stability of the absolutely continuous spectrum of random schrodinger operators on tree graphs the subject of this work is random schrodinger operators on regular rooted tree graphs with stochastically homogeneous disorder the operators are of the form h t u v acting in l2 with t the adjacency matrix u a radially periodic potential and v a random potential this includes the only class of homogeneously random operators for which it was proven that the spectrum of h exhibits an absolutely continuous ac component a results established by a klein for weak disorder in case u 0 and v given by iid random variables on our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder the method yields the continuity of the ac spectral density of h at 0 the latter is shown to converge in the l 1 sense over closed borel sets in which h 0 has no singular spectrum the analysis extends to random potentials whose values at different sites need not be independent assuming only that their joint distribution is weakly correlated across different tree branches

singular continuous spectrum for the laplacian on certain sparse trees we present examples of rooted tree graphs for which the laplacian has singular continuous spectral measures for some of these examples we further establish fractional hausdorff dimensions the singular continuous components in these models have an interesting multiplicity structure the results are obtained via a decomposition of the laplacian into a direct sum of jacobi matrices

nontrivial solutions for impulsive fractional differential equations via morse theory abstract in this paper we study the existence of nontrivial solutions for an impulsive fractional differential equation with dirichlet boundary conditions by using morse theory coupled with local linking arguments we obtain some new criteria to guarantee that the impulsive fractional differential equations have at least one nontrivial solution

a canonical factorization for meromorphic herglotz functions on the unit disk and sum rules for jacobi matrices we prove a general canonical factorization for meromorphic herglotz functions on the unit disk whose notable elements are that there is no restriction other than interlacing on the zeros and poles for their blaschke product to converge and there is no singular inner function we use this result to provide a significant simplification in the proof of killip simon ann math 158 2003 253 of their result characterizing the spectral measures of jacobi matrices j with j j 0 hilbert schmidt we prove a nonlocal version of case and step by step sum rules

a multiplicity theorem for problems with the p laplacian abstract we consider nonlinear elliptic equations driven by the p laplacian differential operator using degree theoretic arguments based on the degree map for operators of type s we prove the existence of two nontrivial smooth solutions one of which is of constant sign

absolutely continuous spectrum of a schrodinger operator on a tree we give sufficient conditions for the presence of the absolutely continuous spectrum of a schrodinger operator on a regular rooted tree without loops also called regular bethe lattice or cayley tree

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

baer levi semigroups of linear transformations given an infinite dimensional vector space v we consider the semigroup gs m n consisting of all injective linear v v for which codim ran n where dimv m n 0 this is a linear version of the well known baer levi semigroup bl p q defined on an infinite set x where x p q 0 we show that although the basic properties of gs m n are the same as those of bl p q the two semigroups are never isomorphic we also determine all left ideals of gs m n and some of its maximal subsemigroups in this we follow previous work on bl p q by sutov 1966 and sullivan 1978 as well as levi and wood 1984

products of square zero operators we characterize matrices that can be written as a product of two or three square zero matrices we also consider the same questions for bounded operators on an infinite dimensional separable complex hilbert space and in the calkin algebra

mathematical models for the control of a pest population by infected pest in this paper we propose two mathematical models concerning continuous and impulsive pest control strategies respectively therefore our models are the ordinary differential equations and the impulsive differential equations as a result the global asymptotic stability of the equilibria of the ordinary differential equations is studied in the case when an impulsive control is used it is observed that there exists a globally asymptotically stable susceptible pest eradication periodic solution when the amount of infective pests released periodically is larger than some critical value when the amount of infective pests released is less than this critical value the system is shown to be permanent which implies that the trivial susceptible pest eradication solution loses its stability finally by means of numerical simulation we obtain the critical values of the control variable under different methods of release of infected pests

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions abstract in this paper we study the existence of multiple solutions for the integral boundary value problems of fractional differential equations by the method of upper and lower solutions and leray schauder degree theory the sufficient conditions about the existence of at least three solutions are obtained moreover it is proved that the integral boundary value problem has at least three positive solutions under the conditions of m 0 and f is nonnegative by given two upper and lower solutions which can be easily obtained through our methods we can present the existence theorem of at least three solutions two examples are also included to illustrate the effectiveness of the proposed results

nilpotent local automorphisms of an independence algebra let a be a strong independence algebra of infinite rank m let a be the inverse monoid of all local automorphisms of a the baer levi semigroup b over the algebra a is defined to be the subsemigroup of a consisting of all the elements with dom a and corank im m let k m be the subsemigroup of a generated by b 1 b then k m is inverse and it is generated as a semigroup by n 2 the subset of a consisting of all nilpotent elements of index 2 the 2 nilpotent depth 2 k m of k m is defined to be the smallest positive integer t such that k m n 2 n 2 t in fact 2 k m is either 2 or 3 and a criterion is found which distinguishes between the two cases if n denotes the set of all nilpotents in a then the subsemigroup generated by n is also k m in fact k m is proved to be exactly n 2

multiple periodic solutions for asymptotically linear duffing equations with resonance we investigate multiple periodic solutions of asymptotically linear duffing equation with resonance using index theory and morse theory and obtain a new result

the congruences on the semigroup of balanced transformations of an infinite set abstract in 1966 j m howie j london math soc 41 707 716 showed that the semigroup generated by all non identity idempotent transformations of an infinite set x is the disjoint union of two semigroups one of which is denoted by h and consists of all balanced transformations of x that is all transformations whose defect shift and collapse are equal and infinite subsequently j m howie 1981 proc roy soc edinburgh sect a88 159 167 169 184 and marques 1983 proc roy soc edinburgh sect a93 245 257 showed that certain rees quotient semigroups associated with h are congruence free here we describe all congruences on h

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

positive solutions to superlinear attractive singular impulsive differential equation abstract in this paper we study positive periodic solutions to impulsive differential equation with the attractive singular perturbation the existence theorem is proved using the leray schauder alternative principle and the fixed point theorem the perturbation term in the equation we are mainly interested in is that it has not only an attractive singularity but also the superlinearity

absolutely continuous spectra of quantum tree graphs with weak disorder we consider the laplacian on a rooted metric tree graph with branching number k 2 and random edge lengths given by independent and identically distributed bounded variables our main result is the stability of the absolutely continuous spectrum for weak disorder a useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the weyl titchmarsh function to trees the proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder

solvability of three point boundary value problems for second order differential equations with deviating arguments the monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi solutions

sum rules for jacobi matrices and divergent lieb thirring sums abstract let e j be the eigenvalues outside 2 2 of a jacobi matrix with a n 1 l 2 and b n 0 and the density of the a c part of the spectral measure for the vector 1 we show that if b n l 4 b n 1 b n l 2 then j e j 2 5 2 and if b n l 4 b n 1 b n l 2 then 2 2 ln x 4 x 2 3 2 dx we also show that if a n 1 b n l 3 then the above integral is finite if and only if a n 1 a n b n 1 b n l 2 we prove these and other results by deriving sum rules in which the a c part of the spectral measure and the eigenvalues appear on opposite sides of the equation

monotone iterative method for the second order three point boundary value problem with upper and lower solutions in the reversed order abstract this article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order the sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method meantime the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given some results of previous literature are extended and improved a numerical example is also included to illustrate the effectiveness of the proposed results

existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order abstract this paper studies the existence and uniqueness of solutions of second order three point boundary value problems with lower and upper solutions in the reversed order obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution

nilpotents and congruences on semigroups of transformations with fixed rank in 1988 howie and marques smith studied p m a rees quotient semigroup of transformations associated with a regular cardinal m and described the elements which can be written as a product of nilpotents in p m in 1981 marques proved that if m denotes the malcev congruence on p m then p m m is congruence free for any infinite m in this paper we describe the products of nilpotents in p m when m is nonregular and determine all the congruences on p m when m is an arbitrary infinite cardinal we also investigate when a nilpotent is a product of idempotents

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

subordination and superordination for univalent solutions for fractional differential equations in this article we establish the existence and uniqueness of univalent solution for fractional differential equation moreover we illustrate some properties of this solution containing differential and integral subordination properties

sums of commuting square zero transformations abstract we show that a linear transformation on a vector space is a sum of two commuting square zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of im t ker t in ker t is greater than or equal to the dimension of the space im t 2 we also characterize products of two commuting unipotent transformations with index 2

existence results for fractional differential systems through a local minimization principle in this paper we investigate the existence of one weak solution for a class of fractional differential systems the approach is based on variational methods for smooth functionals defined on reflexive banach spaces the main result is also demonstrated with examples

spectral properties of jacobi matrices and sum rules of special form abstract in this article we relate the properties of elements of a jacobi matrix from certain class to the properties of its spectral measure the main tools we use are the so called sum rules introduced by case in orthogonal polynomials from the viewpoint of scattering theory j math phys 15 1974 2166 2174 orthogonal polynomials ii j math phys 16 1975 1435 1440 later the sum rules were efficiently applied by killip simon sum rules for jacobi matrices and their applications to spectral theory ann math 158 2003 253 321 to the spectral analysis of jacobi matrices we use a modification of the method that permits us to work with sum rules of higher orders as a corollary of the main theorem we obtain a counterpart of a result of molchanov novitskii vainberg first kdv integrals and absolutely continuous spectrum for 1 d schrodinger operator comm math phys 216 2001 195 213 for a continuous schrodinger operator on a half line

existence of multiple solutions for second order boundary value problems we give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y f x y y 0 x epsilon 0 1 y 0 0 y 1 in the special case f x y y f y greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and of lakshmikantham et al

existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions abstract this paper is concerned with the existence and approximation of solutions for a class of first order functional differential equations with periodic boundary conditions we present a new comparison result and extend previous results

sum rules and the szegő condition for orthogonal polynomials on the real line we study the case sum rules especially c 0 for general jacobi matrices we establish situations where the sum rule is valid applications include an extension of shohat s theorem to cases with an infinite point spectrum and a proof that if lim n a n 1 and lim nb n exist and 2 then the szeg condition fails

existence of solutions for some fourth order boundary value problems with parameters abstract in this paper existence and multiplicity results for solutions are obtained for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by using the critical point theory and morse theory we obtain that if satisfy 4 2 1 then the above bvp has solutions where is in some different intervals

variational methods for the fractional sturm liouville problem abstract this article is devoted to the regular fractional sturm liouville eigenvalue problem by applying the methods of fractional variational analysis we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues moreover we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

boundary value problems involving upper and lower solutions in reverse order in this paper we discuss the existence of extreme solutions of the boundary value problem for a class of first order functional equations with a nonlinear boundary condition in the presence of a lower solution a and an upper solution b with email protected email protected we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

a congruence free semigroup associated with an infinite cardinal number let x be a set with infinite cardinality m and let qm be the semigroup of balanced elements in t x as described by howie if i is the ideal qm x m then the rees factor pm qm i is 0 bisimple and idempotent generated its minimum non trivial homomorphic image p m has both these properties and is congruence free moreover p m has depth 4 in the sense that e p m 4 p m and e p m 3 6 p m

existence of solutions for a fractional boundary value problem via the mountain pass method and an iterative technique in this paper we consider the existence of solution to the following fractional boundary value problem ddx p 0d x b u x q xd 1 b u x f x u x 0 email protected 0 1 u 0 u 1 0 where the constants email protected 0 1 0d x b and xd 1 b denote left and right riemann liouville fractional integrals of order b respectively 0r is continuous due to the general assumption on the constants p and q the problem does not have a variational structure despite that here we study it performing variational methods combining with an iterative technique and give an existence criteria of solution for the problem under suitable assumptions the results extend the results in f jiao y zhou existence of solutions for a class of fractional boundary value problems via critical point theory comput math appl 62 2011 1181 1199

on superlinear p x laplacian problems without ambrosetti and rabinowitz condition abstract in this paper we consider the p x laplacian equations on the bounded domain the nonlinearity is superlinear but does not satisfy the usual ambrosetti rabinowitz condition near infinity or its dual version near zero existence and multiplicity results are obtained via morse theory and modified functional methods in a sense we expand a recent result of gasinski and papageorgiou l gasinski n s papageorgiou anisotropic nonlinear neumann problems calc var partial differential equations 42 2011 323 354

numerical and analytical solutions of new generalized fractional diffusion equation our current paper is devoted to studying the numerical and analytical solutions for a class of generalized fractional diffusion equations gfdes with new generalized time fractional derivative gtfd the gtfd we propose here is defined in the caputo sense we consider the gfdes on a bounded domain the numerical solutions are obtained by using the finite difference method fdm of full discretization the stability of fdm is discussed and the order of convergence is evaluated numerically numerical experiments are given which illustrate that the fdm is simple and effective for solving the gfdes with different coefficients and source functions an interesting phenomenon is that we can observe the period like solution in gfdes with a particular positive periodic weight function using the method of separation of variables we convert the homogeneous gfde into two ordinary differential equations and solve them via the help of solutions of the initial value problem with caputo derivative in the analytical solution we observe that the weight function in the denominator and scale function mapping the response domain differently since the derivative considered in this article is new many existing results of fdes are generalized

semigroups generated by nilpotent transformations in a seminal paper published in 1966 john howie characterised the elements of tx the semigroup under composition of all total transformations of a set x into itself which can be written as a product of idempotents in tx we now initiate the study of the subsemigroup of px the semigroup of all partial transformations of x which is generated by the nilpotents of px

monotone iterative technique for impulsive integro differential equations with periodic boundary conditions this paper investigates the existence of minimal and maximal solutions of impulsive integro differential equations with periodic boundary conditions by establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

resonance problems for the p laplacian with a nonlinear boundary condition abstract consider resonance problems at an arbitrary eigenvalue of the elliptic equations p u u p 2 u g 1 u h 1 x in u p 2 u u p 2 u g 2 u h 2 x on some existence theorems are obtained under the landesman lazer type conditions by using variational method

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

on the discrete boundary value problem for anisotropic equation abstract in this paper we consider the discrete anisotropic boundary value problem using critical point theory firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions

doubly resonant semilinear elliptic problems via nonsmooth critical point theory abstract we consider the existence of weak solutions for classical doubly resonant semilinear elliptic problems we show how the main technical assumptions can be used to define appropriate metrics on the underlying function space so that extensions of the results already known in the literature can be obtained using only basic facts from critical point theory for continuous functionals on complete metric spaces

nontrivial solutions of superlinear p laplacian equations we consider p laplacian equations on a bounded domain where the nonlinearity is superlinear but dose not satisfy the usual ambrosetti rabinowitz condition near infinity or its dual version near zero nontrivial solutions are obtained by computing the critical groups and morse theory

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

a general variational principle and some of its applications abstract in this paper given a reflexive real banach space x and two sequentially weakly lower semicontinuous functionals on x with strongly continuous and coercive we are mainly interested in the existence of infinitely many local minima of the functional for each sufficiently large r

absolutely continuous spectrum for the anderson model on a tree a geometric proof of klein s theorem we give a new proof of a version of klein s theorem on the existence of absolutely continuous spectrum for the anderson model on the bethe lattice at weak disorder

a nilpotent generated semigroup associated with a semigroup of full transformations let x be a set with infinite regular cardinality m and let x be the semigroup of all self maps of x the semigroup q m of balanced elements of x plays an important role in the study by howie 3 5 6 of idempotent generated subsemigroups of x as does the subset s m of stable elements which is a subsemigroup of q m if and only if m is a regular cardinal the principal factor p m of q m corresponding to the maximum class j m contains s m and has been shown in 7 to have a number of interesting properties let n 2 be the set of all nilpotent elements of index 2 in p m then the subsemigroup n 2 of p m generated by n 2 consists exactly of the elements in p m s m moreover p m s m has 2 nilpotent depth 3 in the sense that

monotone iterative method for first order functional difference equations with nonlinear boundary value conditions in this paper we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first order functional difference equations with nonlinear boundary value conditions an example is given to illustrate the results obtained

on monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations abstract in this paper we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations we obtain some new results relative to the lower solution and upper solution with

multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods in this paper a coupled system of nonlinear fractional differential equations is considered the existence of at least three distinct weak solutions is obtained by means of the variational methods and the critical point theory due to bonanno and marano in addition an example is presented to illustrate the feasibility and effectiveness of the main results

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

some critical point theorems and their applications to periodic solution for second order hamiltonian systems abstract some critical point theorems without the compactness assumptions are obtained by the reduction method the perturbation argument and the least action principle as applications some existence results of periodic solutions are obtained for nonautonomous second order hamiltonian systems which unify and generalize some recent corresponding results

on some discrete boundary value problem with parameters abstract we consider the existence and multiplicity of solutions to discrete boundary value problems with the use of critical point theory combined with some monotonicity argument we consider both sub and super linear problems

solvability and numerical simulations for bvps of fractional coupled systems involving left and right fractional derivatives abstract we study the three point boundary value problems for the fractional differential coupled systems with both left and right fractional derivatives in the paper firstly the upper lower solutions method for the systems is presented and some existence results to the systems solutions are also obtained secondly we also give out an iterative approach for finding approximate solutions of the systems finally we consider a concrete example where we obtain the existence result of the solution and also give out its numerical solution and the graphical simulation

existence of solutions for sequential fractional integro differential equations and inclusions with nonlocal boundary conditions abstract we investigate the existence of solutions for caputo type sequential fractional integro differential equations and inclusions subject to nonlocal boundary conditions involving riemann liouville and riemann stieltjes integrals for the proofs of our main theorems we use the contraction mapping principle and the krasnosel skii fixed point theorem for the sum of two operators in the case of fractional equations and the nonlinear alternative of leray schauder type for kakutani maps and the covitz nadler fixed point theorem in the case of fractional inclusions some examples are presented to illustrate our results

three nonnegative solutions for fractional differential equations with integral boundary conditions in this paper we study multiple nonnegative solutions for the fractional differential equation with integral boundary conditions cd ax t f t x t 0 email protected 0 1 x 0 email protected 0 1g 0 s x s ds x 1 email protected 0 1g 1 s x s ds x k 0 email protected 0 1g k s x s ds k 2 3 a by means of leggett williams fixed point theorem some new results on the existence of at least three nonnegative solutions are obtained

some results for impulsive problems via morse theory abstract we use morse theory to study impulsive problems first we consider asymptotically piecewise linear problems with superlinear impulses and prove a new existence result for this class of problems using the saddle point theorem next we compute the critical groups at zero when the impulses are asymptotically linear near zero in particular we identify an important resonance set for this problem as an application we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity our results here are based on the simple observation that the underlying sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects

products of idempotent linear transformations in 1966 j m howie characterised the transformations of an arbitrary set that can be written as a product under composition of idempotent transformations of the same set in 1967 j a erdos considered the analogous problem for linear transformations of a finite dimensional vector space and in 1983 r j dawlings investigated the corresponding idea for bounded operators on a separable hilbert space in this paper we study the case of arbitrary vector spaces