jordan triple endomorphisms and isometries of unitary groups abstract in this paper we present the general form of all continuous endomorphisms of the group u n of n n complex unitary matrices with respect to the jordan triple product these are the continuous maps u n u n which satisfy v w v v w v v w u n the result is applied to determine the structure of certain isometries of u n these include the isometries relative to any metric given by a unitarily invariant norm on the space m n of all n n complex matrices and also the isometries relative to any member of a new class of metrics on u n recently introduced by chau li poon and sze 6

multiple extremal eigenpairs by the power method we report the production and benchmarking of several refinements of the power method that enable the computation of multiple extremal eigenpairs of very large matrices in these refinements we used an observation by booth that has made possible the calculation of up to the 10th eigenpair for simple test problems simulating the transport of neutrons in the steady state of a nuclear reactor here we summarize our techniques and efforts to date on determining mainly just the two largest or two smallest eigenpairs to illustrate the effectiveness of the techniques we determined the two extremal eigenpairs of a cyclic matrix the transfer matrix of the two dimensional ising model and the hamiltonian matrix of the one dimensional hubbard model

automorphisms of b h with respect to minus partial order abstract let h be an infinite dimensional hilbert space and b h the algebra of all bounded linear operators on h we discuss possible extensions of the concept of the minus partial order from matrix to operator algebras in particular we show that mitra s unified theory of matrix partial orders based on generalized inverses can be modified in such a way that we get a unified approach to partial orders on b h or even more general algebras and rings then we choose the most natural among possible definitions of minus partial order on b h and describe the structure of corresponding automorphisms

centrally stable algebras abstract we define an algebra a to be centrally stable if for every epimorphism from a to another algebra b the center z b of b is equal to z a the image of the center of a after providing some examples and basic observations we consider in somewhat greater detail central stability in tensor products of algebras and finally establish our main result which states that a finite dimensional unital algebra a over a perfect field f is centrally stable if and only if a is isomorphic to a direct product of algebras of the form c i f i a i where f i is a field extension of f c i is a commutative f i algebra and a i is a central simple f i algebra

minimal locally linearly dependent spaces of operators abstract let s be an n dimensional minimal locally linearly dependent space of operators acting between vector spaces u and v we obtain the sharp lower bound and the sharp upper bound for dim s u and give a complete description of those minimal locally linearly dependent spaces at which the upper bound is attained

the solution of tingley s problem for the operator norm unit sphere of complex n n matrices abstract in this paper we study geometric structures and isometries of the unit sphere of n n complex matrices we introduce the notion of proper exposed points of the unit ball of a banach space which provides a refinement of that of exposed point provided that the space is separable as a new geometric interpretation of unitary matrices it is shown that the set of all proper exposed points of the unit ball of m n c coincides with its unitary group u n where m n c denotes the algebra of all n n complex matrices moreover we show that every maximal convex subset of the unit sphere of m n c can be identified with the unit ball of m n 1 c by an affine bijection preserving the unitary matrices this map is isometric with respect to any unitarily invariant norm as an application we completely determine the forms of surjective isometries on the unit sphere of m n c with respect to the operator norm which provides an affirmative answer to tingley s problem on m n c

hyper tauberian algebras and weak amenability of figa talamanca herz algebras abstract we study certain commutative regular semisimple banach algebras which we call hyper tauberian algebras we first show that they form a subclass of weakly amenable tauberian algebras then we investigate the basic and hereditary properties of them moreover we show that if a is a hyper tauberian algebra then the linear space of bounded derivations from a into any banach a bimodule is reflexive we apply these results to the figa talamanca herz algebra a p g of a locally compact group g for p 1 we show that a p g is hyper tauberian if the principal component of g is abelian finally by considering the quantization of these results we show that for any locally compact group g a p g equipped with an appropriate operator space structure is a quantized hyper tauberian algebra this in particular implies that a p g is operator weakly amenable

orthogonally additive polynomials on banach function algebras abstract for a banach function algebra a we consider the problem of representing a continuous d homogeneous polynomial p a x where x is an arbitrary banach space that satisfies the property p f g p f p g whenever f g a are such that supp f supp g we show that such a polynomial can be represented as p f t f d f a for some continuous linear map t a x for a variety of banach function algebras such as the algebra of continuous functions c 0 for any locally compact hausdorff space the algebra of lipschitz functions lip k for any compact metric space k and 0 1 the figa talamanca herz algebra a p g for some locally compact groups g and p 1 the algebras a c a b and b v c a b of absolutely continuous functions and of continuous functions of bounded variation on the interval a b in the case where a c n a b p can be represented as p f t n 1 n d f n 1 f n d where the sum is taken over n 1 n d z d with 0 n 1 n d n for appropriate continuous linear maps t n 1 n d c n n d a b x

iterated networks and the spectra of renormalizable electromechanical systems beginning with anlrc network with impedance functionz a sequence of iterated networksnk with impedance functionsz k k 1 2 3 is introduced the asymptotic comportment ofz k and the spectra ofnk are analyzed in terms of the julia set ofz an example is given of an iterated network associated with a cascade of period doubling bifurcations

isometries of the special orthogonal group abstract in this paper we describe all isometries on the special orthogonal group as an application we give a form of spectrally multiplicative map on the special orthogonal group

time series analysis and monte carlo methods for eigenvalue separation in neutron multiplication problems a time series approach has been applied to the nuclear fission source distribution generated by monte carlo mc particle transport in order to calculate the non fundamental mode eigenvalues of the system the novel aspect is the combination of the general technical principle of projection pursuit for multivariate data with the neutron multiplication eigenvalue problem in the nuclear engineering discipline proof is thoroughly provided that the stationary mc process is linear to first order approximation and that it transforms into one dimensional autoregressive processes of order one ar 1 via the automated choice of projection vectors the autocorrelation coefficient of the resulting ar 1 process corresponds to the ratio of the desired mode eigenvalue to the fundamental mode eigenvalue all modern mc codes for nuclear criticality calculate the fundamental mode eigenvalue so the desired mode eigenvalue can be easily determined this time series approach was tested for a variety of problems including multi dimensional ones numerical results show that the time series approach has strong potential for three dimensional whole reactor core the eigenvalue ratio can be updated in an on the fly manner without storing the nuclear fission source distributions at all previous iteration cycles for the mean subtraction lastly the effects of degenerate eigenvalues are investigated and solutions are provided

condensed julia sets with an application to a fractal lattice model hamiltonian on considere l ensemble de julia pour l application rationnelle complexe z z 2 e z ou et e sont des parametres complexes dans la limite e 0

derivations and right ideals of algebras abstract let r be a k algebra acting densely on v d where k is a commutative ring with unity and v is a right vector space over a division k algebra d let be a nonzero right ideal of r and let f x 1 x t be a nonzero polynomial over k with constant term 0 such that r 0 for some coefficient of f x 1 x t suppose that d r r is a nonzero derivation it is proved that if rank d f x 1 x t m for all x 1 x t and for some positive integer m then either is generated by an idempotent of finite rank or d ad b for some b end v d of finite rank in addition if f x 1 x t is multilinear then b can be chosen such that rank b 2 6 t 13 m 2

determining elements in banach algebras through spectral properties abstract let a be a unital c algebra and a its second dual by a and r a we denote the spectrum and the spectral radius of a a respectively the following two statements hold for arbitrary a b a 1 a c b c 0 for every c a if and only if there exists a central projection z a such that a z b 2 r a c r b c for every c a if and only if there exists a central element z in a such that a z b and z 1

the complex potential generated by the maximal measure for a family of rational maps the exact complex potential generated by the maximal measure for a family of rational maps is given the results are of analytical nature because the complex potential does not change nicely if the coordinates of a rational map are changed there exist applications of this result to the theory of moments

on n ϵ pseudospectra of operators on banach spaces abstract in this paper we extend the concept of the n pseudospectra of hansen to the case of bounded linear operators on banach spaces and prove several relations to the usual spectrum we particularly discuss the approximation by rectangular finite sections and the impact of the fundamental result of shargorodsky on jumping pseudospectra

monte carlo determination of multiple extremal eigenpairs we present a monte carlo algorithm that allows the simultaneous determination of a few extremal eigenpairs of a very large matrix without the need to compute the inner product of two vectors or store all the components of any one vector the algorithm a monte carlo implementation of a deterministic one we recently bench marked is an extension of the power method in the implementation presented we used a basic monte carlo splitting and termination method called the comb incorporated the weight cancellation method of arnow et al and exploited a sampling method the sewing method that does a large state space sampling as a succession of small state space samplings we illustrate the effectiveness of the algorithm by its determination of the two largest eigenvalues of the transfer matrices for variously sized two dimensional zero field ising models while very likely useful for other transfer matrix problems the algorithm is however quite general and should find application to a larger variety of problems requiring a few dominant eigenvalues of a matrix

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

on the composition of q skew derivations in banach algebras abstract let a be a banach algebra let be an automorphism of a and let d be q skew derivations of a we show that if d a is quasi nilpotent for any a a then d a 3 lies in the radical of a for all a a this result simultaneously generalizes the previous results obtained by bresar and semrl 5 chebotar et al 7 and abdelali 1

trace characterizations and socle identifications in banach algebras abstract as a follow up to a paper of d petz and j zemanek 1 a number of equivalent conditions which characterize the trace among linear functionals on matrix algebras finite rank operators and the socle elements of semisimple banach algebras in general are given moreover the converse problem is also addressed that is given the equivalence of certain conditions which characterize the trace what can be said about the structure of the socle in particular we characterize those socles isomorphic to matrix algebras in this manner as well as those socles which are minimal two sided ideals

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

properties and preservers of the pseudospectrum abstract the interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated it is shown that one can characterize hermitian matrices positive semi definite matrices orthogonal projections unitary matrices etc in terms of the pseudospectrum also characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum difference or product of matrix pairs it is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix taking the matrix transpose or multiplying by a scalar in 1 1

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

infinite dimensional numerical linear algebra theory and applications we present a new method for computing spectra and pseudospectra of bounded operators on separable hilbert spaces the core in this theory is a generalization of the pseudospectrum called the n pseudospectrum

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

extension of modified power method to two dimensional problems in this study the generalized modified power method was extended to two dimensional problems a direct application of the method to two dimensional problems was shown to be unstable when the number of requested eigenmodes is larger than a certain problem dependent number the root cause of this instability has been identified as the degeneracy of the transfer matrix in order to resolve this instability the number of sub regions for the transfer matrix was increased to be larger than the number of requested eigenmodes and a new transfer matrix was introduced accordingly which can be calculated by the least square method the stability of the new method has been successfully demonstrated with a neutron diffusion eigenvalue problem and the 2d c5g7 benchmark problem

maps preserving operator pairs whose products are projections abstract let b h be the algebra of all bounded linear operators on a complex hilbert space h with dim h 2 it is proved that a surjective map on b h preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti unitary operator u on h such that a u au for all a in b h for some constants with 2 1 related results for surjective maps preserving operator pairs whose triple jordan products are nonzero projections in both directions are also obtained these show that the operator pairs whose products or triple jordan products are nonzero projections are isometric invariants of b h

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

locally linearly dependent operators and reflexivity of operator spaces we obtain two results on the existence of large subspaces of operators of small rank in locally linearly dependent spaces of operators as a consequence we obtain an upper bound for the rank of operators belonging to a minimal locally linearly dependent space of operators it has been known that the only obstruction to the reflexivity of a finite dimensional operator space comes from the operators with small ranks our results improve known bounds on the minimal rank that guarantees the reflexivity

a general solution strategy of modified power method for higher mode solutions a general solution strategy of the modified power iteration method for calculating higher eigenmodes has been developed and applied in continuous energy monte carlo simulation the new approach adopts four features 1 the eigen decomposition of transfer matrix 2 weight cancellation for higher modes 3 population control with higher mode weights and 4 stabilization technique of statistical fluctuations using multi cycle accumulations the numerical tests of neutron transport eigenvalue problems successfully demonstrate that the new strategy can significantly accelerate the fission source convergence with stable convergence behavior while obtaining multiple higher eigenmodes at the same time the advantages of the new strategy can be summarized as 1 the replacement of the cumbersome solution step of high order polynomial equations required by booth s original method with the simple matrix eigen decomposition 2 faster fission source convergence in inactive cycles 3 more stable behaviors in both inactive and active cycles and 4 smaller variances in active cycles advantages 3 and 4 can be attributed to the lower sensitivity of the new strategy to statistical fluctuations due to the multi cycle accumulations the application of the modified power method to continuous energy monte carlo simulation and the higher eigenmodes up to 4th order are reported for the first time in this paper modified power method is applied to continuous energy monte carlo simulation transfer matrix is introduced to generalize the modified power method all mode based population control is applied to get the higher eigenmodes statistic fluctuation can be greatly reduced using accumulated tally results fission source convergence is accelerated with higher mode solutions

pseudospectra of special operators and pseudospectrum preservers abstract denote by b h the banach algebra of all bounded linear operators on a complex hilbert space h let a b h and denote by a the spectrum of a for e 0 define the e pseudospectrum e a of a as e a z a e e b h e e in this paper the pseudospectra of several special classes of operators are characterized as an application complete descriptions are given of the maps of b h leaving invariant the pseudospectra of a b for different kinds of binary operations on operators such as the difference a b the operator product ab and the jordan product a b b a

compressions of linearly independent selfadjoint operators abstract the following question is considered what is the smallest number k with the property that for every family x 1 x k of k selfadjoint and linearly independent operators on a real or complex hilbert space h there exists a subspace h 0 h of dimension k such that the compressions of x 1 x k to h 0 are still linearly independent upper and lower bounds for k are established for any k and the exact value is found for k 2 3 it is also shown that the set of all k dimensional subspaces h 0 with the desired property is open and dense in the respective grassmannian the k 3 case is used to prove that the ratio numerical range w a b of a pair of operators on a hilbert space either has a non empty interior or lies in a line or a circle

extension of the noise propagation matrix method for higher mode solutions abstract the noise propagation matrix method npmm has been extended to get higher mode solutions previous studies show that the npmm can be used to compute the dominance ratio of a system it is essentially the same as the coarse mesh projection method cmpm both of which use the noise propagation matrix npm to determine the dominance ratio either after finishing the monte carlo simulation or on the fly during the simulation since only the fundamental fission source information is explicitly utilized while the higher mode information is implicitly contained in the statistical noises the npmm can usually only give an approximate estimation of the dominance ratio after thousands of cycles in this study the npmm is extended by simulating the higher modes explicitly so that the dominance ratio estimation can be more accurate and efficient besides the higher mode solutions can be obtained at the same time with good accuracy and efficiency

duality between some linear preserver problems iii c spectral norms and skew symmetric matrices with fixed singular values abstract let f denote either the complex field c or the real field r let v be sn f or kn f the vector spaces of all n n symmetric and skew symmetric matrices respectively over f for c c1 cn 0 with c1 cn 0 the c spectral norm of a matrix a v is the quantity a c i l n c i i a where 1 a n a are the singular values of a let d d1 dn 0 with d1 dn 0 we study the linear isometries between the normed spaces v c and v d by using the fact that they are dual transformations of the linear operators which map d onto c where c x v x has singular values c 1 c n it is shown that such isometries and hence their dual transformations exist if and only if c and d are scalar multiples of each other in such case we completely determine the structure of such isometries and prove that they and their dual transformations belong to a same class of operators in the proof we obtain characterizations of the extreme points of the unit ball in v for different cases with respect to c which is of independent interest

the n ϵ pseudospectrum of an element of a banach algebra abstract let a be a complex unital banach algebra a a n z and 0 the n pseudospectrum n a of a is defined as n a a a a 2 n 1 2 n 1 here a denotes the spectrum of a the usual pseudospectrum a of a is a special case of this namely 0 a it is proved that n pseudospectrum approximates the closed neighbourhood of spectrum for large n further it has been shown that n pseudospectrum has no isolated points has a finite number of connected components and each component contains an element from a some examples are given to illustrate these results

a note on generalized derivations in banach algebras abstract in this note we characterize a complex banach algebra a admitting a generalized derivation g such that the cardinality of the spectrum g x is exactly one for all x a

on the solvability complexity index the n pseudospectrum and approximations of spectra of operators we show that it is possible to compute spectra and pseudospectra of linear operators on separable hilbert spaces given their matrix elements the core in the theory is pseudospectral analysis and in particular the n pseudospectrum and the residual pseudospectrum we also introduce a new classification tool for spectral problems namely the solvability complexity index this index is an indicator of the difficultness of different computational spectral problems

moments of balanced measures on julia sets by a theorem of s demko there exists a balanced measure on the julia set of an arbitrary nonlinear rational transformation on the riemann sphere it is proved here that if the transformation admits an attractive or indifferent cycle then there is a point with respect to which all the moments of a balanced measure exist moreover these moments can be calculated exactly an explicit balanced measure is exhibited in an example where the julia set is the whole sphere and for which the moments with respect to any point do not all exist 1 introduction the julia set for a nonlinear rational transformation on the riemann sphere is the closure of the set of all repulsive cycles of the transformation br fa j the location or distribution of the set is of interest in the theory of iterated maps and it may be characterized by an associated invariant measure in this paper we extend previous results by showing that when there is an attractive or indifferent cycle certain moments of such a measure exist and can be calculated exactly from the rational transformation this means for example that when the julia set is contained strictly in the real line all of the orthogonal polynomials on the set can be calculated as can the entries of the corresponding infinite dimensional jacobi matrices the latter are of the type which have recently been shown to furnish model schrodinger equations for crystals with almost periodic potentials bbm this leads us to the following speculation which provides one reason for our interest in the present topic we note first the frequent occurrence in the physical universe of objects steady state systems which are fractals from some point of

orthogonally additive polynomials on convolution algebras associated with a compact group abstract let g be a compact group let x be a banach space and let p l 1 g x be an orthogonally additive continuous n homogeneous polynomial then we show that there exists a unique continuous linear map l 1 g x such that p f f n f for each f l 1 g we also seek analogues of this result about l 1 g for various other convolution algebras including l p g for 1 p and c g

on the invariant sets of a family of quadratic maps the julia setb for the mappingz z 2 is considered where is a complex parameter for 2 a new upper bound for the hausdorff dimension is given and the monic polynomials orthogonal with respect to the equilibrium measure onb are introduced a method for calculating all of the polynomials is provided and certain identities which obtain among coefficients of the three term recurrence relations are given a unifying theme is the relationship betweenb and chains which is explored for 1 4 2 and for with 1 4 with the aid of the bottcher equation thenb is shown to be a holder continuous curve for 1 4

uniqueness under spectral variation in the socle of a banach algebra abstract let a be a complex semisimple banach algebra with identity and denote by x and x the nonzero spectrum and spectral radius of an element x a respectively we explore the relationship between elements a b a that satisfy one of the following conditions 1 a x b x for all x a 2 a x b x for all x a the latter problem was identified by bresar and spenko in 7 in particular we use these conditions to spectrally characterize prime banach algebras amongst the class of banach algebras with nonzero socles as well as to obtain spectral characterizations of socles which are minimal two sided ideals

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

transformations of the unitary group on a hilbert space abstract let h be an infinite dimensional separable complex hilbert space and u be the group of all unitary operators on h motivated by the algebraic properties of surjective isometries of u that have recently been revealed and also by some classical results related to automorphisms of the unitary groups of operator algebras we determine the structures of bijective transformations of u that respect certain algebraic operations these are among others the usual product of operators the jordan triple product the inverted jordan triple product and the multiplicative commutator our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group

commutators commutativity and dimension in the socle of a banach algebra a generalized wedderburn artin and shoda s theorem abstract as a follow up to work done in 7 some new insights to the structure of the socle of a semisimple banach algebra are obtained in particular it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals remarkably the finite dimensional case here reduces to the classical wedderburn artin theorem and this approach does not use any continuous irreducible representations of the algebra in question furthermore the structure of the socles for which the classical shoda s theorem for matrices can be extended is characterized exactly as those socles which are minimal two sided ideals it is then shown that the set of commutators in the socle i e x y y x x y soc a is a vector subspace finally we characterize those socles which belong to the center of a banach algebra and obtain results which suggest that the dimension of certain subalgebras of the socle in fact provides a measure to some extent of commutativity

tingley s problem on finite von neumann algebras abstract in this paper we give an affirmative answer to tingley s problem between finite von neumann algebras to do this we see that every norm closed proper face of the unit ball of a unital c algebra can be represented as the intersection of a family of maximal proper faces

orthogonally additive polynomials on spaces of continuous functions we show that for every orthogonally additive homogeneous polynomial p on a space of continuous functions c k with values in a banach space y there exists a linear operator s c k y such that p f s f n this is the c k version of a related result of sundaresam for polynomials on l p spaces

truncation and spectral variation in banach algebras abstract let a and b be elements of a semisimple complex and unital banach algebra a using subharmonic methods we show that if the spectral containment a x b x holds for all x a then ax belongs to the bicommutant of bx for all x a given the aforementioned spectral containment the strong commutation property then allows one to derive for a variety of scenarios a precise connection between a and b the current paper gives another perspective on the implications of the above spectral containment which was also studied not long ago by j alaminos m bresar et al

maps preserving the idempotency of products of operators abstract let b x be the algebra of all bounded linear operators on the banach space x and let n x be the set of nilpotent operators in b x suppose b x b x is a surjective map such that a b b x satisfy ab n x if and only if a b n x if x is infinite dimensional then there exists a map f b x c 0 such that one of the following holds a there is a bijective bounded linear or conjugate linear operator s x x such that has the form a s f a a s 1 b the space x is reflexive and there exists a bijective bounded linear or conjugate linear operator s x x such that has the form a s f a a s 1 if x has dimension n with 3 n b x is identified with the algebra m n of n n complex matrices then there exist a map f m n c 0 a field automorphism c c and an invertible s m n such that has one of the following forms a a ij f a s a ij s 1 or a a ij f a s a ij t s 1 where a t denotes the transpose of a the results are extended to the product of more than two operators and to other types of products on b x including the jordan triple product a b aba furthermore the results in the finite dimensional case are used to characterize surjective maps on matrices preserving the spectral radius of products of matrices

zero product preserving maps on banach algebras of lipschitz functions abstract let k d be a non empty compact metric space and 0 1 let a be either lip k or lip k and let b be a commutative unital banach algebra we show that every continuous linear map t a b with the property that t f t g 0 whenever f g a are such that f g 0 is of the form t w for some invertible element w in b and some continuous epimorphism a b

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

more elementary operators that are spectrally bounded we study spectrally bounded elementary operators of length two on a complex unital banach algebra a some related conditions are investigated as well in particular we show that if s is an elementary operator of length two such that s x is quasi nilpotent for every x a then s x 3 rada for every x a

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

orthogonally additive polynomials on fourier algebras abstract we show that an n homogeneous polynomial p on the fourier algebra a g of a locally compact group g can be represented in the form p f t f n f a g for some t in the group von neumann algebra vn g of g if and only if it is orthogonally additive and completely bounded

on the approximation of spectra of linear operators on hilbert spaces abstract we present several new techniques for approximating spectra of linear operators not necessarily bounded on an infinite dimensional separable hilbert space our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized to an infinite dimensional setting to provide approximations of spectra of elements in a large class of operators we conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n pseudospectrum and argue how that can be used as an approximation to the spectrum