on maps preserving operators of local spectral radius zero abstract let l x be the algebra of all bounded linear operators on a complex banach space x we describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x we also describe surjective linear maps on l x that satisfy r t x 0 r t x 0 for every x x and t l x furthermore we characterize maps not necessarily linear nor surjective on l x which satisfy r t s x 0 if and only if r t s x 0 for every x x and t s l x

surjective linear maps preserving local spectra let b x be the algebra of all bounded linear operators on an infinite dimensional complex banach space x we characterize linear surjective and continuous maps on b x preserving different local spectral quantities at a nonzero fixed vector

electromagnetic analysis of large scale periodic arrays using a two level cbfs method accelerated with fmm fft to realize efficient analysis of large scale planar periodic arrays a two level characteristic basis functions method accelerated with fast multipole method fast fourier transform fmm fft is developed in this paper the characteristic basis functions cbfs for each element are constructed in a two level framework in which the cbfs construction based on plane wave derivation for single element at child level and the cbfs construction based on local interaction between array elements at father level are combined the number of unknowns is reduced significantly by the two level cbfs further fmm fft is applied to expedite the computation of interaction between elements numerical results of large scale planar periodic arrays are given to demonstrate the validity and efficiency of the present method

multiplicative mappings at some points on matrix algebras let mn be the algebra of all n n matrices and let mn mn be a linear mapping we say that is a multiplicative mapping at g if st s t for any s t mn with st g fix g mn we say that g is an all multiplicative point if every multiplicative linear bijection at g with in in is a multiplicative mapping in mn where in is the unit matrix in mn we mainly show in this paper the following two results 1 if g mn with detg 0 then g is an all multiplicative point in mn 2 if is an multiplicative mapping at in then there exists an invertible matrix p mn such that either s psp 1 for any s mn or t pttrp 1 for any t mn

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

biseparating maps between operator algebras we prove that a biseparating map between spaces b e and some other banach algebras is automatically continuous and a multiple of an algebra isomorphism

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

accurate analysis of large scale periodic structures using an efficient sub entire domain basis function method an efficient sub entire domain sed basis function method has been proposed to analyze large scale periodic structures with finite sizes accurately the sed basis function is defined on the support of each single cell of the periodic structure after introducing dummy cells with respect to an observation cell the real physics of sed basis function is captured accurately by solving a small size problem further analysis has shown that all kinds of sed basis functions used in the periodic structure can be obtained by solving a single small problem hence the original large scale problem involving n n sub 0 m unknowns is decomposed into two small size problems one of which contains 9m unknowns and the other of which contains only n sub 0 unknowns here n sub 0 is the total cell number in the periodic structure and m is the number of subdomain basis functions in each unit cell several examples are given to test the validity and efficiency of the proposed method numerical results from the new method have excellent agreements with those from the conventional method of moments however the cpu time has been greatly reduced

electromagnetic scattering by finite periodic arrays using the characteristic basis function and adaptive integral methods we introduce a novel technique that combines the aim algorithm with the characteristic basis function method cbfm to solve the problem of electromagnetic scattering by large but finite periodic arrays an important advantage of using the cbfm for this problem is that we only need to analyze a single unit cell to construct the characteristic basis functions cbfs for the entire array the cbfs are generated by illuminating a single unit cell with a plane wave incident from different angles for both the and polarizations the initial set of cbfs generated in the manner described above are then downselected by applying a singular value decomposition svd procedure and retaining only the left singular vectors whose corresponding singular values fall above a threshold next in the conventional cbfm we derive a reduced matrix by applying the galerkin procedure and solve it directly if its size is manageable however when solving an array problem which precludes the direct solve option we can utilize the adaptive integral method aim algorithm detailed below not only to accelerate the solution but to reduce memory requirements as well numerical examples are included in this communication to demonstrate the accuracy and the numerical efficiency of the proposed technique

from quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

maps on matrices compressing the local spectrum in the spectrum abstract let m n c be the algebra of all complex n n matrices let x 0 be a nonzero vector in c n we describe maps on m n c not linear or surjective satisfying t s x 0 t s for all t s m n c we also obtain a similar description by supposing that is surjective and t s x 0 t s for all t s m n c

ito formula for one dimensional continuous time quantum random walk abstract in the paper we first extend the discrete time quantum random walk dqrw to continuous time quantum random walk cqrw then we establish an ito formula for the one dimensional cqrw as an application we present tanaka s formula with the ito formula and characterize the relation between quantum local time l t a and dirac delta function finally we discuss quantum stochastic calculus for the cqrw

operators preserving the strong birkhoff james orthogonality on b h abstract we characterize the class of surjective conjugate linear mappings b h b h that preserve the strong birkhoff james orthogonality in both directions we also give characterizations of rank one operators as well as coisometries in terms of the strong birkhoff james orthogonality

maps on the positive definite cone of a c algebra preserving certain quasi entropies abstract we describe the structure of those bijective maps on the cone of all positive invertible elements of a c algebra with a normalized faithful trace which preserve certain kinds of quasi entropy it is shown that essentially any such map is equal to a jordan isomorphism of the underlying algebra multiplied by a central positive invertible element

additive spectrum compressors abstract additive bijections a b which compress the spectrum between two unital standard operator algebras are characterized applications to local approximate anti multiplications are also given

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

a localization technique for linear preserver problems abstract we introduce a new general technique for solving linear preserver problems the idea is to localize a given linear preserver at each non zero vector in such a way we get vector valued linear maps on the space of matrices which inherit certain properties from if we can prove that such induced maps have a standard form then itself has either a standard form or a very special degenerate form we apply this technique to characterize linear preservers of full rank using this technique we further reprove two classical results describing the general form of linear preservers of rank one and linear preservers of the unitary or orthogonal group

spectrum preservers revisited abstract we classify spectrum preserving additive bijections which map the set of linear possibly unbounded maps on a complex banach space x to the set of linear possibly unbounded maps on a complex banach space y

solution of electrically large problems with multilevel characteristic basis functions we present a multilevel version to the characteristic basis function method cbfm this technique extends the range of applicability of the conventional cbfm and allows a direct solution of problems even when they exceed 250 000 degrees of freedom in this paper it is shown how we can use the multilevel cbfm which is a natural extension of the conventional approach and which saves both time and memory and yet retains excellent accuracy in the computed far fields also the adaptive cross algorithm is used for fast calculation of the interaction between the blocks in the process of generating the reduced matrix

linear maps preserving the minimum and surjectivity moduli of hilbert space operators abstract let b h be the algebra of all bounded linear operators on a complex infinite dimensional hilbert space h for every t b h let m t and q t denote the minimum modulus and surjectivity modulus of t respectively let b h b h be a surjective linear map in this paper we prove that the following assertions are equivalent i m t m t for all t b h ii q t q t for all t b h iii there exist two unitary operators u v b h such that t u t v for all t b h this generalizes the result of mbekhta 7 theorem 3 1 to the non unital case

a spectral characterization of operators having rank k abstract let a b x the algebra of all bounded linear operators on a complex banach space x and let k be a positive integer it is proved that a has rank k if and only if i k 1 m 1 t c m a t for every t b x and some nonzero scalars c 1 c k 1 and ii k m 1 t c m a t for some t b x and some distinct scalars c 1 c k

characterizations of jordan isomorphisms of c algebras by weighted geometric mean related operations and quantities abstract in this paper we consider three operations on positive definite cones of c algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum renyi relative entropy we show how jordan isomorphisms between c algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves we also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity

spectral preservers and approximate spectral preservers on operator algebras abstract let a and b be unital primitive banach algebras with minimal idempotents we prove that every surjective spectral isometry from a onto b is of the form where c with 1 and is either an isomorphism or an anti isomorphism from a onto b as an application we show that for all banach spaces x and y the spectral nearisometries and the approximate spectrum preserving maps from l x onto l y are perturbations of actual spectral isometries and spectrum preserving maps respectively

additive maps on standard operator algebras preserving invertibilities or zero divisors let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be an additive surjection from a onto b we prove that if is unital and preserves any one of left right invertibility left right zero divisors left right topological divisors of zero quasi affinity injectivity surjectivity range density lower boundedness and left right maximal ideals in both directions then it has one of the following forms isomorphism conjugate isomorphism anti isomorphism and conjugate anti isomorphism we also show that preserves zero products in both directions if and only if is either an isomorphism or a conjugate isomorphism

linear maps preserving the minimum and reduced minimum moduli abstract we describe linear maps from a c algebra onto another one preserving different spectral quantities such as the minimum modulus the surjectivity modulus and the reduced minimum modulus

isometries of the unitary groups and thompson isometries of the spaces of invertible positive elements in c algebras abstract we show that the existence of a surjective isometry which is merely a distance preserving map between the unitary groups of unital c algebras implies the existence of a jordan isomorphism between the algebras in the case of von neumann algebras we describe the structure of those isometries showing that any of them is extendible to a real linear jordan isomorphism between the underlying algebras multiplied by a fixed unitary element we present a result of similar spirit for the surjective thompson isometries between the spaces of all invertible positive elements in unital c algebras

quantum decomposition of random walk on cayley graph of finite group in the paper a quantum decomposition qd for short of random walk on cayley graph of finite group is introduced which contains two cases one is qd of quantum random walk operator qrwo for short another is qd of quantum random walk state qrws for short using these findings i finally obtain some applications for quantum random walk qrw for short which are of interest in the study of qrw highlighting the role played by qrwo and qrws

additive idempotence preservers abstract let a be a local matrix algebra over a field of characteristic different from 2 b an arbitrary algebra and x y banach spaces additive mappings a b which preserve idempotents are characterized the result is then used in classifying additive surjections b x b y which preserve idempotents and do not annihilate finite rank operators

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays ii generalization to 3 d cases for pt i see ibid vol 48 no 3 p 594 600 2000 this paper deals with the generalization to three dimensional 3 d arrays of the truncated floquet wave tfw diffraction method for the full wave analysis of large arrays this generalization potentially includes arrays consisting of microstrip excited slots cavity backed apertures and patches the formulation is carried out first by deriving an appropriate fringe integral equation ie and next by defining entire domain basis functions in terms of global array functions shaped as tfw diffracted rays whose analytical expression is derived on the basis of prototype canonical problems the efficiency and the accuracy of this method is demonstrated by comparison with the results of an element by element full wave approach for a rectangular slot array

a note on spectrum preserving maps abstract let a and b be unital semisimple banach algebras if m 2 a b is a bijective spectrum preserving linear map then is a jordan homomorphism

limit theorem for continuous time quantum walk on the line concerning a discrete time quantum walk x t d with a symmetric distribution on the line whose evolution is described by the hadamard transformation it was proved by the author that the following weak limit theorem holds x t d t ensuremath rightarrow dx ensuremath pi 1 ensuremath x 2 sqrt 1 ensuremath 2 x 2 as t ensuremath rightarrow ensuremath infty the present paper shows that a similar type of weak limit theorem is satisfied for a continuous time quantum walk x t c on the line as follows x t c t ensuremath rightarrow dx ensuremath pi sqrt 1 ensuremath x 2 as t ensuremath rightarrow ensuremath infty these results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete and continuous time classical random walks y t sqrt t ensuremath rightarrow e ensuremath x 2 2 dx sqrt 2 ensuremath pi as t ensuremath rightarrow ensuremath infty the work deals also with the issue of the relationship between discrete and continuous time quantum walks this topic subject of a long debate in the previous literature is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases

2 local automorphisms of operator algebras abstract a not necessarily continuous linear or multiplicative function from an algebra a into itself is called a 2 local automorphism if agrees with an automorphism of a at each pair of points in a in this paper we study when a 2 local automorphism of a c algebra or a standard operator algebra on a locally convex space is an automorphism

maps completely preserving spectral functions abstract let x y be infinite dimensional complex banach spaces and a b be standard operator algebras on x and y respectively in this paper we show that surjective maps completely preserving certain spectral function from a to b are isomorphisms where stands for any one of 13 spectral functions l r l r p c p c p c ap s and ap s

characterizations of automorphisms of operator algebras on banach spaces abstract let x be a complex banach space of dimension greater than one and denote by b x the algebra of all the bounded linear operators on x it is shown that if b x b x is a multiplicative map not assumed linear and if is sufficiently close to a linear automorphism of b x in some uniform sense then it is actually an automorphism

invertibility preserving linear maps on j subspace lattice algebras it is shown that every unital surjective linear map between j subspace lattice algebras which preserves invertibility in both directions is a jordan isomorphism this result can apply to atomic boolean subspace lattice algebras and pentagon subspace lattice algebras

spherical isometries of finite dimensional c algebras abstract in this paper it is shown that every surjective isometry between the unit spheres of two finite dimensional c algebras extends to a real linear jordan isomorphism followed by multiplication operator by a fixed unitary element this gives an affirmative answer to tingley s problem between two finite dimensional c algebras moreover we show that if two finite dimensional c algebras have isometric unit spheres then they are isomorphic

unital and multiplicatively spectrum preserving surjections between semi simple commutative banach algebras are linear and multiplicative abstract let t be a surjective map from a unital semi simple commutative banach algebra a onto a unital commutative banach algebra b suppose that t preserves the unit element and the spectrum f g of the product of any two elements f and g in a coincides with the spectrum t f t g then b is semi simple and t is an isomorphism the condition that t is surjective is essential an example of a non linear and non multiplicative unital map from a commutative c algebra into itself such that t f t g f g holds for every f g are given we also show an example of a surjective unital map from a commutative c algebra onto itself which is neither linear nor multiplicative such that t f t g f g holds for every f g

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

on linear maps preserving generalized invertibility and related properties let h be an infinite dimensional complex hilbert space b h be the algebra of all bounded linear operators on h we study surjective linear maps on b h preserving generalized invertibility we also investigate surjective linear maps preserving fredholm respectively semi fredholm operators our results improve those of mbekhta rodman and semrl

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

precorrected fft solution of the volume integral equation for 3 d inhomogeneous dielectric objects this work presents a fast solution to the volume integral equation for electromagnetic scattering from three dimensional inhomogeneous dielectric bodies by using the precorrected fast fourier transform fft method the object is modeled using tetrahedral volume elements and the basis functions proposed by schaubert et al are employed to expand the unknown electric flux density the basis functions are then projected onto a fictitious uniform grid surrounding the nonuniform mesh enabling the fft to be used to speed up the matrix vector multiplies in the iterative solution of the matrix equation the resultant method greatly reduces the memory requirement to o n and the computational complexity to o nlogn where n is the number of unknowns as a result this method is capable of computing electromagnetic scattering from large complex dielectric objects

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

approximately spectrum preserving maps let x and y be superreflexive complex banach spaces and let b x and b y be the banach algebras of all bounded linear operators on x and y respectively if a bijective linear map b x b y almost preserves the spectra then it is almost multiplicative or anti multiplicative furthermore in the case where x y is a separable complex hilbert space such a map is a small perturbation of an automorphism or an anti automorphism

three routes to the exact asymptotics for the one dimensional quantum walk we demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one coin quantum walk on the infinite line via the jacobi polynomials that arise in the path integral representation we calculate the asymptotics using a method that is significantly easier to use than the darboux method it also provides a single integral representation for the wavefunction that works over the full range of positions n including throughout the transitional range where the behaviour changes from oscillatory to exponential previous analyses of this system have run into difficulties in the transitional range because the approximations on which they were based break down here the fact that there are two different kinds of approach to this problem path integral versus schrodinger wave mechanics is ultimately a manifestation of the equivalence between the path integral formulation of quantum mechanics and the original formulation developed in the 1920s we also discuss how and why our approach is related to the two methods that have already been used to analyse these systems

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

numerical analysis of large scale finite periodic arrays using a macro block characteristic basis function method numerical analysis of a large scale finite periodic array is accurate but quite costly when the array is analyzed as a finite array the computational cost for the numerical analysis can be reduced greatly when the array is approximated as an infinite periodic structure however the edge effect that strongly affects the active impedance or current distribution of the array elements near the edge is neglected in this paper a macro block characteristic basis function method mb cbfm for numerical analysis of a large scale finite periodic array with a uniform amplitude and linear phased excitation is proposed the mb cbfm utilizes blocks and macro blocks to group the elements in order to reduce its computational cost without degrading accuracy numerical simulations demonstrate that the cpu time and computer memory of the mb cbfm are o n when the size of array is sufficiently large

preservers of local spectra of matrix sums abstract let m n c be the set of n n complex matrices and for a nonzero vector e c n and t m n c let t e denote the local spectrum of t at e we characterize surjective maps on m n c which satisfy t s e t s e for all matrices t and s we arrive at the same conclusion by supposing that t s e t s e for all matrices t and s without the surjectivity assumption on continuous maps from m n c onto itself that preserve the local spectral radius of the sum of matrices at a nonzero fixed vector are also characterized

an iterative solution for electrically large problems combining the characteristic basis function method and the multilevel fast multipole algorithm an iterative scheme for the rigorous computation of electrically large problems is presented the approach is based on a combination of the characteristic basis function method cbfm and the multilevel fast multipole algorithm mlfma that can deal with very large problems that require an iterative solution process even considering that the application of the cbfm entails an important reduction of the number of unknowns when compared to method of moments approaches based on subdomain functions this reduction is due to the fact that the number of macro basis functions called characteristic basis functions cbfs is lesser than the number of low level subsectional functions used to sample the geometry in addition the use of the mlfma avoids the need to calculate and store the coupling terms in the reduced matrix that are not on or close to the diagonal thereby optimizing the cpu time and the memory storage requirements non uniform rational b splines nurbs surfaces are employed for the representation of the geometry and the cbfs are described in terms of curved rooftops generated in the parametric space the associated macro testing functions are defined as aggregations of curved razor blade functions

monte carlo simulation of electromagnetic scattering from two dimensional random rough surfaces the fast multipole method fast fourier transform fmm fft method is developed to compute the scattering of an electromagnetic wave from a two dimensional 2 d rough surface the resulting algorithm computes a matrix vector multiply in o n log n operations this algorithm is shown to be more efficient than another o n log n algorithm the multilevel fast multipole algorithm mlfma for surfaces of small height for surfaces with larger roughness the mlfma is found to be more efficient using the mlfma monte carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from 2 d random rough surfaces using a workstation for the rougher surface backscattering enhancement is clearly observable as a pronounced peak in the backscattering direction of the computed bistatic scattering coefficient for the smoother surface the monte carlo results compare well with the results of the approximate kirchhoff theory

linear rank and corank preserving maps on b h and an application to semigroup isomorphisms of operator ideals abstract we characterize linear rank k nonincreasing rank k preserving and corank k preserving maps on b h the algebra of all bounded linear operators on the hilbert space h this unifies and extends finite dimensional results and results on linear rank 1 non increasing and rank 1 preserving maps in the infinite dimensional case we conclude with an application to semigroup isomorphisms of operator ideals

linear maps on mn c preserving the local spectrum let x0 cn be a nonzero vector we prove that if a linear map mn c mn c preserves the local spectrum at x0 i e t x0 t x0 for all t mn c then there exists an invertible matrix a such that a x0 x0 and t ata 1 for every t

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

on a spectral characterization of rank one matrices abstract let a be an n n complex matrix with a 2 0 it is proved that a has rank one if and only if t a t ca t for every rank one matrix t and some fixed scalar c 0 or 1 the condition a 2 0 cannot be dropped in case n 4

on positive linear maps preserving invertibility theorem a positive linear map 4 between two c algebras is a jordan homomorphism if d preserves invertibility and the range of is a c algebra a counterexample is given for the case that the range of q is not assumed to be a c algebra this answers a question raised by b russo proc amer math sot 17

quantum renyi relative entropies on density spaces of c algebras their symmetries and their essential difference abstract we extend the definitions of different types of quantum renyi relative entropy from the finite dimensional setting of density matrices to density spaces of c algebras we show that those quantities which trivially coincide in the classical commutative case are essentially different on non commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces besides we determine the symmetry groups of density spaces corresponding to each of those quantum renyi relative entropies and find that they are identical similar results concerning the umegaki and the belavkin staszewksi relative entropies are also presented

invertibility preservers on central simple algebras abstract we solve kaplansky s problem concerning the structure of linear preservers of invertibility in the special case of maps on central simple algebras

characterizing isomorphisms in terms of completely preserving invertibility or spectrum let a and b be standard operator algebras on complex banach spaces x and y respectively in this paper we characterize the surjective maps completely preserving the invertibility in both directions and the surjective maps completely preserving the spectrum from a to b we show that a surjective map from a to b is a ring isomorphism if and only if it is unital and completely preserves the invertibility of operators in both directions is an isomorphism if and only if it completely preserves the spectrum of operators

bures isometries between density spaces of c algebras abstract we describe the precise structure of all surjective a priori nonlinear or better say nonaffine bures isometries between density spaces of c algebras equipped with faithful traces it turns out that those maps are closely related to linear jordan isomorphisms between the underlying algebras beside density spaces we consider the problem also on the positive definite and positive semidefinite cones of c algebras

quantum walks on cayley graphs we address the problem of the construction of quantum walks on cayley graphs our main motivation is the relationship between quantum algorithms and quantum walks in particular we discuss the choice of the dimension of the local hilbert space and consider various classes of graphs on which the structure of quantum walks may differ we completely characterize quantum walks on free groups and present partial results on more general cases some examples are given including a family of quantum walks on the hypercube involving a clifford algebra