyhedral cone comprising semipositive matrices when the powers a k k 0 1 have a common semipositivity vector then a has a positive eigenvalue if a has a sole peripheral eigenvalue and the powers of a have a common semipositivity vector with a non vanishing term in the direction of the left eigenspace of then a leaves a proper cone invariant

geometric conditions for the existence of positive eigenvalues of matrices finite dimensional theorems of perron frobenius type are proved for a e e nn and a nonnegative integer k we let w t a be the cone generated by ai ak l in en we show that a satisfies the perron schaefer condition if and only if the closure wk a of w t a is a pointed cone this theorem is closely related to several known results if k po a the index of the eigenvalue 0 in spec a we prove that a has a positive eigenvalue if and only if w t a is a pointed nonzero cone or equivalently w t a is not a real subspace of en our proofs are elementary and based on a method of birkhoff s we discuss the relation of this method to pringsheim s theorem

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

reflectionless potentials for difference schrodinger equations as a part of the program discrete quantum mechanics we present general reflectionless potentials for difference schrodinger equations with pure imaginary shifts by combining contiguous integer wave number reflectionless potentials we construct the discrete analogues of the potential with the integer h which belong to the recently constructed families of solvable dynamics having the q ultraspherical polynomials with as the main part of the eigenfunctions for the general scattering theory for these potentials we need the connection formulas for the basic hypergeometric function with which is not known the connection formulas are expected to contain the quantum dilogarithm functions as the counterparts of the q gamma functions we propose a conjecture of the connection formula of the function with based on the conjecture we derive the transmission and reflection amplitudes which have all the desirable properties they provide a strong support to the conjectured connection formula

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

exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice we address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice as kay and moses showed in 1956 reflectionless potentials in one dimensional quantum mechanics are exactly solvable with an additional time dependence these potentials are identified as the soliton solutions of the korteweg de vries kdv hierarchy an n soliton potential has the time t and 2n positive parameters k1 kn and cj j 1 n corresponding to n discrete eigenvalues kj2 the eigenfunctions are elementary functions expressed by the ratio of determinants the darboux crum krein adler transformations or the abraham moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters cj we explore various identities satisfied by the eigenfunctions of the soliton potentials which reflect the uniqueness theorem of gel fand levitan marchenko equations for separable degenerate kernels

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

extensions of solvable potentials with finitely many discrete eigenstates we address the problem of rational extensions of six examples of shape invariant potentials having finitely many discrete eigenstates the overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions which are an essential ingredient for multi indexed and iso spectral extensions of these potentials they have exactly the same form as the eigenfunctions but their degrees are much higher than nmax so that their energies are lower than the groundstate energy

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

linear equations over cones and collatz wielandt numbers let k be a proper cone in r n let a be an n x n real matrix that satisfies ak k let b be a given vector of k and let be a given positive real number the following two linear equations are considered in this paper i i n a x b x k and ii a l n x b x k we obtain several equivalent conditions for the solvability of the first equation for the second equation we give an equivalent condition for its solvability in case when b a and we also find a necessary condition when b a and also when b a sufficiently close to b a where b a denotes the local spectral radius of a at b with fixed we also consider the questions of when the set a l n k k equals 0 or k and what the face of k generated by the set is then we derive some new results about local spectral radii and collatz wielandt sets or numbers associated with a cone preserving map and extend a known characterization of m matrices among z matrices in terms of alternating sequences

on the equivalence of the integral and the differential exact solution generation methods for the one dimensional schrodinger equation it is demonstrated that the integral exact solution generation methods for the one dimensional schrodinger equation based on the gelfand levitan formalism are in some cases equivalent to the differential ones based on the n order darboux transformation some new exact solvable potentials are generated from the effective coulomb potential and the harmonic oscillator potential a new form of n soliton potential i e reflectionless potential with n discrete energy levels disposed in a desirable manner based on an explicit expression for an n order wronski determinant constructed from hyperbolic functions and its orthonormal discrete spectrum eigenfunctions are given

on the invariant faces associated with a cone preserving map for an n n nonnegative matrix p an isomorphism is obtained between the lattice of initial subsets of 1 n for p and the lattice of p invariant faces of the nonnegative orthant r motivated by this isomorphism we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone preserving map on a polyhedral cone formulated in terms of its invariant faces in particular we obtain the following extension of the famous rothblum index theorem for a nonnegative matrix if a leaves invariant a polyhedral cone k then for each distinguished eigenvalue ofa fork there is a chain of m distinct a invariant join irreducible faces of k each containing in its relative interior a generalized eigenvector of a corresponding to referred to as semi distinguished a invariant faces associated with where m is the maximal order of distinguished generalized eigenvectors of a corresponding to but there is no such chain with more than m members we introduce the important new concepts of semi distinguished a invariant faces and of spectral pairs of faces associated with a cone preserving map and obtain several properties of a cone preserving map that mostly involve these two concepts when the underlying cone is polyhedral perfect or strictly convex and or smooth or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval plentiful illustrative examples are provided some open problems are posed at the end 1991 mathematics subject classification primary 15a48 secondary 47b65 47a25 46b42

invariant ellipsoidal cones abstract conditions on the spectrum of a matrix a which are equivalent to the existence of a proper convex cone k such that a is k nonnegative k positive k irreducible or k strongly nonnegative are known we study the effect of adding the requirement that k be ellipsoidal it is shown that the conditions are unchanged except in the k nonnegative case

an algebraic analytic approach to nonnegative bases abstract rothblum and independently richman and schneider have used a combination of graph theoretic and algebraic approaches based on the frobenius normal form of a matrix to show that if a is an n n nonnegative matrix then the generalized eigenspace of a corresponding to its perron root a has a nonnegative basis moreover they showed that among the jordan chains corresponding to a a has a chain of maximal length consisting entirely of nonnegative generalized eigenvectors in this paper we develop an algebraic analytic approach to the proofs of these results which is based on the asymptotic behavior of the resolvent operator the approach allows us to suggest an algorithm for computing a nonnegative basis

on the collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

generalizes darboux transformations classification of inverse scattering methods for the radial schrodinger equation a wide class of darboux transformations of a sturm liouville equation providing a unified treatment of exactly solvable models in quantum scattering theory is considered systematically a classification of darboux transformations is given and the relations between darboux transformations and standard inverse scattering procedures for the radial schrodinger equation including the matrix method of newton sabatier are studied in particular the definitions properties and matrix generalizations of darboux transformations associated with marchenko type integral equations are studied in detail

perron frobenius theory for finite dimensional spaces with a hyperbolic cone abstract if x is a real n dimensional space provided with a subnorm then the inequality x defines a so called hyperbolic cone ine r x in this case the perron frobenius theory admits some special features a relevant characterization of nonnegative operators in a matrix form is given first auxiliary information from spectral theory and from the geometry of subnormed spaces is collected as preparation

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

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

on matrices leaving invariant a nontrivial convex set abstract a real square matrix a leaves a nontrivial convex set invariant if there exists a convex set c which is not a linear subspace such that a c c it is shown that this is equivalent to the statement that a has an eigenvalue with 0 or 1

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

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

wold type decompositions and wandering subspaces for operators close to isometries for operators t satisfying certain inequalities we obtain a weak analog of the classical wold kolmogorov decomposition theorem representing t as a direct sum of a unitary operator and a shift operator acting in some hilbert space of analytic functions the concept of a dual operator is introduced which reflects relationships between shift operators acting in two hilbert spaces of analytic functions dual to each other with respect to the cauchy pairing among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels

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

berezin transforms on noncommutative varieties in polydomains abstract let q be a set of polynomials in noncommutative indeterminates z i j i 1 k j 1 n i in this paper we study noncommutative varieties v q h x x i j d h g x 0 for all g q where d h is a regular polydomain in b h n 1 n k and b h is the algebra of bounded linear operators on a hilbert space h under natural conditions on q we show that there is a universal model s s i j such that g s 0 g q acting on a subspace of a tensor product of full fock spaces we characterize the variety v q h and its pure part in terms of the universal model and a class of completely positive linear maps we obtain a characterization of those elements in v q h which admit characteristic functions and prove that the characteristic function is a complete unitary invariant for the class of completely non coisometric elements we study the universal model s its joint invariant subspaces and the representations of the universal operator algebras it generates the variety algebra a v q the hardy algebra f v q and the c algebra c v q using noncommutative berezin transforms associated with each variety we develop an operator model theory and dilation theory for large classes of varieties in noncommutative polydomains this includes various commutative cases which are closely connected to the theory of holomorphic functions in several complex variables and algebraic geometry

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

non polynomial extensions of solvable potentials a la abraham moses abraham moses transformations besides darboux transformations are well known procedures to generate extensions of solvable potentials in one dimensional quantum mechanics here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the abraham moses transformations for typical solvable potentials e g the radial oscillator the darboux poschl teller and some others these seed solutions are simple generalisations of the virtual state wavefunctions which are obtained from the eigenfunctions by discrete symmetries of the potentials the virtual state wavefunctions have been an essential ingredient for constructing multi indexed laguerre and jacobi polynomials through multiple darboux crum transformations in contrast to the darboux transformations the virtual state wavefunctions generate non polynomial extensions of solvable potentials through the abraham moses transformations

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them