limits of hypercyclic and supercyclic operators abstract an operator t on a complex separable infinite dimensional hilbert space h is hypercyclic if there is a vector y in h such that the orbit y ty t2y is dense in h it is shown that if t is hypercyclic then t does not have any eigenvalue and that the union of the spectrum of t and the unit circle is a connected set this result is used to obtain a spectral characterization of the norm closure of the class hc h of all hypercyclic operators acting on h as well as a formula for the distance from a given operator to hc h analogous results are obtained for the closely related class sc h of all supercyclic operators this paper includes some information about the structure of the set of all hypercyclic supercyclic vectors of a given hypercyclic supercyclic resp operator t

hermitian analogues of hilbert s 17 th problem abstract we pose and discuss several hermitian analogues of hilbert s 17 th problem we survey what is known offer many explicit examples and some proofs and give applications to cr geometry we prove one new algebraic theorem a non negative hermitian symmetric polynomial divides a non zero squared norm if and only if it is a quotient of squared norms we also discuss a new example of putinar scheiderer

chaotic polynomials on banach spaces we show the existence of chaotic in the sense of devaney polynomials on banach spaces of q summable sequences such polynomials p consist of composition of the backward shift with a certain fixed polynomial p of one complex variable on each coordinate in general we also prove that p is chaotic in the sense of auslander and yorke if and only if 0 belongs to the julia set of p

nonlinear maps preserving the minimum and surjectivity moduli abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe maps from b x onto b y satisfying c s t c s t for all s t b x where c stands either for the minimum modulus or the surjectivity modulus or the maximum modulus we also obtain analog results for the finite dimensional case

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

the structure of almost invariant half spaces for some operators abstract given a banach space x and an operator t l x an almost invariant half space aihs is a closed subspace y satisfying t y y f for some finite dimensional subspace f and dim y dim x y in this paper we study the connection between aihs and noncompactness of some subsets of the unit ball chains of t invariant subspaces and different parts of spectrum of t and t we describe a simple template for such subspaces and show that if t is quasinilpotent or weakly compact then it admits an aihs y

every operator has almost invariant subspaces abstract we show that any bounded operator t on a separable reflexive infinite dimensional banach space x admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension in the non reflexive spaces we show that the same is true for operators which have non eigenvalues in the boundary of their spectrum in the hilbert space our methods produce perturbations that are also small in norm improving on an old result of brown and pearcy

there is no bound on sizes of indecomposable banach spaces abstract assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable banach spaces i e such that whenever they are decomposed as x y then one of the closed subspaces x or y must be finite dimensional it requires alternative techniques compared to those which were initiated by gowers and maurey or argyros with the coauthors this is because hereditarily indecomposable banach spaces always embed into l and so their density and cardinality is bounded by the continuum and because dual banach spaces of densities bigger than continuum are decomposable by a result due to heinrich and mankiewicz the obtained banach spaces are of the form c k for some compact connected hausdorff space and have few operators in the sense that every linear bounded operator t on c k for every f c k satisfies t f g f s f where g c k and s is weakly compact or equivalently strictly singular in particular the spaces carry the structure of a banach algebra and in the complex case even the structure of a c algebra

hypercyclic bilinear operators on banach spaces abstract we study the dynamics induced by an m linear operator we answer a question of bes and conejero showing an example of an m linear hypercyclic operator acting on a banach space moreover we prove the existence of m linear hypercyclic operators on arbitrary infinite dimensional separable banach spaces we also prove an existence result about symmetric bihypercyclic bilinear operators answering a question by grosse erdmann and kim

spaces of nuclear and compact operators without a complemented copy of c ωω abstract this paper deals with spaces of nuclear operators n x y and spaces of compact operators k z w containing no complemented copy of c for fixed ordinals 1 and 1 of the same cardinality we provide additional conditions on the banach spaces x y z and w ensuring that each of the following statements is equivalent to 1 n x c y c is isomorphic to n x c y c 2 k z c w c is isomorphic to k z c w c these results are generalizations of the classical isomorphic classification of spaces c 1 due to bessaga and pelczynski 7 to the setting of the spaces of operators on x c spaces the generalization 1 covers the case where x is an l 1 predual and y contains no complemented copy of c the generalization 2 covers the case z l p 1 p and w contains no copy of c 0

initial monomial invariants of holomorphic maps we study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals we start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics giving a readily computable invariant in this important case for example the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball next we associate to each subspace x subset mathcal o u a generic initial monomial subspace which is invariant under biholomorphic transformations and multiplication by nonzero functions the generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics

invariant subspaces for non normable frechet spaces abstract a frechet space x satisfies the hereditary invariant subspace resp subset property if for every closed infinite dimensional subspace m in x each continuous operator on m possesses a non trivial invariant subspace resp subset in this paper we exhibit a family of non normable separable infinite dimensional frechet spaces satisfying the hereditary invariant subspace property and we show that many non normable frechet spaces do not satisfy this property we also state sufficient conditions for the existence of a continuous operator without non trivial invariant subset and deduce among other examples that there exists a continuous operator without non trivial invariant subset on the space of entire functions h c

on spectrally bounded linear maps on b x abstract let x be a complex banach space and denote by b x the algebra of all linear bounded operators on x for t b x denote by r t its spectral radius we prove that if b x b x is a linear and surjective map such that i given any a b b x there exists a positive real number m a b such that r a b m a b r a b for all complex numbers satisfying 1 ii r t 0 implies r t 0 then there exist a linear functional f b x c a nonzero complex number c and either a bounded invertible linear map u x x such that t c u t u 1 f t i t b x or a bounded invertible linear map u x x such that t c u t u 1 f t i t b x also the linear functional f satisfies the property that given any a b b x there exists n a b 0 such that f a f b n a b r a b for all complex numbers satisfying 1

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

coarse topological transitivity on open cones and coarsely j class and d class operators abstract we generalize the concept of coarse hypercyclicity introduced by feldman in 13 to that of coarse topological transitivity on open cones we show that a bounded linear operator acting on an infinite dimensional banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive mixing operator on an open cone is topologically transitive mixing resp we also localize these concepts by introducing two new classes of operators called coarsely j class and coarsely d class operators and we establish some results that may make these classes of operators potentially interesting for further studying namely we show that if a backward unilateral weighted shift on l 2 n is coarsely j class or d class on an open cone then it is hypercyclic then we give an example of a bilateral weighted shift on l z which is coarsely j class hence it is coarsely d class and not j class note that concerning the previous result it is well known that the space l z does not support j class bilateral weighted shifts see 10 finally we show that there exists a non separable banach space which supports no coarsely d class operators on open cones some open problems are added

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

bounding the rank of hermitian forms and rigidity for cr mappings of hyperquadrics using green s hyperplane restriction theorem we prove that the rank of a hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds as an application we prove a rigidity theorem for cr mappings between hyperquadrics in the spirit of the results of baouendi huang and baouendi ebenfelt huang given a real analytic cr mapping of a hyperquadric not equivalent to a sphere to another hyperquadric q a b either the image of the mapping is contained in a complex affine subspace or a is bounded by a constant depending only on b finally we prove a stability result about existence of nontrivial cr mappings of hyperquadrics that is as long as both a and b are sufficiently large and comparable then there exist cr mappings whose image is not contained in a hyperplane the rigidity result also extends when mapping to hyperquadrics in infinite dimensional hilbert space

on the simplicial cohomology of banach operator algebras abstract we investigate the simplicial cohomology of certain banach operator algebras the two main examples considered are the banach algebra of all bounded operators on a banach space and its ideal of approximable operators sufficient conditions will be given forcing banach algebras of this kind to be simplicially trivial

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

hypercyclic tuples of operators and somewhere dense orbits in this paper we prove that there are hypercyclic n 1 tuples of diagonal matrices on cn and that there are no hypercyclic n tuples of diagonalizable matrices on cn we use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions we then show that on real hilbert spaces there are tuples with somewhere dense orbits that are not dense but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit on a real or complex space must be dense

operators on l spaces abstract it is shown that the space of bounded linear operators on certain l spaces is non separable these spaces include the spaces of j bourgain and f delbaen and spaces constructed by s a argyros and r haydon

interpolating hereditarily indecomposable banach spaces a banach space x is said to be hereditarily indecomposable h i if for any pair of closed subspaces y z of x with y z 0 y z is not a closed subspace throughout this section by the term subspace we mean a closed infinite dimensional subspace of x the h i spaces form a new and as we believe fundamental class of banach spaces the celebrated example of a banach space with no unconditional basic sequence due to w gowers and b maurey gm is the first construction of a h i space it is easily seen that every h i space does not contain any unconditional basic sequence actually the concept of h i spaces came after w johnson s observation that this was a property of the gowers maurey example to describe even further the peculiar structure of a h i space we recall an alternative definition of such a space a banach space x is a h i space if and only if for every pair of subspaces y z and e 0 there exist y y z z with y z 1 and y z e thus h i spaces are structurally irrelevant to classical banach spaces in particular to hilbert spaces other constructions of h i spaces already exist we mention argyros and deliyanni s construction of h i spaces which are asymptotic spaces ad2 v ferenczi s example of a uniformly convex h i space f2 and h i modified asymptotic spaces contained in adkm other examples of banach spaces which are h i or which have a h i subspace are given in g1 h os1 the construction of such a space requires several steps and it uses two fundamental ideas the first is tsirelson s recursive definition of saturated norms ts and the second is maurey rosenthal s construction of weakly null sequences without unconditional basic subsequences mr an important ingredient in the gowersmaurey construction is the schlumprecht space this is a tsirelson type banach space which is arbitrarily distortable and has been used in the solution of important problems thus beyond its use in the constructions of h i spaces it plays a central role in the solution of the distortion problem for hilbert spaces os the essential difference between schlumprecht and tsirelson spaces became more transparent in ad2 where the mixed tsirelson spaces were introduced it is natural to expect that h i spaces share special and interesting properties not found in the previously known banach spaces indeed the following theorem is proven in

the unconditional basic sequence problem we construct a banach space that does not contain any infinite un conditional basic sequence and investigate further properties of this space for example it has no subspace that can be written as a topological direct sum of two infinite dimensional spaces this property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity in par ticular it is either strictly singular or fredholm with index zero this implies that the space is not isomorphic to any proper subspace trinity college cambridge cb2 1tq england current address department of mathematics university college london gower street lon don wc1e 6bt england e mail address wtg l o phx cam ac uk universit paris vii u f r de mathtmatiques 2 place jussieu 75251 paris cedex 05 france current address equipe d analyse et mathematiques appliquees universite de mamne la vallee 93160 noisy le grand france e mail address maurey logiquejussieu fr this content downloaded from 157 55 39 105 on fri 07 oct 2016 04 30 48 utc all use subject to http about jstor org terms

embedding strictly pseudoconvex domains into balls every relatively compact strictly pseudocc nvex domain d with c2 boundary in a stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball however for each n 2 there exist bounded strictly pseudoconvex domains d in cn with real analytic boundary such that no proper holomorphic map from d into any finite dimensional ball extends smoothly to d 0 introduction in this paper we study the representations of bounded strictly pseudoconvex domains d c cn if the boundary of d is of class ck k e 2 3 oo then by a theorem of fornaess 8 and khenkin 13 d can be mapped biholomorphically onto the intersection xn q of a bounded strictly convex domain q c cn with ck boundary and a closed complex submanifold x defirxed in a neighborhood of q in cn x intersecting the boundary of q transversally moreover the map f d x n q extends to a holomorphic map on a neighborhood of d the convex domain q depends on d hence a natural question is whether a similar result holds with q replaced by the unit ball b z z1l zn e cn z 2 e zjl2 n that intersects bbn transversally such that d is biholomorphically equivalent to xn e3ff this question has been mentioned by lempert 14 pinduk 20 bedford 2 and others our main result is that the answer to this question is negative in general if d is as above then every biholomorphism of d onto xne3n extends smoothly to d according to 3 however we will show that not all such domains d admit a proper holomorphic map into a finite dimensional ball that is smooth on d theorem 1 1 a similar local result was obtained independently by faran 7 we shall show that the answer to the question q is positive if we allow the intersections of complex submanifolds with strictly convex domains q c cn with real analytic boundaries theorem 1 2 thetheoremsoffornaess 8 andkhenkin 13 onlygiveanqwith smooth boundary received by the editors april 4t 1985 1980 mathematics subject classification primary 32h05 1research supported by a fellowship from the alfred p sloan foundation t 1986 american mathematical society 0002 9947 86 1 00 25 per page

bihypercyclic bilinear mappings abstract we propose a notion of orbit for bilinear mappings on banach spaces this then leads us to the concept of bihypercyclic bilinear mappings that is of bilinear mappings with a dense orbit we provide some examples of such mappings and we obtain some basic properties of bihypercyclicity in our main result we show that every separable real or complex banach space supports a bihypercyclic bilinear mapping we also formulate some open problems

the universality of l1 as a dual space let x be a banach space with a separable dual we prove that x embeds isomorphically into a mathcal l infty space z whose dual is isomorphic to l1 if moreover u is a space with separable dual so that u and x are totally incomparable then we construct such a z so that z and u are totally incomparable if x is separable and reflexive we show that z can be made to be somewhat reflexive

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

rota s universal operators and invariant subspaces in hilbert spaces a hilbert space operator is called universal in the sense of rota if every operator on the hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces we exhibit an analytic toeplitz operator whose adjoint is universal in the sense of rota and commutes with a quasi nilpotent injective compact operator with dense range in particular this new universal operator invites an approach to the invariant subspace problem that uses properties of operators that commute with the universal operator

boundary characterization of holomorphic isometric embeddings between indefinite hyperbolic spaces abstract we provide in this paper a boundary characterization in terms of the boundary cr invariants or pseudo hermitian invariants for holomorphic isometric embeddings between indefinite hyperbolic spaces of general codimensions

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

when is multiplication in a banach algebra open abstract we develop the theory of banach algebras whose multiplication regarded as a bilinear map is open we demonstrate that such algebras must have topological stable rank 1 however the latter condition is strictly weaker and implies only that products of non empty open sets have non empty interior we then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in l 1 n 0 is open by appealing to ultraproduct techniques we demonstrate that neither in l 1 z nor in l 1 q convolution is uniformly open the problem of openness of multiplication in banach algebras of bounded operators on banach spaces and their calkin algebras is also discussed

factorization with genus 2 curves the elliptic curve method ecm is one of the best factorization methods available it is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower we use special hyperelliptic curves and kummer surfaces to reduce the complexity of the algorithm our implementation gmp hecm is faster than gmp ecm for factoring large numbers

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method