characterization of real inner product spaces by means of symmetrical bilinear forms abstract we investigate those normed spaces e for which the norm of any continuous symmetric bilinear form on e has the same value as the norm of the associated homogeneous polynomial in the real case we show that this property characterizes an inner product space i p s we also give some other characterizations of an i p s through symmetric bilinear forms

weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

supercyclic subspaces spectral theory and weighted shifts abstract a vector x in a banach space b is called hypercyclic for a bounded operator t if the orbit t n x n 0 is dense in b if the scalar multiples of the elements in the orbit are dense then the vector x is called supercyclic we give a general sufficient condition for a bounded operator on a banach space to have an infinite dimensional closed subspace of supercyclic vectors as a consequence we also obtain a spectral sufficient condition for the existence of such a subspace for an operator these results allow us to characterize unilateral and bilateral weighted shifts that have an infinite dimensional closed subspace of supercyclic vectors surprisingly there are weighted shift operators that have supercyclic vectors but in which all closed subspaces of supercyclic vectors are finite dimensional our results complement recent work on hypercyclic subspaces and supercyclic subspaces they also suggest that the problem of the existence of infinite dimensional closed subspaces of supercyclic vectors is not only different but also more difficult to handle than the corresponding problem for hypercyclic vectors

n weakly hypercyclic and n weakly supercyclic operators if x is a locally convex topological vector space over a scalar field f r or c and if e is a subset of x then we define e to be n weakly dense in x if for every onto continuous linear operator f x fn we have that f e is dense in fn if x is a hilbert space this is equivalent to requiring that e have a dense orthogonal projection onto every subspace of dimension n we then consider continuous linear operators on x that have orbits or scaled orbits that are n weakly dense in x we show that on a separable hilbert space there are non trivial examples of such operators and establish many of their basic properties a fundamental tool is ball s solution of the complex plank problem which implies that certain sets are 1 weakly closed

on some new discrete generalizations of gronwall s inequality abstract this paper derives new discrete generalizations of the gronwall bellman integral inequality these generalizations should have wide application in the study of finite difference equations and numerical analysis the main result theorem 3 concerns a very general form of linear bellman type discrete inequalities in one independent variable it is a discrete analogue of an integral inequality obtained by the present author in j math anal appl 103 1984 184 197 theorem 4 and it extends many discrete inequalities of agarwal and thandapani pachpatte and sugiyama two nonlinear extensions of theorem 3 are also established here

gaussian variables polynomials and permanents abstract we prove that given n unit vectors aj nj 1 in a complex hilbert space there exists a unit vector x such that x a 1 x a n n n 2 this answers a question about the best constant in an inequality for polynomials in several variables the proof depends on a relation between gaussian variables and permanents

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

lower bounds for norms of products of polynomials via bombieri inequality in this paper we give a different interpretation of bombieri s norm this new point of view allows us to work on a problem posed by beauzamy about the behavior of the sequence sn p supqn pqn 2 where p is a fixed m homogeneous polynomial and qn runs over the unit ball of the hilbert space of n homogeneous polynomials we also study the factor problem for homogeneous polynomials defined on c and we obtain sharp inequalities whenever the number of factors is no greater than n in particular we prove that for the product of homogeneous polynomials on infinite dimensional complex hilbert spaces our inequality is sharp finally we use these ideas to prove that any set zk k 1 of unit vectors in a complex hilbert space for which sup z 1 z z1 z zn is minimum must be an orthonormal system

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

multiobjective control problem with generalized invexity abstract a multiobjective control problem is introduced the definition of invexity for continuous functions is extended to invexity pseudoinvexity and quasi invexity duality results are established for wolfe as well as mond wier type duals

geometry of 2 homogeneous polynomials on lp spaces 1 p abstract we characterise extreme 2 homogeneous polynomials of diagonal form on l p with p 2 and give a class of examples of smooth points we describe the unit ball of p 2 l p 2 for both the cases p 2 and p we apply the results thus found to study the geometry of the predual of p 2 l p 2 and to show that for p 2 an extreme 2 homogeneous polynomial on l p 2 remains extreme when regarded as a polynomial on l p

weak closure and polarization constant by gaussian measure let x be a banach space and x n subset x we study under which condition on xn the set xn n 0 is weakly closed our results depend on the geometry of x

geometry of three homogeneous polynomials on real hilbert spaces let h be a two dimensional real hilbert space we give a characterisation of the extreme and the smooth points of the unit ball of the space of three homogeneous polynomials on h in terms of the coefficients of the polynomial we also determine the smooth points of the unit ball of the predual of the space of three homogeneous polynomials

semi fredholm theory hypercyclic and supercyclic subspaces a vector x in a banach space cal b is called hypercyclic for a bounded linear operator t cal b rightarrow cal b if the orbit t n x n geq 1 is dense in cal b we prove that if t is hereditarily hypercyclic and its essential spectrum meets the closed unit disk then there is an infinite dimensional closed subspace consisting except for zero entirely of hypercyclic vectors for t the converse is also true even if t is not hereditarily hypercyclic in this way we improve and extend to banach spaces a recent result for hilbert spaces we also investigate the corresponding problem for supercyclic operators in this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite dimensional closed subspace of hypercyclic vectors moreover for certain kinds of supercyclic operators we can characterize when they have an infinite dimensional closed subspace of supercyclic vectors 1991 mathematics subject classification 47b38 47b99

on some new discrete inequalities of the bellman bihari type pour etudier la stabilite et le comportement asymptotique de certaines equations aux differences on etablit de nouvelles inegalites discretes du type bellman bihari

hypercyclic and cyclic vectors abstract let x denote an arbitrary separable banach space over the field of complex numbers and b x the banach algebra of all bounded linear operators on x we prove the following results 1 an element of the space x is hypercyclic supercyclic for all positive powers tn of an operator t in b x if it is hypercyclic supercyclic for t 2 under some condition on the spectrum of the adjoint of a cyclic operator the set of all cyclic vectors of the operator is dense this result extends to any cyclic commutative subset of b x 3 under a mild condition on the spectrum of a cyclic operator t the set of all separating vectors for the commutant t of t is dense this also extends to any cyclic commutative subset of b x 4 a slightly stronger version of a theorem of k f clancey and d d rogers on cyclic vectors finally we define and discuss hereditarily hypercyclic operators

weakly supercyclic operators inspired by a recent result that a weakly hypercyclic operator may fail to be norm hypercyclic we show there exists a weakly supercyclic operator that fails to be norm supercyclic moreover despite a classical result of hilden and wallen that every unilateral weighted backward shift is supercyclic we show such an operator may have a weakly supercyclic vector that is not a norm supercyclic vector in addition to these results we extend a result of kitai by showing a hyponormal operator cannot be weakly hypercyclic

squared chaotic random variables new moment inequalities with applications abstract we prove a new family of inequalities involving squares of random variables belonging to the wiener chaos associated with a given gaussian field our result provides a substantial generalization as well as a new analytical proof of an estimate by frenkel 2007 10 and also constitutes a natural real counterpart to an inequality established by arias de reyna 1998 2 in the framework of complex gaussian vectors we further show that our estimates can be used to deduce new lower bounds on homogeneous polynomials thus partially improving results by pinasco 2012 19 as well as to obtain a novel probabilistic representation of the remainder in hadamard inequality of matrix analysis

discrete nonlinear integral inequalities in two variables and their applications in this paper some discrete integral inequalities in two variables with more than one nonlinear term are established our results are also applied to show the boundedness of certain boundary value problem for difference equations

common hypercyclic vectors for the conjugate class of a hypercyclic operator abstract given a separable in nite dimensional hilbert space it was recently shown by the authors that there is a path of chaotic operators which is dense in the operator algebra with the strong operator topology and along which every operator has the exact same dense set of hypercyclic vectors in the present work we show that the conjugate set of any hypercyclic operator on a separable in nite dimensional banach space always contains a path of operators which is dense with the strong operator topology and yet the set of common hypercyclic vectors for the entire path is a dense set as a corollary the hypercyclic operators on such a banach space form a connected subset of the operator algebra with the strong operator topology 1 introduction throughout the present paper let be a separable in nite dimensional banach space over the scalar eld or and let denote the algebra of bounded linear operators an operator in is hypercyclic if there is a vector in for which its orbit 0 is dense in such a vector is called a hypercyclic vector for an operator in is hypercyclic if and only if the set of hypercyclic vectors for denoted by is a dense set see kitai 23 for a countable family of hypercyclic operators a direct application of the baire category theorem implies that the set

non sequential weak supercyclicity and hypercyclicity abstract a bounded linear operator t acting on a banach space b is called weakly hypercyclic if there exists x b such that the orbit t n x n 0 1 is weakly dense in b and t is called weakly supercyclic if there is x b for which the projective orbit t n x c n 0 1 is weakly dense in b if weak density is replaced by weak sequential density then t is said to be weakly sequentially hypercyclic or supercyclic respectively it is shown that on a separable hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic this is achieved by constructing a borel probability measure on the unit circle for which the fourier coefficients vanish at infinity and the multiplication operator m f z z f z acting on l 2 is weakly supercyclic it is not weakly sequentially supercyclic since the projective orbit under m of each element in l 2 is weakly sequentially closed this answers a question posed by bayart and matheron it is proved that the bilateral shift on l p z 1 p is weakly supercyclic if and only if 2 p and that any weakly supercyclic weighted bilateral shift on l p z for 1 p 2 is norm supercyclic it is also shown that any weakly hypercyclic weighted bilateral shift on l p z for 1 p 2 is norm hypercyclic which answers a question of chan and sanders

mixed type duality in multiobjective programming problems two mixed type duals are introduced for multiobjective programming and for multiobjective fractional programming respectively results about weak duality strong duality and strictly reverse duality are presented under more suitable conditions

on some new integral inequalities in n independent variables abstract integral inequalities of the gollwitzer type in n independent variables are established which generalize some known results obtained by gollwitzer bondge pachpatte shih and yeh

the plank problem for symmetric bodies given a symmetric convex bodyc andn hyperplanes in an euclidean space there is a translate of a multiple ofc at least frac 1 n 1 times as large insidec whose interior does not meet any of the hyperplanes the result generalizes bang s solution of the plank problem of tarski and has applications to diophantine approximation

geometry of integral polynomials m ideals and unique norm preserving extensions we use the aron berner extension to prove that the set of extreme points of the unit ball of the space of integral k homogeneous polynomials over a real banach space x is k x 1 with this description we show that for real banach spaces x and y if x is a nontrivial m ideal in y then ek sk sx the k th symmetric tensor product of x endowed with the injective symmetric tensor norm is never an m ideal in ek sk sy this result marks up a difference with the behavior of nonsymmetric tensors since when x is an m ideal in y it is known that ekkx the k th tensor product of x endowed with the injective tensor norm is an m ideal in ekky nevertheless if x is also asplund we prove that every integral k homogeneous polynomial in x has a unique extension to y that preserves the integral norm other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given

on some systems of two discrete inequalities of gronwall type abstract in this paper we discuss some systems of two discrete inequalities of gronwall type some applications of our results are given

a comparison theorem for difference inequalities in the study of growth and decay properties of nonlinear evolution equations in particular hyperbolic and parabolic partial differential equations as t the following difference inequality which is satisfied by an energy expression arises mathop sup limits t leq s leq t 1 u s 1 alpha leq c 1 t r u t u t 1 g t

polynomials on riesz spaces abstract in this paper we investigate polynomial mappings on riesz spaces we give a characterization of positivity of homogeneous polynomials in terms of forward differences finally we prove hahn banach type extension theorems for positive and regular polynomial mappings

on the coefficient problem a version of the kahane katznelson de leeuw theorem for spaces of matrices it is known that for every an l2 z there exists a functionf c t such that an f n for everyn z we prove a noncommutative version for every matrixa aij such that supi verbar aij j l2and supj aij i l2are finite there exists a matrix bij defining a bounded operator onl2 such that aij bij for everyi j we extend this to other norms on matrices and present an abstract version of the coefficient problem

a geometric estimate on the norm of product of functionals abstract the open problem of determining the exact value of the n th linear polarization constant c n of r n has received considerable attention over the past few years this paper makes a contribution to the subject by providing a new lower bound on the value of sup y 1 x 1 y x n y where x 1 x n are unit vectors in r n the new estimate is given in terms of the eigenvalues of the gram matrix x i x j and improves upon earlier estimates of this kind however the intriguing conjecture c n n n 2 remains open

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

plank type problems for polynomials on banach spaces abstract in this work we prove plank type problems for polynomials on banach spaces we give estimates in the case of complex banach spaces and in particular in the case of complex hilbert spaces moreover we get optimal results in the case of 2 homogeneous polynomials on real hilbert spaces we have also derived estimates in the case of the classical l p spaces

optimality criteria in nonlinear programming involving nonconvex functions abstract in this paper a new class of functions called convex quasiconvex and pseudoconvex are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions

mixed type duality for multiobjective variational problems the concept of mixed type duality has been extended to the class of multiobjective variational problems a number of duality relations are proved to relate the efficient solutions of the primal and its mixed type dual problems the results are obtained for convex generalized convex functions these studies have been generalized to the case of invex generalized invex functions our results apparently generalize a fairly large number of duality results previously obtained for finite dimensional nonlinear programming problems under various convexity assumptions

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

geometry of spaces of polynomials abstract connections between the shape of the unit ball of a banach space and analytic properties of the banach space have been studied for many years in this article some geometric properties of spaces related ton homogeneous polynomials are considered in particular the rotundity and smoothness of spaces of continuousn homogeneous polynomials and its preduals are studied furthermore an inequality relating the product of the norms of linear functionals on a banach space with the norm of the continuousn homogeneous polynomial determined by the product of the linear functionals is derived this inequality is used to study the strongly exposed points of the predual of the space of continuous 2 homogeneous polynomials

linear polarization constants of hilbert spaces abstract this paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms the purpose of this work is to investigate the so called n th linear polarization constant c n x of a finite dimensional banach space x and in particular of a hilbert space note that c n x is an isometric invariant of the space it has been proved by j arias de reyna linear algebra appl 285 1998 395 408 that if h is a complex hilbert space of dimension at least n then c n h n n 2 the same value of c n h for real hilbert spaces is only conjectured but estimates were obtained in many cases in particular it is known that the n th linear polarization constant of a d dimensional real or complex hilbert space h is of the approximate order d n 2 for n large enough and also an integral form of the asymptotic quantity c h that is the linear polarization constant of the hilbert space h where dim h d was obtained together with an explicit form for real spaces here we present another proof we find the explicit form even for complex spaces and we elaborate further on the values of c n h and c h in particular we answer a question raised by j c garcia vazquez and r villa mathematika 46 1999 315 322 also we prove the conjectured c n h n n 2 for real hilbert spaces of dimension n 5 a few further estimates have been also derived

geometric theory of spaces of integral polynomials and symmetric tensor products abstract we investigate the nature of extreme weak exposed and weak strongly exposed points of the unit ball of spaces of n homogeneous integral polynomials on and n fold symmetric products of a banach space e for the space of integral polynomials we show the set of extreme points is contained in the set n e 1 we give smul yan type theorems for spaces of n homogeneous polynomials and n fold symmetric tensors that characterise weak exposed resp weak strongly exposed points in terms of g teaux resp frechet differentiability of the norm on various spaces of tensor products and polynomials our study of the geometry of these spaces has many applications when e has the radon nikod m property we show that the spaces of n homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n when the dimensions of e and n are both at least 2 then the space of n homogeneous polynomials on e is neither smooth nor rotund for a certain class of reflexive banach space the space of n homogeneous approximable polynomials on e is either reflexive or is not isometric to a dual banach space we conclude with a choquet theorem for a space of homogeneous polynomials

notes on subspace hypercyclic operators a bounded linear operator t on a banach space x is called subspace hypercyclic for a subspace m if orb t x m is dense in m for a vector x m in this case we show that p t has a relatively dense range for every real or complex polynomial p which in turn answers a question posed in madore and martinez avendano 2011 8 as a consequence the algebraic structure of the set of subspace hypercyclic vectors can be described