complex structures on banach spaces with a subsymmetric basis we prove that a real banach space e with a subsymmetric basis and isomorphic to the sequence space e e has a unique complex structure we also show that if a real banach space x has the property p then all its complex structures also satisfies p when p is any of the following properties bounded approximation property g l l u st being injective and being complemented in a dual space

complex structures on real vector lattices the main result of this paper states that a universally complete real vector lattice admits a band preserving complex structure if and only if it contains no locally one dimensional bands a description of order bounded disjointness preserving complex structures on archimedean semiprime f algebras is also given

diffusion approximation for slow motion in fully coupled averaging in systems which combine fast and slow motions it is usually impossible to study directly corresponding two scale equations and the averaging principle suggests to approximate the slow motion by averaging in fast variables we consider the averaging setup when both fast and slow motions are diffusion processes depending on each other fully coupled and show that there exists a diffusion process which approximates the slow motion in the l 2 sense much better than the averaged motion prescribed by the averaging principle

on stable oscillations and equilibriums induced by small noise we consider the motion of a light particle when the force field is perturbed by a small nose if a certain relation between the mass of the particle and the noise intensity holds the motion of the particle will be close to periodic oscillations or to a stable equilibrium which do not exist without the noise we study various classes of random perturbations in particular we consider the question of computer simulation of these effects and calculate the correction term which appears when the gaussian perturbations are replaced by the simple random walk these are the stochastic resonance type effects and their mathematical description is based on the large deviation theory

quantities characterizing semi fredholm operators and perturbation radii abstract we show that none of the operational quantities that have been introduced to characterize the upper or the lower semi fredholm operators is equivalent to the corresponding perturbation radius

lp strong convergence of the averaging principle for slow fast spdes with jumps abstract the averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component this paper concerns the l p strong convergence of the averaging principle for two time scales stochastic partial differential equations spdes driven by wiener processes and poisson jumps to achieve this a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation where the dissipative conditions are needed furthermore it is shown that under suitable assumptions the slow component l p strongly converges to the solution of the averaged equation the rate of the convergence is also obtained as a byproduct

on the number of permutatively inequivalent basic sequences in a banach space let x be a banach space with a schauder basis en n n the relation e0 is borel reducible to permutative equivalence between normalized block sequences of en n n or x is c0 or lp saturated for some 1 p if en n n is shrinking unconditional then either it is equivalent to the canonical basis of c0 or lp 1 p or the relation e0 is borel reducible to permutative equivalence between sequences of normalized disjoint blocks of x or of x if en n n is unconditional then either x is isomorphic to l2 or x contains 2 subspaces or 2 quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases

on asymptotic properties of maximum likelihood estimators for parabolic stochastic pde s we investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic spdes of the form du t x a 0 theta a 1 u t x dt dw t x wherea 0 anda 1 are partial differential operators andw is a cylindrical brownian motion we introduce a spectral method for computing mles based on finite dimensional approximations to solutions of such systems and establish criteria for consistency asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity we derive the asymptotic properties of the mle from a condition on the order of the operators in particular the mle is consistent if and only if ord a 1 1 2 ord a 0 a 1 d

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

an infinite ramsey theorem and some banach space dichotomies a problem of banach asks whether every infinite dimensional banach space which is isomorphic to all its infinite dimensional subspaces must be isomorphic to a separable hilbert space in this paper we prove a result of a ramsey theoretic nature which implies an interesting dichotomy for subspaces of banach spaces combined with a result of komorowski and tomczakjaegermann this gives a positive answer to banachxs problem we then generalize the ramsey theoretic result and deduce a further dichotomy for banach spaces with an unconditional basis

averaging principle for one dimensional stochastic burgers equation abstract in this paper we consider the averaging principle for one dimensional stochastic burgers equation with slow and fast time scales under some suitable conditions we show that the slow component strongly converges to the solution of the corresponding averaged equation meanwhile when there is no noise in the slow component equation we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 r for any 0 r 1

stochastic three dimensional rotating navier stokes equations averaging convergence and regularity we consider stochastic three dimensional rotating navier stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems

reconstructing initial data using observers error analysis of the semi discrete and fully discrete approximations a new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in ramdani et al automatica 46 10 1616 1625 2010 based on the concept of observers also called luenberger observers this algorithm covers a large class of abstract evolution pde s in this paper we are concerned with the convergence analysis of this algorithm more precisely we provide a complete numerical analysis for semi discrete in space and fully discrete approximations derived using finite elements in space and an implicit euler method in time the analysis is carried out for abstract schrodinger and wave conservative systems with bounded observation locally distributed

a c0 saturated banach space with tight structure it is shown that variants of the hi methods could yield objects closely connected to the classical banach spaces thus we present a new c0 saturated space denoted as x0 with rather tight structure the space x0 is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure everyone is either of type i namely contains an isomorph of x0 itself or else is isomorphic to a subspace of c0 type ii furthermore for any analytic decomposition of x0 into two subspaces one is of type i and the other is of type ii the operators of x0 share common features with those of hi spaces

on the hereditary proximity to l1 in the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to l1 the second part is devoted to a complete separation of the hereditary local proximity to l1 from the asymptotic one more precisely for every countable ordinal we construct a separable reflexive space x such that every infinite dimensional subspace of it has bourgain l1 index greater than and the space itself has no l1 spreading model we also present a reflexive hi space admitting no lp as a spreading model

rolewicz type chaotic operators abstract in this article we introduce a new class of rolewicz type operators in l p 1 p we exhibit a collection f having the cardinality of the continuum consisting of operators of this type which are chaotic and remain so under almost all finite linear combinations provided that the linear combination has sufficiently large norm as a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm

renorming banach spaces with the mazur intersection property in this paper we give new sufficient and necessary conditions for a banach space to be equivalently renormed with the mazur intersection property as a consequence several examples and applications of these results are obtained among them it is proved that every banach space embeds isometrically into a banach space with the mazur intersection property answering a question asked by giles gregory and sims we also prove that for every treet the spacec0 t admits a norm with the mazur intersection property solving a problem posed by deville godefroy and zizler finally assuming the continuum hypothesis we find an example of an asplund space admitting neither an equivalent norm with the above property nor a nicely smooth norm

finite representability of operators abstract we introduce operator local supportability as a new type of operator finite representability that generalizes bellenot finite representability we prove that local supportability and local representability are mutually independent new examples of both types of finite representability are given for instance for every operator t we prove that t u is locally supportable in t u we also prove that given an operator t with range in c 0 1 t is locally representable in t l 1

on the existence and uniqueness of complex structure and spaces with few operators we construct a 2 n dimensional real normed space whose banach mazur distance to the set of spaces admitting complex structure is of order nl 2 and two complex n dimensional normed spaces which are isometric as real spaces but whose complex banach masur distance is of order n both orders of magnitude are the largest possible we also construct finite dimensional spaces with the property that all well bounded operators on them are rather small in the sense of some ideal norm perturbations of multiples of identity we also state some metatheorem which can be used to produce spaces with various pathological properties 1 introduction and the main results the main purpose of this paper is to prove the following results theorem 1 1 given n there exists a 2n dimensional real normed space x such that whenever y is an n dimensional complex normed space and yr is y treated as a real space then the banach mazur distance d x yr c4 where c is a numerical constant given a complex normed space y denote by y the space obtained from y by replacing the multiplication by scalar x y xy by k y a y xy and preserving the rest of the structure addition norm we then have theorem 1 2 given n there ejcists an n dimensional space y over c such that d y y cn where c is a numerical constant here d denotes distance of complex spaces the orders of magnitude o 4 and o n in theorems 1 1 and 1 2 are of course the largest possible in the first case the n dimensional complex hilbert space always gives the upper estimate in the second d yl y2 n if dimyl dim y2 n in both the real and complex case a weaker version of theorem 1 1 with estimate o 5 n log n was shown recently by p mankiewicz 8 a version of theorem 1 2 with estimate o n log n was shown independently of this paper and approximately at the same time by j bourgain 1 who also settled the infinite dimensional problem constructing a complex banach space y such that y and y are not isomorphic with all the consequences see corollary 1 3 below similar problems were considered recently by n j kalton 5 who constructed an analogous example received by the editors november 1 1984 and in revised form march 4 1985 1980 mathematics subject classification primary 46b20 51a99 secondary 46l99 47b10 47d15 52a22 52a40 60e15 supported in part by nsf dms 8401906 c 1986 american mathematical society 0002 9947 86 1 00 25 per page

realizing irreducible semigroups and real algebras of compact operators let b x be the algebra of bounded operators on a complex banach space x viewing b x as an algebra over r we study the structure of those irreducible subalgebras which contain nonzero compact operators in particular irreducible algebras of trace class operators with real trace are characterized this yields an extension of brauer type results on matrices to operators in infinite dimensions answering the question is an irreducible semigroup of compact operators with real spectra realizable i e simultaneously similar to a semigroup whose matrices are real

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

operators on c 0 1 preserving copies of asymptotic l1 spaces given separable banach spaces x y z and a bounded linear operator t x y then t is said to preserve a copy of z provided that there exists a closed linear subspace e of x isomorphic to z and such that the restriction of t to e is an into isomorphism it is proved that every operator on c 0 1 which preserves a copy of an asymptotic l1 space also preserves a copy of c 0 1

uniqueness of complex structure and real hereditarily indecomposable banach spaces there exists a real hereditarily indecomposable banach space x x c respectively x x h such that the algebra l x s x is isomorphic to c respectively to the quaternionic division algebra h up to isomorphism x c has exactly two complex structures which are conjugate totally incomparable and both hereditarily indecomposable so there exist two banach spaces which are isometric as real spaces but totally incomparable as complex spaces this extends results of j bourgain and s szarek j bourgain real isomorphic complex banach spaces need not be complex isomorphic proc amer math soc 96 2 1986 221 226 s szarek on the existence and uniqueness of complex structure and spaces with few operators trans amer math soc 293 1 1986 339 353 s szarek a superreflexive banach space which does not admit complex structure proc amer math soc 97 3 1986 437 444 and proves that a theorem of g godefroy and n j kalton g godefroy n j kalton lipschitz free banach spaces studia math 159 1 2003 121 141 about isometric embeddings of separable real banach spaces does not extend to the complex case the quaternionic example x h on the other hand has unique complex structure up to isomorphism other examples with a unique complex structure are produced including a space with an unconditional basis and non isomorphic to l2 this answers a question of s szarek in s szarek a superreflexive banach space which does not admit complex structure proc amer math soc 97 3 1986 437 444

two time scales hyperbolic parabolic equations driven by poisson random measures existence uniqueness and averaging principles abstract in this article we are concerned with averaging principle for stochastic hyperbolic parabolic equations driven by poisson random measures with slow and fast time scales we first establish the existence and uniqueness of weak solutions of the stochastic hyperbolic parabolic equations then under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the stochastic wave equation is an average with respect to the stationary measure of the fast varying process finally we derive the rate of strong convergence for the slow component towards the solution of the averaged equation

a c_0 saturated banach space with no long unconditional basic sequences we present a banach space mathfrak x with a schauder basis of length omega 1 which is saturated by copies of c 0 and such that for every closed decomposition of a closed subspace x x 0 oplus x 1 either x 0 or x 1 has to be separable this can be considered as the non separable counterpart of the notion of hereditarily indecomposable space indeed the subspaces of mathfrak x have few operators in the sense that every bounded operator t x rightarrow mathfrak x from a subspace x of mathfrak x into mathfrak x is the sum of a multiple of the inclusion and a omega 1 singular operator i e an operator s which is not an isomorphism on any non separable subspace of x we also show that while mathfrak x is not distortable being c 0 saturated it is arbitrarily omega 1 distortable in the sense that for every lambda 1 there is an equivalent norm cdot on mathfrak x such that for every non separable subspace x of mathfrak x there are x y in s x such that cdot cdot ge la

on large indecomposable banach spaces abstract hereditarily indecomposable banach spaces may have density at most continuum plichko and yost 2000 19 argyros and tolias 2004 1 in this paper we show that this cannot be proved for indecomposable banach spaces we provide the first example of an indecomposable banach space of density 2 2 the space exists consistently is of the form c k and it has few operators in the sense that any bounded linear operator t c k c k satisfies t f g f s f for every f c k where g c k and s c k c k is weakly compact strictly singular

extremely non complex c k spaces abstract we show that there exist infinite dimensional extremely non complex banach spaces i e spaces x such that the norm equality id t 2 1 t 2 holds for every bounded linear operator t x x this answers in the positive question 4 11 of v kadets m martin j meri norm equalities for operators on banach spaces indiana univ math j 56 2007 2385 2411 more concretely we show that this is the case of some c k spaces with few operators constructed in p koszmider banach spaces of continuous functions with few operators math ann 330 2004 151 183 and g plebanek a construction of a banach space c k with few operators topology appl 143 2004 217 239 we also construct compact spaces k 1 and k 2 such that c k 1 and c k 2 are extremely non complex c k 1 contains a complemented copy of c 2 and c k 2 contains a 1 complemented isometric copy of l

examples of improjective operators it has been an open question for some time whether improjective operators are always inessential here we give some examples that answer in the negative this question as well as some other related ones posed in 2 3 11 12 the description of the examples uses a indecomposable space constructed by gowers and maurey 5 and a characterization of the indecomposable banach spaces in terms of improjective operators

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

the lattice of closed ideals in the banach algebra of operators on certain banach spaces very few banach spaces e are known for which the lattice of closed ideals in the banach algebra of all bounded linear operators on e is fully understood indeed up to now the only such banach spaces are up to isomorphism hilbert spaces and the sequence spaces c0 and lp for 1p we add a new member to this family by showing that there are exactly four closed ideals in for the banach space e l2n c0 that is e is the c0 direct sum of the finite dimensional hilbert spaces l21 l22 l2n

sums of hypercyclic operators abstract every bounded operator on a complex infinite dimensional separable hilbert space can be written as the sum of two hypercyclic operators and also as the sum of two chaotic operators

a banach space with a countable infinite number of complex structures abstract we give examples of real banach spaces with exactly infinite countably many complex structures and with 1 many complex structures

large deviations and approximations for slow fast stochastic reaction diffusion equations abstract a large deviation principle is derived for a class of stochastic reaction diffusion partial differential equations with slow fast components the result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small gaussian perturbation this result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow fast stochastic partial differential equations

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

banach spaces of continuous functions with few operators we present two constructions of infinite separable compact hausdorff spaces k for which the banach space c k of all continuous real valued functions with the supremum norm has remarkable properties in the first construction k is zero dimensional and c k is non isomorphic to any of its proper subspaces nor any of its proper quotients in particular it is an example of a c k space where the hyperplanes one co dimensional subspaces of c k are not isomorphic to c k in the second construction k is connected and c k is indecomposable which implies that it is not isomorphic to any c k for k zero dimensional all these properties follow from the fact that there are few operators on our c k s if we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator t c k c k is of the form gi s where g c k and s is weakly compact or equivalently in c k spaces strictly singular

averaging principle for a class of stochastic reaction diffusion equations we consider the averaging principle for stochastic reaction diffusion equations under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component we calculate limiting slow motion the study of solvability of kolmogorov equations in hilbert spaces and the analysis of regularity properties of solutions allow to generalize the classical approach to finite dimensional problems of this type in the case of spde s

on essentially incomparable banach spaces we introduce the concept of essentially incomparable banach spaces and give some examples then for two essentially incomparable banach spaces x and y we prove that a complemented subspace of the product x x y is isomorphic to the product of a complemented subspace of x and a complemented subspace of y if additionally x and y are isomorphic to their respective hyperplanes then the group of invertible operators in x x y is not connected the results can be applied to some classical banach spaces

the perturbation classes problem in fredholm theory abstract we give a negative answer to the perturbation classes problem the perturbation class of the upper semi fredholm operators contains properly the strictly singular operators and the perturbation class of the lower semi fredholm operators contains properly the strictly cosingular operators

unconditional basic sequences in spaces of large density abstract we study the problem of the existence of unconditional basic sequences in banach spaces of high density we show in particular the relative consistency of the statement that every banach space of density contains an unconditional basic sequence

long time behavior of weakly coupled oscillators we consider small perturbations of a simple completely integrable system with many degrees of freedom a collection of independent one degree of freedom oscillators in the perturbed system the individual oscillators are no longer independent we show that the long time behavior of such a system even in the case of purely deterministic perturbations should in general be described as a stochastic process the limiting stochastic process is a markov process on an open book space corresponding to the collection of first integrals of the non perturbed system

averaging principle for the higher order nonlinear schrodinger equation with a random fast oscillation this work concerns the problem associated with averaging principle for a higher order nonlinear schrodinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction diffusion equation evolving with respect to the fast time this model can be translated into a multiscale stochastic partial differential equations stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear schrodinger equation is an average with respect to the stationary measure of the fast varying process finally by using the khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation and as a consequence the system can be reduced to a single higher order nonlinear schrodinger equation with a modified coefficient

strong convergence in stochastic averaging principle for two time scales stochastic partial differential equations abstract the theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time scales and is relatively mature for stochastic ordinary differential equations in this paper we study the averaging principle for a class of stochastic partial differential equations with two separated time scales driven by scalar noises under suitable assumptions it is shown that the slow component strongly converges to the solution of the corresponding averaged equation

stochastic homogenization of multicontinuum heterogeneous flows abstract we consider a multicontinuum model in porous media applications which is described as a system of coupled flow equations the coupling between different continua depends on many factors and its modeling is important for porous media applications the coefficients depend on particle deposition that is described in terms of a stochastic process solution of an sde the stochastic process is considered to be faster than the flow motion and we introduce time space scales to model the problem our goal is to pass to the limit in time and space and to find an associated averaged system this is an averaging homogenization problem where the averages are computed in terms of the invariant measure associated to the fast motion and the spatial variable we use the techniques developed in our previous paper bessaih et al 2019 to model the interactions between the continua and derive the averaged model problem that can be used in many applications

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

an averaging principle for the time dependent abstract stochastic evolution equations with infinite delay and wiener process in this paper averaging principle for the time dependent stochastic evolution equations tdsees with infinite delay is investigated in proper non lipschitz conditions we prove that the mild solution of the averaged stochastic evolution equations asees governed by time independent family of linear operators converges to that of the original one in l 2 sense at last an application example is presented to prove the obtained theory

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

countable groups of isometries on banach spaces a group g is representable in a banach space x if g is isomorphic to the group of isometries on x in some equivalent norm we prove that a countable group g is representable in a separable real banach space x in several general cases including when g 1 1 h h finite and dim x h or when g contains a normal subgroup with two elements and x is of the form co y or l p y 1 p this is a consequence of a result inspired by methods of s bellenot 1986 and stating that under rather general conditions on a separable real banach space x and a countable bounded group g of isomorphisms on x containing id there exists an equivalent norm on x for which g is equal to the group of isometries on x we also extend methods of k jarosz 1988 to prove that any complex banach space of dimension at least 2 may be renormed with an equivalent complex norm to admit only trivial real isometries and that any complexification of a banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries it follows that every real banach space of dimension at least 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry and that every real cartesian square may be renormed to admit a unique complex structure up to isometry

even infinite dimensional real banach spaces abstract this article is a continuation of a paper of the first author v ferenczi uniqueness of complex structure and real hereditarily indecomposable banach spaces adv math 213 1 2007 462 488 about complex structures on real banach spaces we define a notion of even infinite dimensional real banach space and prove that there exist even spaces including hi or unconditional examples from v ferenczi uniqueness of complex structure and real hereditarily indecomposable banach spaces adv math 213 1 2007 462 488 and c k examples due to plebanek g plebanek a construction of a banach space c k with few operators topology appl 143 2004 217 239 we extend results of v ferenczi uniqueness of complex structure and real hereditarily indecomposable banach spaces adv math 213 1 2007 462 488 relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space and give characterizations of even spaces in terms of this group we also generalize results of v ferenczi uniqueness of complex structure and real hereditarily indecomposable banach spaces adv math 213 1 2007 462 488 about totally incomparable complex structures to essentially incomparable complex structures while showing that the complex version of a space defined by s argyros and a manoussakis s argyros a manoussakis an indecomposable and unconditionally saturated banach space studia math 159 1 2003 1 32 provides examples of essentially incomparable complex structures which are not totally incomparable

strong averaging principle for two time scale non autonomous stochastic fitzhugh nagumo system with jumps in this paper we study an averaging principle for stochastic fitzhugh nagumo system with different time scales driven by cylindrical wiener processes and poisson jumps where the slow equation is non autonomous and the fast equation is autonomous case under suitable assumptions we show that the slow component mean square strongly converges to the solution of the corresponding averaging equation and the rate of the convergence as a by product is also affirmed finally we give some open problems which are derived from this paper

on the convergence of galerkin approximation schemes for second order hyperbolic equations in energy and negative norms given certain semidiscrete and single step fully discrete galerkin approximations to the solution of an initial boundary value problem for a second order hyperbolic equation h and l2 error estimates are obtained these estimates are valid simultaneously when the approximation to the initial data is taken to be the projection onto the approximating space with respect to the inner product which induces the energy norm that is naturally associated with the problem the l2 estimate is obtained as a by product of the analysis of convergence in certain negative norms estimates are also obtained for the convergence of higher order time derivatives in the presence of sufficiently smooth data

on isometry groups and maximal symmetry we study problems of maximal symmetry in banach spaces this is done by providing an analysis of the structure of small subgroups of the general linear group gl x where x is a separable reflexive banach space in particular we provide the first known example of a banach space x without any equivalent maximal norm or equivalently such that gl x contains no maximal bounded subgroup moreover this space x may be chosen to be super reflexive in memory of greg hjorth 1963 2011 and nigel kalton 1946 2010

the convergence of galerkin approximation schemes for second order hyperbolic equations with dissipation in this paper we consider certain semidiscrete and fully discrete galerkin approximations to the solution of an initial boundary value problem for a second order hyperbolic equation with a dissipative term estimates are obtained in the energy and negative norms associated with the problem yielding in particular hland l2 error estimates the approximation to the initial data is taken in this case as the projection with respect to the energy inner product onto the approximating space we also obtain estimates for higher order time derivatives

stochastic averaging for a class of two time scale systems of stochastic partial differential equations abstract this paper focuses on systems of stochastic partial differential equations that have a slow component driven by a fractional brownian motion and a fast component driven by a fast varying diffusion we establish an averaging principle in which the fast varying diffusion process acts as a noise and is averaged out in the limit the slow process is shown to have a limit in the l 2 sense which is characterized by the solution of a stochastic partial differential equation driven by a fractional brownian motion whose coefficients are averages of that of the original slow process with respect to the stationary measure of the fast varying diffusion this averaging principle paves a way for reduction of computational complexity the implication is that one can ignore the complex original systems and concentrate on the average systems instead

equivalence after extension for compact operators on banach spaces in recent years the coincidence of the operator relations equivalence after extension and schur coupling was settled for the hilbert space case by showing that equivalence after extension implies equivalence after one sided extension in the present paper we investigate consequences of equivalence after extension for compact banach space operators we show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension in analogy with the necessary and sufficient conditions for compact hilbert space operators to be equivalent after extension in terms of their singular values we prove under certain additional conditions the necessity of a similar relationship between the s numbers of two compact banach space operators that are equivalent after extension for arbitrary s functions we investigate equivalence after extension for operators on lp spaces we show that two operators that act on different lp spaces cannot be equivalent after one sided extension such operators can still be equivalent after extension for instance all invertible operators are equivalent after extension however if one of the two operators is compact then they cannot be equivalent after extension this contrasts the hilbert space case where equivalence after one sided extension and equivalence after extension are in fact identical relations finally for general banach spaces x and y we investigate consequences of an operator on x being equivalent after extension to a compact operator on y we show that in this case a closed finite codimensional subspace of y must embed into x and that certain general banach space properties must transfer from x to y we also show that no operator on x can be equivalent after extension to an operator on y if x and y are essentially incomparable banach spaces

operator spaces with few completely bounded maps we construct several examples of hilbertian operator spaces with few completely bounded maps in particular we give an example of a separable 1 hilbertian operator space x 0 such that whenever x is an infinite dimensional quotient of x 0 x is a subspace of x and t x rightarrow x is a completely bounded map then t i x s where s is compact hilbert schmidt and s 2 16 s cb s 2 moreover every infinite dimensional quotient of a subspace of x 0 fails the operator approximation property we also show that every banach space can be equipped with an operator space structure without the operator approximation property

average and deviation for slow fast stochastic partial differential equations abstract averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes this article derives an averaged equation for a class of stochastic partial differential equations without any lipschitz assumption on the slow modes the rate of convergence in probability is obtained as a byproduct importantly the stochastic deviation between the original equation and the averaged equation is also studied a martingale approach proves that the deviation is described by a gaussian process this gives an approximation to errors of order o instead of order o attained in previous averaging

averaging principle for stochastic korteweg de vries equation abstract averaging principle is a powerful tool for studying qualitative analysis of nonlinear dynamical systems in this paper we will establish an averaging principle for stochastic korteweg de vries equation under a general averaging condition with the help of this averaging principle we can establish an effective approximation for the solution of stochastic korteweg de vries equation this can tell us the asymptotic behavior of the solution and make the interaction between nonlinearity uncertainty and multiple scales more clear in order to obtain this averaging principle we need to establish the smoothing effect for the third order operator

positional graphs and conditional structure of weakly null sequences abstract we prove that unless assuming additional set theoretical axioms there are no reflexive spaces without unconditional sequences of the density continuum we show that for every integer n there are normalized weakly null sequences of length n without unconditional subsequences this together with a result of dodos et al 2011 7 shows that is the minimal cardinal that could possibly have the property that every weakly null sequence has an infinite unconditional basic subsequence we also prove that for every cardinal number which is smaller than the first erd s cardinal there is a normalized weakly null sequence without subsymmetric subsequences finally we prove that mixed tsirelson spaces of uncountable densities must always contain isomorphic copies of either c 0 or l p with p 1

numerical methods for stochastic partial differential equations with multiple scales a new method for solving numerically stochastic partial differential equations spdes with multiple scales is presented the method combines a spectral method with the heterogeneous multiscale method hmm presented in w e d liu e vanden eijnden analysis of multiscale methods for stochastic differential equations commun pure appl math 58 11 2005 1544 1585 the class of problems that we consider are spdes with quadratic nonlinearities that were studied in d blomker m hairer g a pavliotis multiscale analysis for stochastic partial differential equations with quadratic nonlinearities nonlinearity 20 7 2007 1721 1744 for such spdes an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales our method based on micro and macro solvers allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem numerical experiments illustrate the behavior of the proposed method

banach spaces with projectional skeletons a projectional skeleton in a banach space is a directed family of projections onto separable subspaces covering the entire space the class of banach spaces with projectional skeletons is strictly larger than the class of plichko spaces i e banach spaces with a countably norming markushevich basis we show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to spaces we characterize the existence of a projectional skeleton in terms of elementary substructures providing simple proofs of known results concerning weakly compactly generated spaces and plichko spaces we prove a preservation result for plichko banach spaces involving transfinite sequences of projections as a corollary we show that a banach space is plichko if and only if it has a commutative projectional skeleton