unbounded subnormal weighted shifts on directed trees ii abstract a new method of verifying the subnormality of unbounded hilbert space operators based on an approximation technique is proposed diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established an approach to this issue via consistent systems of probability measures is invented the role played by determinate stieltjes moment sequences is elucidated lambert s characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi analytic vectors which is a new phenomenon in this area

derivations vanishing on s lets be the group of finite permutations on countably many symbols we exhibit an embedding ofs into a uhf algebra mathfrak a of glimm typen such that if is a derivation vanishing ons and satisfying 0 where is the unique trace on mathfrak a then admits an extension which is the generator of ac dynamics

on quadratic integral equations in orlicz spaces abstract in this paper we study the quadratic integral equation of the form x t g t x t a b k t s f s x s d s several existence theorems for a e monotonic solutions in orlicz spaces are proved for strongly nonlinear functions f the presented method of the proof can be easily extended to different classes of solutions

subnormality of unbounded composition operators over one circuit directed graphs exotic examples abstract a recent example of a non hyponormal injective composition operator in an l 2 space generating stieltjes moment sequences invented by three of the present authors was built over a non locally finite directed tree the main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and in the affirmative case to find the simplest possible graph with these properties simplicity refers to local valency the problem is solved affirmatively for the locally finite directed graph g 2 0 which consists of two branches and one loop the only simpler directed graph for which the problem remains unsolved consists of one branch and one loop the consistency condition the only efficient tool for verifying subnormality of unbounded composition operators is intensively studied in the context of g 2 0 which leads to a constructive method of solving the problem the method itself is partly based on transforming the krein and the friedrichs measures coming either from shifted al salam carlitz q polynomials or from a quartic birth and death process

nonuniformly elliptic equations of p laplacian type abstract this paper deals with the existence of a generalized solution in w 0 1 p to a nonuniformly nonlinear elliptic equation of the form div a x u f x u in a bounded domain of r n here a satisfies a x c 0 h 0 x h 1 x p 1 for any in r n a e x h 0 l p p 1 and h 1 l loc 1

unbounded derivations commuting with compact group actions ii abstract let be a closed derivation in a c algebra 21 which commuteswith an ergodic action of a compact group on 5 then generates a c dynamics of 91 similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators 1 introduction in 9 sakai showed that a non zero closed derivation in c t commuting withtranslations is a constant multiple of the derivative following this it wasshown in 4 7 that if g is a locally compact group and is a closed derivationin c 0 g commuting with left translations by elements of g then is the generatorof a c dynamics i e strongly continuous one parameter group of automor phisms of c 0 g a like result holds for c 0 g h h a closed sub group of g provided that g is either separable or the projective limit of lie groups in thisnote we assume g is compact and we obtain similar results for an ergodic action ofg on an arbitrary c algebra theorem 2 1 and for an arbitrary action of g on anabelian c algebra theorem 3 2 or on the algebra of compact operators theorem 4 1 we refer to 1 3 9 for background on unbounded derivations in c algebras

non commutative moment problems we are concerned with two problems called the generalized moment problem gmp and the quantum moment problem qmp both problems can be considered as non commutative versions of the classical moment problem et g be a connected lie group and 7b e g the universal enveloping algebra of the lie algebra of g suppose f is a given linear functional on 7b e g and is a fixed unitary representation of g we then ask gmp when do there exist a unitary representation u of g and a cyclic c vector for u such that f x x 7b e g qmp when is f of the form f x trdv x x 7b e g where is an operator such that dv x is of trace class for every x 7b e g

iterated collocation methods for volterra integral equations with delay arguments in this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for volterra integral equations with constant delay this analysis includes continuous collocation based volterra runge kutta methods as well as iterated collocation methods and their discretizations

unbounded derivations tangential to compact groups of automorphisms ii let g be a compact abelian group and an action of g on a c algebra u such that u u u 0 u for all g where u is the spectral subspace of u corresponding to the character on g derivations which are defined on the algebra uf of g finite elements are considered in the special case u 0 these derivations are characterized by a cocycle on g with values in the relative commutant of u in the multiplier algebra of u and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded under various restrictions on g and properties of the cocycle are deduced which again give characterizations of in terms of decompositions into generators of one parameter subgroups of g and approximately inner derivations finally a perturbation technique is devised to reduce the case uf uf to the case uf uf and u 0 this is used to show that any derivation with d uf is wellbehaved and if furthermore g t1 and uf uf the closure of generates a one parameter group of automorphisms of u in the case g td d 2 3 finite and uf uf it is shown that extends to a generator of a group of automorphisms of the weak closure of u in any g covariant representation

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

existence theorems for some quadratic integral equations using the theory of measures of noncompactness we prove a few existence theorems for some quadratic integral equations the class of quadratic integral equations considered below contains as a special case numerous integral equations encountered in the theories of radiative transfer and neutron transport and in the kinetic theory of gases in particular the well known chandrasekhar integral equation also belongs to this class

commuting self adjoint extensions of symmetric operators defined from the partial derivatives we consider the problem of finding commuting self adjoint extensions of the partial derivatives 1 i xj j 1 d with domain cc where the self adjointness is defined relative to l2 and is a given open subset of rd the measure on is lebesgue measure on rd restricted to the problem originates with segal and fuglede and is difficult in general in this paper we provide a representation theoretic answer in the special case when i 2 and i is an open interval we then apply the results to the case when is a d cube id and we describe possible subsets rd such that e id is an orthonormal basis in l2 id

existence theorems for an integral equation of the chandrasekhar h equation with perturbation abstract we prove existence of positive solutions in thaces c 0 1 and c c 0 1 of an integral equation of the chandrasekhar h equation with perturbation

on a class of urysohn stieljes quadratic integral equations and their applications abstract we investigate a class of nonlinear quadratic integral equation of urysohn stieltjes type such a class contains a number of classical nonlinear quadratic integral equations such as the chandrasekhar quadratic integral equation we consider the solvability of the equations in question in the space of continuous functions under very general assumptions several particular cases of the assumed hypotheses are discussed and numerous applications are indicated

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

fields over unsharp coordinates it has been shown that space time coordinates can exhibit only very few types of short distance structures if described by linear operators they can be continuous discrete or unsharp in one of two ways in the literature various quantum gravity models of space time at short distances point towards one of these two types of unsharpness here we investigate the properties of fields over such unsharp coordinates we find that these fields are continuous but possess only a finite density of degrees of freedom similar to fields on lattices as a special case we recover the shannon sampling theorem of information theory

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

a non local perturbation of the logistic equation in irn abstract a logistic equation in the whole space is considered in this problem a non local perturbation is included we establish a new sub supersolution method for general nonlocal elliptic equations and consequently we obtain the existence of positive solutions of a nonlocal logistic equation

study on existence of solutions for some nonlinear functional integral equations abstract using the technique of measures of noncompactness in banach algebra we employ abstract fixed point theorems such as darbo s theorem to study the existence solution in banach algebra c 0 a for some functional integral equations which include many key integral and functional equations that arise in nonlinear analysis

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains abstract the goal of this paper is to study the existence of non trivial weak solutions for the nonuniformly nonlinear elliptic equation div h x u q x u f x u in an unbounded domain r n n 3 where h x l loc 1 the solutions will be obtained in a subspace of the sobolev space h 0 1 and the proofs rely essentially on a variation of the mountain pass theorem in d m duc nonlinear singular elliptic equations j london math soc 40 2 1989 420 440

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

collocation and continuous implicit runge kutta methods for a class of delay volterra integral equations abstract in this paper we present an analysis of the global convergence properties of certain piecewise polynomial collocation methods for a class of volterra integral equations with constant delay this analysis also encompasses continuous implicit m stage volterra runge kutta methods based on collocation techniques

commutative algebras of unbounded operators abstract we consider algebras u of unbounded operators defined on a common invariant dense domain d in a hilbert space h we assume that u is generated by a finite number of operators s1 sd satisfying the inequality x s x x s x x 0 for all finite sets of vectors x indexed by 1 d i 0 1 1 i d where s s 1 1 s d d and denotes the inner product in h we show that every such algebra dilates to a commutative algebra in a generally bigger hilbert space the latter algebra is generated by commuting formally normal extension operators n1 nd such that sjx njx x d 1 j d moreover the adjoint operators n j are represented as multiplication operators on a quotient of the space of d valued polynomials in complex variables z1 zd c d n j z j z d z j u z j z d if it is further assumed that d contains a dense space of vectors analytic for each of the operators sj then we prove that the nj s are essentially normal and there is a spectral measure e on the product of the spectrum of the closed normal operators n j such that n i f de and sjx jde x for x d as an application we consider representations of the canonical commutation relations in d degrees of freedom if p1 pd q1 qd denote the momentum resp position operators and sj pj iqj we show that s1 sd admits a representation as above we also show using our analytic vector theorem that the corresponding extension operators n 1 n d are commuting and normal for the schrodinger representation finally we calculate the spectral representation for the operators n j and prove absolute continuity and uniform multiplicity equal infinity the exposition of the paper begins with the case d 1 this is the unbounded version of the subnormal operators which have been studied extensively in the past in the case of bounded operators we stress the differences between the theory for the bounded and the unbounded case even in the case d 1 the applications of the unbounded theory are surprising

numerical solution of non linear volterra integral equations this paper deals with non linear volterra integral equations of the type y x f x 0x h t x y t y x dt convergence criteria are given in the same sense of the maximum and ca norms for the numerical solution of this type of volterra integral equation several numerical methods are compared

monotonic solutions of a quadratic integral equation of volterra type abstract we study a nonlinear quadratic integral equation of volterra type in the banach space of real functions defined and continuous on a bounded and closed interval with the help of a suitable measure of noncompactness we show that the mentioned integral equation has monotonic solutions

lie algebras of unbounded derivations abstract in this paper some new results on analytic domination of operators and on integrability of lie algebras of operators are proved and then our methods are applied to the study of lie algebras of unbounded derivations in c algebras

numerical solutions of volterra integral equations with a solution dependent delay on demontre l existence et l unicite des solutions pour l equation ainsi qu une limitation a priori de la solution on definit une solution numerique et on demontre que la solution numerique converge vers la solution de l equation integrale

anticommuting selfadjoint operators abstract several equivalent definitions of anticommutativity for selfadjoint operators are presented it is shown that the anticommutativity of two selfadjoint operators a and b satisfying af bg bf ag 0 for all and g in the intersection of the domains of a and b is equivalent to the selfadjointness of their sum and difference in studying the structure of anticommuting selfadjoint operators a and b attention may be restricted to the case where both a and b are injective in that case a is unitarily equivalent to a moreover a and b are uniquely determined by a triplet b p q where b is unitary and p and q are commuting injective positive and selfadjoint operators this is a natural correspondence

existence of solutions for a class of nonlinear volterra singular integral equations in this paper we prove some results concerning the existence of solutions for a large class of nonlinear volterra singular integral equations in the space c 0 1 consisting of real functions defined and continuous on the interval 0 1 the main tool used in the proof is the concept of a measure of noncompactness we also present some examples of nonlinear singular integral equations of volterra type to show the efficiency of our results moreover we compare our theory with the approach depending on the use of the theory of volterra stieltjes integral equations we also show that the results of the paper are applicable in the study of the so called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems

comparison of commuting one parameter groups of isometries abstract let a s be two commuting strongly continuous one parametergroups of isometries on a banach space j with generators 5q and 5 andanalytic elements srff srff respectively then it is easy to show that if oais relatively bounded by 5 then srff c saf and in this paper we establishthe inverse implication for unitary one parameter groups on hubert spaces andfor one parameter groups of automorphisms of abelian c algebras it is notknown in general whether the inverse implication holds or not but it does nothold for one parameter semigroups of contractions 1 introduction our intention in this paper is to examine the extent to which the analyticelements determine a strongly continuous one parameter group of isometries ona banach space for example if if v are two one parameter unitary groups on the hubert space sv with analytic elements a jfia v and generators h k and if u commutes with v then we prove that s w u c s jv if

existence of nondecreasing solutions of a quadratic integral equation of volterra type nondecreasing solutions of the quadratic integral equations is an interesting subject in the area of mathematics sciences and engineering using the theory of measures of noncompactness and applying a new method we prove the existence of nondecreasing solutions of a quadratic integral equation of volterra type in c i

monotonic solutions of a quadratic integral equation of fractional order abstract in the paper we indicate an error made in the proof of the main result of the paper m a darwish on quadratic integral equation of fractional orders j math anal appl 311 2005 112 119 moreover we provide correct proof of a slightly modified version of the mentioned result the main tool used in our proof is the technique associated with the hausdorff measure of noncompactness

derivations commuting with abelian gauge actions on lattice systems let be an action of a compact abelian groupg on ac algebraa and assume that the fixed point subalgebraa is an af algebra we show that if is a closed derivation ona commuting with and the restriction of toa generates a one parameter group of automorphisms then itself is a generator in particular the result applies if is an infinite product action ofg on a uhf algebra furthermore if in this situation 1 and 2 are two derivations both satisfying the hypotheses on and 1 and 2 have the same restriction toa then there exists a one parameter subgroup of the action with generator 0 such thatd 1 d 2 d 0 is a joint core for the three derivations and 2 1 0 on this core

spectral sets whose spectrum is a lattice with a base abstract let l be a finite subset of euclidean d dimensional space r d and let s be a lattice in r d such that l l l l l s 0 a classification of all lebesgue measurable subsets of r d for which the functions e x exp i 2 x x l s l s l l s s form an orthogonal basis for l 2 is given in terms of geometric and combinatorial data the data is derived from l and s

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

unbounded derivations commuting with compact group actions let be a closed derivation in ac algebra mathfrak a which commutes with an ergodic action of a compact group on mathfrak a then generates ac dynamics of mathfrak a similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators

spectral theory of finite volume domains in rn abstract let be an arbitrary open subset of r n of finite positive measure and assume the existence of a subset r n such that the exponential functions e exp i 1x1 nxn 1 n form an orthonormal basis for l 2 with normalized measure assume 0 and define subgroups k and a of r n by k 0 r n 2 z a a r n ua m u a m where ut is the unitary representation of r n on l 2 given by ute eit e t r n and where m is the multiplication algebra of l on l2 assume that a is discrete then there is a discrete subgroup d a of dimension n a fundamental domain d for d and finite sets of representers r r r each containing 0 r for a k in k0 and r for a k in a such that is disjoint union of translates of d a r a d neglecting null sets and r d0 if r is a set of representers for d a in d then r k is a translation set for i e r n direct sum neglecting null sets the case a r n corresponds to d d0 and k this last case corresponds in turn to a function theoretic assumption of forelli

harmonic analysis and fractal limit measures induced by representations of a certain c algebra we describe a class of measurable subsets in rd such that l2 has an orthogonal basis of frequencies e x ei2 x x indexed by rd we show that such spectral pairs have a self similarity which may be used to generate associated fractal measures typically with cantor set support the hilbert space l2 does not have a total set of orthogonal frequencies but a harmonic analysis of may be built instead from a natural representation of the cuntz c algebra which is constructed from a pair of lattices supporting the given spectral pair we show conversely that such a pair may be reconstructed from a certain cuntz representation given to act on l2

spectral theory of commuting self adjoint partial differential operators let denote a connected and open subset of rn the existence of n commuting self adjoint operators h1 hn on l2 such that each hj is an extension of i xj acting on cc is shown to be equivalent to the existence of a measure on rn such that f tf the fourier transform of f is unitary from l2 onto it is shown that the support of can be chosen as a subgroup of rn iff h1 hn can be chosen such that the unitary groups generated by h1 hn act multiplicatively on l2 this happens iff after correction by a null set forms a system of representatives for the quotient of rn by some subgroup i e iff is essentially a fundamental domain

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

analytic vectors and integrability of lie algebra representations abstract we prove that a representation of a finite dimensional lie algebra in a banach space is integrable if it admits a suitable invariant set of analytic vectors

the heat semigroup and integrability of lie algebras abstract let du x 1 s t c t 1 2 0 denote a continuous representation u of a lie group g acting on a banach space b and s the semigroup generated by the closed laplacian associated with a basis of the lie algebra g in the derived representation du we prove there is a c 0 such that du x i s t c t 1 2 0 for all xi in the basis of g and we also prove that this property is characteristic for integrability of a lie algebra of operators satisfying the usual dissipativity requirements for generators of one parameter groups the basic estimate is established from bounds on the heat kernel associated with s

numerical solutions of a perturbed chandrasekhar h equation abstract numerical solutions of perturbed chandrasekhar h type integral equations are analysed convergence criteria in the sense of the maximum norm and the c norm are studied several numerical methods are compared