a general method for the existence of periodic solutions of differential systems in the plane abstract we propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations which can be used in many different circumstances applications are given to some nonresonant cases even for systems with superlinear growth in some direction or with a singularity systems at resonance are also considered provided a landesman lazer type of condition is assumed

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

resonant semilinear problems with nonlinear term depending on the derivative we study the existence of solution for a boundary value problem at resonance where the nonlinearity depends only on the derivative in a sense we can say that the problem considered is strongly resonant our proofs make use of the lyapunov schmidt reduction in so doing we are led with the asymptotic estimate of the corresponding bifurcation equation

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

interpolation of weighted and vector valued hardy spaces real and complex interpolation methods when applied to the couple hpo e0 wo hp el wl give what is expected if eo and e1 are quasi banach lattices of measurable functions satisfying certain mild conditions and if log wi powj l p1 e bmo wo w1 being weights on the unit circle the last condition is in fact necessary it is expected of course that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple lpo eo wo lp e1 w1

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

holomorphic martingales and interpolation between hardy spaces the complex method a probabilistic proof is given to identify the complex interpolation space of h1 t and h t as hp t introduction in this note a soft propabilistic proof of p w jones s theorem on the complex interpolation space between h1 and hoc is given we shall work with n th varopoulos s space of holomorhpic martingales the observation presented in this article is that the use of a stopping time decomposition simplifies constructions of serguei v kislyakov and quanhua xu to obtain the following theorem the complex interpolation space th t h t 0 o 0 e 1 coincides with hp t provided that 1 1 03 p as is well known this result has been obtained by p w jones using l estimates to the a problem see j his work also contains the description of the real interpolation spaces for the couple hl h at about the same time jean bourgain obtained a marcinkiewicz type decomposition using completely different techniques see b1 recently in a series of papers jean bourgain b2 serguei kislyakov kl k2 k x gilles pisier p and quanhua xu xl x2 obtained deep results concerning real and complex interpolation methods between vector valued weighted hardy spaces of analytic functions s kislyakov s paper kl contains the following idea to approximate the characteristic function 1 1f1 2 by analytic functions on the circle t he starts a max l itl and then considers 1 a iha where h denotes the hilbert transform on t received by the editors august 11 1992 and in revised form february 1 1994 originally communicated to the proceedings of the ams by dale alspach 1991 mathematics subject classification primary 60g46 42b30

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

quadratic forms weighted eigenfunctions and boundary value problems for non linear second order ordinary differential equations some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised different approaches are compared with one another using topological and variational methods and the theory of weighted eigenvalue problems

existence and multiplicity for solutions of neumann problem for semilinear elliptic equations the existence and multiplicity results are obtained for solutions of neumann problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

integral operators mapping into the space of bounded analytic functions abstract we address the problem of studying the boundedness compactness and weak compactness of the integral operators t g f z 0 z f g d acting from a banach space x into h we obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of x

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

extensions of multilinear operators and banach space properties a new characterization of the dunford pettis property in terms of the extensions of multilinear operators to the biduals is obtained for the first time multilinear characterizations of the reciprocal dunford pettis property and pelczynski s property v are also found polynomial and holomorphic versions of these properties are given as well

landesman lazer conditions for resonant p laplacian problems with jumping nonlinearities we study the existence of solutions of the p laplacian dirichlet problem 1 p u p u h x u f x u u x 0 1 2 u 0 u 1 0 where r p 1 s s 1sgn for s 1 r the function h 0 1 r r has the form h x a x p a x p x 0 1 r with max 0 and a l1 0 1 and the function f 0 1 r2 r is continuous and satisfies f x k x 1 q 1 x 0 1 r2 for some q 1 p and k l1 0 1 the dominant asymptotic behaviour of equation 1 as u is determined by the coefficients a and we allow a a in which case the problem is said to be jumping if the positively homogeneous problem obtained from 1 2 by setting f 0 has a non trivial solution then the problem is said to be resonant and is said to be a half eigenvalue assuming that the problem 1 2 is both jumping and resonant we will obtain a solution under certain landesman lazer conditions on f

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

theorems of hardy and paley for vector valued analytic functions and related classes of banach spaces we investigate the classes of banach spaces where analogues of the classical hardy inequality and the paley gap theorem hold for vector valued functions we show that the vector valued paley theorem is valid for a large class of banach spaces necessarily of cotype 2 which includes all banach lattices of cotype 2 all banach spaces whose dual is of type 2 and also the preduals of c algebras for the trace class s1 and the dual of the algebra of all bounded operators on a hilbert space a stronger result holds namely the vector valued analogue of the fefferman theorem on multipliers from h into in particular for the latter spaces the vector valued hardy inequality holds this inequality is also true for every banach space of type 1 bourgain 0 introduction if f z oajeiit is an analytic trigonometric polynomial then j o j elajl j 1 1 c1 r if t i dt

resonance in nonlinear wave equations with x dependent coefficients abstract this paper is devoted to the existence results for the one dimensional wave equation with x dependent coefficients when resonance occurs at the eigenvalue r n by using the topological degree theory the authors get a result which is similar to that in the literature

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

multiple solutions of a class of neumann problem for semilinear elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations with unbounded nonlinearity by the implicit function theorem and the morse theory

l estimates for the bar partial problem in a half planeproblem in a half plane l approche de dualite pour trouver des solutions de f est suffisante pour plusieurs problemes apparaissant en theorie h p on donne des operateurs solution qui permettent de resoudre des problemes non resolubles par la dualite

roots of unity and unbounded motions of an asymmetric oscillator r 2 z the general questions that we have in mind are under what conditions ona b and f t can we say that 1 has unbounded solutions are there equationsin the class 1 where unbounded and periodic solutions can coexist equation 1 was first considered by dancer in 2 3 and 4 and byfucik in 7 they studied the periodic and dirichlet boundary valueproblems for 1 and looked at this equation as a model of the so called equations with jumping nonlinearities we sum up some of the results in 2 3 consider the family of curves in the a b parameter space 1 a 1 b 2p p 1 2 2 when a b is not in any of these curves then 1 has at least one 2 periodic solution however if a b lies in one of these curves there existfunctions f t for which 1 has no periodic solution more results on theperiodic problem for 1 have been obtained by lazer and mckenna 8 fabry 5 and fabry and fonda 6 the known results on the existenceof periodic solutions already shed some light on the boundedness problem actually the second theorem of massera implies that if 1 has no periodic

on the resonance problem with nonlinearity which has arbitrary linear growth abstract this paper deals with the nonlinear two point boundary value problem at resonance even nonlinearities g with an arbitrary linear growth in resp may be considered but only on the cost of the corresponding bound on their linear growth at resp it generalizes the previous results in this direction obtained by m schechter j shapiro and m snow trans amer math soc 241 1978 69 78 l cesari and r kannan proc amer math soc 88 1983 605 613 and s ahmad proc amer math soc 93 1984 381 384

nonlinear perturbations of differential operators with nontrivial kernel and applications to third order periodic boundary value problems this paper deals with the solvability of the nonlinear operator equations in normed spaces yx egx j where dp is a linear map with possible nontrivial kernel applications are given to the existence of periodic solutions for the thirdorder scalar differential equation x ax bx cx g t x p t under various conditions on the interaction of g t x x with spectral contigurations of a b and c

arens regularity of banach algebras and the geometry of banach spaces we show that a banach algebra a such that a has the property v of a pelczynski is arens regular a strong uniqueness property of the extension of the product on a unital c algebra a to a is proved the algebraic structure of the bidual of a c algebra can be obtained through the local reflexivity principle we give examples which show that the results are sharp

periodic solutions of some resonant hamiltonian systems abstract based on the galerkin approximation and the morse inequality an existence theorem of periodic solutions of the hamiltonian system j z h t z is given where h t z is asymptotically linear and resonant at infinity existence of periodic solutions of the resonant asymmetric p laplacian equation is also discussed

zero separating invariants for finite groups abstract we fix a field k of characteristic p for a finite group g denote by g and g respectively the minimal number d such that for every finite dimensional representation v of g over k and every v v g 0 or v v 0 respectively there exists a homogeneous invariant f k v g of positive degree at most d such that f v 0 let p be a sylow p subgroup of g which we take to be trivial if the group order is not divisible by p we show that g p if n g p p is cyclic we show g n g p if g is p nilpotent and p is not normal in g we show g g l where l is the smallest prime divisor of g these results extend known results in the non modular case to the modular case

hankel operators on bounded analytic functions for u a domain in the complex plane and g a bounded measurable function on u the generalized hankel operator sg on h u is the operator of multiplication by g followed by projection into l h under certain conditions on u we show that either sg is compact or there is an embedded on which sg is bicontinuous we characterize those g s for which sg is compact in the case that u is a behrens roadrunner domain

nonlinear boundary value problems at resonance soit un espace de mesure fini et h l 2 l espace des fonctions a valeurs reelles de carre integrable mesurables de lebesgue sur on considere l equation lu r n u g u h ou h h l dom l h h est un operateur lineaire auto adjoint densement defini r n appartient au spectre l g r r est une fonction qui peut etre non lineaire on donne des resultats d existence generaux

double resonance with landesman lazer conditions for planar systems of ordinary differential equations abstract we prove the existence of periodic solutions for first order planar systems at resonance the nonlinearity is indeed allowed to interact with two positively homogeneous hamiltonians both at resonance and some kind of landesman lazer conditions are assumed at both sides we are thus able to obtain as particular cases the existence results proposed in the pioneering papers by lazer and leach 1969 27 and by frederickson and lazer 1969 18 our theorem also applies in the case of asymptotically piecewise linear systems and in particular generalizes fabry s results in fabry 1995 10 for scalar equations with double resonance with respect to the dancer fucik spectrum

separating invariants for 2 2 matrices abstract a minimal separating set is found for the algebra of matrix invariants of several 2 2 matrices over an infinite field of arbitrary characteristic

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

landesman lazer condition for nonlinear problems with jumping nonlinearities abstract this paper was motivated by the result of iannacci and nkashama j differential equations 69 1987 289 309 concerning the existence of at least one solution for the differential equation x t m 2 x t g t x t e t with periodic boundary data x 0 x 2 x 0 x 2 0 where m 0 is an integer e is integrable and g satisfies caratheodory s conditions we intend to prove that the landesman lazer type condition is sufficient for solvability of this periodic problem under rather more general assumptions on the growth of the nonlinear term g particularly our hypotheses cover the nonlinearities which may jump over the eigenvalues different from m 2 while the assumptions of iannacci and nkashama allow g only to asymptotically touch the neighbour eigenvalues our approach may be used also for two point boundary value problems the proofs are based on the leray schauder degree theory and on the shooting method for ordinary differential equations

topology and semilinear equations at resonance in hilbert space this work is devoted to the solvability of equations of the form b lu b nu 0 in a real hilbert space b h the kernel ker b l of the linear operator b l b d b l sub b h rarr b h may be infinite dimensional and the range of b l b r b l is such that b r b l ker b l sup perp

the complex convexity of quasi normed linear spaces abstract a notion of uniform convexity is defined for quasi normed complex spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space complex uniformly convex spaces c u c spaces always have cotype in banach lattices possessing cotype and being complex uniformly convex are the same duals of c algebras are c u c and in fact have cotype 2 the behavior of certain martingale difference sequences in c u c spaces is examined this leads to an isomorphic description of the spaces having the property

nonlinear resonance in asymmetric oscillators we study the periodic solutions of equations with asymmetric nonlinearities at resonance with the fucik spectrum we compute the associated topological degree and prove existence multiplicity and stability of large amplitude oscillations for equations with a small friction term such equations can be viewed e g as simple models for investigating vertical oscillations of long span suspension bridges the results are typically of a nonlinear nature as some of the situations observed cannot occur with a linear equation c 1998 academic press

periodic and unbounded motions in asymmetric oscillators at resonance abstract we consider the existence of periodic and unbounded solutions for the asymmetric oscillator x a x b x g x p t where x max x 0 x max x 0 a and b are two positive constants p t is a 2 periodic smooth function and g x satisfies lim x x 1 g x 0 we have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well defined functions p and h in this paper we consider some critical cases and obtain some new sufficient conditions for the existence of 2 periodic and unbounded solutions in particular the obtained periodic solutions are new and cannot be deduced from landesman lazer s and fabry fonda s existence conditions in contrast with many existing results the function g x may be unbounded or oscillatory without asymptotic limits

a joint functional calculus for sectorial operators with commuting resolvents 1 introduction in this paper we are concerned with h 1 functional calculus in the sense of the construction introduced by mcintosh 23 and developed recently by cowling doust mcintosh and yagi 8 given v in 0 p we will use the notation sv fz 2 c n f0g jarg z j vg to denote the open sector of angle 2v around the half line r let x be a complex banach space given a linear operator a on x we denote by d a n a and r a the domain the kernel and the range of a respectively we also denote by j a the spectrum of a and by r a the resolvent set of a

a nonlinear elliptic problem at resonance with a nonsimple eigenvalue where z irn is an open and bounded domain with smooth boundary a is a nonsimple eigenvalue of a g is a real valued function and h e lm f2 after the 1 9 7 0 paper of landesman and lazer i the problem has been intensively researched in several directions for the case in which a is a simple eigenvalue see 2 2 1 1 and the references therein for the case in which a is nonsimple see 6 2 2 2 5 and the references therein it is common in the literature to distinguish the following situations i g t a as t f a with a a 0 o this is the landesman lazer case 2 g t 0 as it1 a and

unbounded perturbations of forced second order ordinary differential equations at resonance we prove the existence of at least one solution for the differential equation x t m2x t g t x t e t with periodicity conditions x 0 x 2 x 0 x 2 0 where m 0 is an integer e is integrable and g satisfies caratheodory conditions our results are obtained for the case when there is resonance at the eigenvalue m2 of the linear second order differential equation t x t 0 r with x 0 x 2 x 0 x 2 0 the function g may be unbounded and touching of the eigenvalue m 1 2 resp m 1 2 if m 0 on a subset of positive measure is allowed our approach also works when periodicity conditions are replaced by dirichlet or neumann boundary conditions the proofs are based on topological degree mawhin s continuation theorem and leray schauder techniques

on resonant neumann problems we consider semilinear neumann problems at resonance and prove existence and multiplicity theorems the existence theorems allow resonance with respect to any eigenvalue of the negative neumann laplacian the multiplicity theorems concern problems resonant at 0 the principal eigenvalue or at the first nonzero eigenvalue our approach uses tools from critical point theory and from morse theory

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

multiple solutions of neumann problem for elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations at resonance or nonresonance with unbounded nonlinearity by the reduction method and the critical point theory in particular a three critical point theorem proposed by brezis nirenberg

positively homogeneous hamiltonian systems in the plane i try to give a general description of the dynamics of the solutions for a planar hamiltonian system with positively homogeneous hamiltonian function and periodic forcing term most of the results obtained are already known in the special case of a scalar second order differential equation with asymmetric nonlinearity

on periodic solutions of nonlinear second order vector differential equations this paper considers existence of periodic solutions for vector lienard differential equations in our main result we write where q t x is a symmetric matrix and h t x is sublinear the key assumption relates the asymptotic behaviour as x of the eigenvalues of q t x to the spectrum of the linear operator d 2 dt 2 several choices for q t x are considered which lead to known theorems and extend others in the case of the duffing equation the assumptions are weakened our approach is based on leray schauder s degree theory and a priori estimates

nonlinear second order elliptic partial differential equations at resonance on etudie la resolubilite de problemes aux valeurs limites pour des equations aux derivees partielles elliptiques d ordre 2 semi lineaires de type resonance dans lesquelles la perturbation non lineaire ne satisfait pas necessairement la condition de landesman lazer ou l hypothese de monotonicite

hahn banach extension of multilinear forms and summability the aim of this paper is to investigate close relations between the validity of hahn banach extension theorems for multilinear forms on banach spaces and summability properties of sequences from these spaces a case of particular importance occurs when we consider banach spaces which have the property that every bilinear form extends to any superspace

structure of subspaces of the compact operators having the dunford pettis property the structure of the subspaces m subset k ell p having the dunford pettis property dpp is studied where k ell p is the space of all compact operators on ell p and 1 p infty the following conditions are shown to be equivalent i m has the dpp ii m is isomorphic to a subspace of c 0 iii the sets sx s in b m subset ell p and s x s in b m subset ell p are relatively compact for all x in ell p and x in ell p the equivalence between i and iii was recently proven in the case of arbitrary hilbert spaces by brown and ulger it is also shown that i and ii are equivalent for subspaces m subset k ell p 1 oplus ldots oplus ell p k this result is optimal in the sense that for 1 p q infty there is a dpp subspace m subset k ell q ell p that fails to be isomorphic to a subspace of c 0

a polynomially bounded operator on hilbert space which is not similar to a contraction let e 0 we prove that there exists an operator te e2 2 such that for any polynomial p we have iip te ii not similar to a contraction i e there does not exist an invertible operator s f2 e f2 such that jis 1tesii tributed to halmos after his well known 1970 paper ten problems in hilbert space we also give some related finite dimensional estimates department of mathematics texas a m university college station texas 77843 universite paris vi equipe d7analyse case 186 75252 paris cedex 05 france e mail address gipqccr jussieu fr this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 52 47 utc all use subject to http about jstor org terms

on the minimal periodic solutions of nonconvex superlinear hamiltonian systems abstract this paper proves a multiplicity result for the minimal periodic solutions of hamiltonian system x t j h x t under a superquadratic hypothesis on h weaker than the ambrosetti rabinowitz type condition we also define a class of homogeneous functions that are more general than the classical ones and prove the rabinowitz s conjecture in the case of h satisfying such new homogeneity

existence uniqueness and stability of oscillations in differential equations with asymmetric nonlinearities we give conditions for the existence uniqueness and asymptotic stability of periodic solutions of a second order differential equation with piecewise linear restoring and 27r periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer with suitable restrictions on the restoring and forcing in the undamped case we give a necessary and sufficient condition

periodic solutions of perturbed isochronous hamiltonian systems at resonance we look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system precise computations of the topological degree are obtained by elementary phase plane analysis c 2005 elsevier inc all rights reserved

rigidity of composition operators on the hardy space h p abstract let be an analytic map taking the unit disk d into itself we establish that the class of composition operators f c f f exhibits a rather strong rigidity of non compact behaviour on the hardy space h p for 1 p and p 2 our main result is the following trichotomy which states that exactly one of the following alternatives holds i c is a compact operator h p h p ii c fixes a linearly isomorphic copy of l p in h p but c does not fix any copies of l 2 in h p iii c fixes a copy of l 2 in h p moreover in case iii the operator c actually fixes a copy of l p 0 1 in h p provided p 1 we reinterpret these results in terms of norm closed ideals of the bounded linear operators on h p which contain the compact operators k h p in particular the class of composition operators on h p does not reflect the quite complicated lattice structure of such ideals

strict singularity of volterra type operators on hardy spaces abstract in this paper we first characterize the boundedness and compactness of volterra type operator s g f z 0 z f g d z d defined on hardy spaces h p 0 p the spectrum of s g is also obtained then we prove that s g fixes an isomorphic copy of l p and an isomorphic copy of l 2 if the operator s g is not compact on h p 1 p in particular this implies that the strict singularity of the operator s g coincides with the compactness of the operator s g on h p at last we post an open question for further study

operators with an absolute functional calculus we study sectorial operators with a special type of functional calculus which we term an absolute functional calculus a typical example of such an operator is an invertible operator a defined on a banach space x considered on the real inter polation space dom a x p where 0 1 and 1 p in general the absolute functional calculus can be characterized in terms of real interpolation spaces we show that operators of this type have a strong form of the h calculus and behave very well with respect to the joint functional calculus we give applications of these results to recent work of arendt batty and bu on the existence of holder continuous solutions for the abstract cauchy problem