asymptotic type for sectorial operators and an integral of fractional powers abstract there is a standard notion of type for a sectorial linear operator acting in a banach space we introduce a notion of asymptotic type for a linear operator v involving estimates on the resolvent i v 1 as 0 we show for example that if v is sectorial and of asymptotic type then the fractional power v is of asymptotic type for a suitable range of positive moreover we establish various properties of the operator 0 1 d v in particular this operator is of asymptotic type 0 for a sectorial operator v this result has an application to the construction of operators satisfying the well known ritt resolvent condition

harmonic functions on hypergroups abstract we initiate a study of harmonic functions on hypergroups in particular we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a liouville theorem for bounded harmonic functions further positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions for arbitrary hypergroups we derive a harnack inequality for positive harmonic functions and prove a liouville theorem for compact hypergroups we discuss an application to harmonic spherical functions

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

time regularity for random walks on locally compact groups let g be a compactly generated locally compact group and let t be the operator of convolution with a probability measure on g our main results give sufficient conditions on for the operator t to be analytic in l p g 1 p where analyticity means that one has an estimate of form vert i t t n vert leq cn 1 for all n 1 2 in l p operator norm counterexamples show that analyticity may not hold if some of the conditions are not satisfied

some remarks on operator norm convergence of the trotter product formula on banach spaces abstract we extend the trotter kato chernoff theory of strong approximation of c 0 semigroups on banach spaces to operator norm approximation of analytic semigroups with error estimate as application we obtain a criterion for the operator norm convergence of the trotter product formula on banach spaces with error estimate n 1 log n provided one of the generators has a bounded h functional calculus for both results we present versions for approximation in operator ideal norms such as the trace norm or the hilbert schmidt norm finally we give some remarks on the operator norm convergence of the trotter product formula for semigroups acting on a scale of l p spaces

on a class of self adjoint elliptic operators in l 2 spaces with respect to invariant measures abstract we consider the operator a u u 2 d u d u where u is a convex real function defined in a convex open set r n and lim x u x setting d x exp 2 u x d x we prove that the realization of a in l 2 with domain u h 2 d u d u l 2 u n 0 at 1 is a self adjoint dissipative operator here 1 is the set of points y in the boundary of such that lim sup x y u x then we discuss several properties of a and of the measure including poincare and log sobolev inequalities in h 1

coercive boundary value problems for regular degenerate differential operator equations abstract this study focuses on nonlocal boundary value problems bvp for degenerate elliptic differential operator equations doe that are defined in banach valued function spaces where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients several conditions obtained that guarantee the maximal l p regularity and fredholmness these results are also applied to nonlocal bvp for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties

functional calculus estimates for tadmor ritt operators abstract we show h functional calculus estimates for tadmor ritt operators also known as ritt operators which generalize and improve results by vitse these estimates are in conformity with the best known power bounds for tadmor ritt operators in terms of the constant dependence furthermore it is shown how discrete square function estimates influence the functional calculus estimates

equations d evolution lineaires associees a des semi groupes de contractions dans les espaces lp on demontre l existence d une solution continue dans les espaces l p sous certaines hypotheses

subordinated discrete semigroups of operators given a power bounded linear operator t in a banach space and a probability f on the non negative integers one can form a subordinated operator s k 0 f k t k we obtain asymptotic properties of the subordinated discrete semigroup s n n 1 2 under certain conditions on f in particular we study probabilities f with the property that s satisfies the ritt resolvent condition whenever t is power bounded examples and counterexamples of this property are discussed the hypothesis of power boundedness of t can sometimes be replaced by the weaker kreiss resolvent condition

fixed point properties of semigroups of non expansive mappings abstract in recent years there have been considerable interests in the study of when a closed convex subset k of a banach space has the fixed point property i e whenever t is a non expansive mapping from k into k then k contains a fixed point for t in this paper we shall study fixed point properties of semigroups of non expansive mappings on weakly compact convex subsets of a banach space or more generally a locally convex space by considering the classes of bicyclic semigroups we answer two open questions one posted earlier by the first author in 1976 dalhousie and the other posted by t mitchell in 1984 virginia we also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non expansive mappings related to another open problem raised by the first author in 1976

finite dimensional invariant subspaces for algebras of linear operators and amenable banach algebras abstract we study a finite dimensional invariant subspace property similar to fan s theorem on semigroups for arbitrary banach algebras a in terms of amenability of x a the closed subalgebra of a generated by the set of all maximal elements in a with respect to a character as a consequence we offer some applications to the measure algebra m g and the generalized fourier algebra a p g of a locally compact group g

a unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces abstract in this paper we consider scalar parabolic equations in a general non smooth setting emphasizing interface conditions and mixed boundary conditions in particular we study dynamics and diffusion on a lipschitz interface and on the boundary where the diffusion coefficients are only assumed to be bounded measurable and positive semidefinite in the bulk we consider diffusion coefficients which may degenerate towards a lipschitz surface for this problem class we introduce a unified functional analytic framework based on sesquilinear forms and show maximal l p regularity and bounded h calculus for the corresponding operator providing well posedness for a large class of initial conditions and external forces

maximal regularity for non autonomous schrodinger type equations abstract in this paper we study the maximal regularity property for non autonomous evolution equations t u t a t u t f t u 0 0 if the equation is considered on a hilbert space h and the operators a t are defined by sesquilinear forms a t we prove the maximal regularity under a holder continuity assumption of t a t in the non hilbert space situation we focus on schrodinger type operators a t m t and prove l p l q estimates for a wide class of time and space dependent potentials m

on maximal l p regularity abstract the aim of this paper is to propose weak assumptions to prove maximal l q regularity for cauchy problem d u d t t l u t f t mainly we only require off diagonal estimates on the real semigroup e t l t 0 to obtain maximal l q regularity the main idea is to use one kind of hardy space h 1 adapted to this problem and then use interpolation results these techniques permit us to prove weighted maximal regularity too

common fixed point properties and amenability of a class of banach algebras abstract let a be a lau algebra and let s a be the semigroup of all positive functionals in a with norm one we obtain some equivalent conditions for left amenability of a in terms of a common fixed point property and the hahn banach property we then apply these results for certain lau algebras on a locally compact group g to give characterizations for amenability of g

on maximal parabolic regularity for non autonomous parabolic operators abstract we consider linear inhomogeneous non autonomous parabolic problems associated to sesquilinear forms with discontinuous dependence of time we show that for these problems the property of maximal parabolic regularity can be extrapolated to time integrability exponents r 2 this allows us to prove maximal parabolic l r regularity for discontinuous non autonomous second order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations

discrete maximal parabolic regularity for galerkin finite element methods the main goal of the paper is to establish time semidiscrete and space time fully discrete maximal parabolic regularity for the time discontinuous galerkin solution of linear parabolic equations such estimates have many applications they are essential for example in establishing optimal a priori error estimates in non hilbertian norms without unnatural coupling of spatial mesh sizes with time steps

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

asymptotically non expansive actions of strongly amenable semigroups and fixed points abstract in this paper we deal with some fixed point properties for a semi topological semigroup s acting on a compact convex subset k of a banach space we first focus on the space l m c s of left multiplicatively continuous functions on s and its strong left amenability the existence of a compact left ideal group in the lmc compactification of s we then study the relation between left amenability and strong left amenability of l m c s with a common fixed point property for non expansive and asymptotically non expansive actions of s our results improve a result of t mitchell in 1970 and answer an open problem of a t m lau in 2010 for the class of strongly left amenable semi topological semigroups

almost everywhere convergence of powers of some positive lp contractions abstract we extend the solution of burkholder s conjecture for products of conditional expectations obtained by delyon and delyon for l 2 and by cohen for l p 1 p to the context of badea and lyubich let t be a finite convex combination of operators t j which are products of finitely many conditional expectations then t n f converges a e for every f l p 1 p with sup n t n f l p the proof uses the work of le merdy and xu on positive l p contractions satisfying ritt s resolvent condition as another application of the work of le merdy and xu we extend a result of bellow jones and rosenblatt proving that if a probability a k k z has bounded angular ratio then for every positive invertible isometry s of an l p space 1 p the operator t k z a k s k is a positive l p contraction such that for every f l p t n f converges a e and sup n t n f l p if a k is supported on n the same result is true when s is only a positive contraction of l p similar results are obtained for averages of bounded continuous representations of a compact lca group by positive operators in one l p space 1 p for a positive contraction t on l p which satisfies ritt s condition and f i t l p 0 1 we prove that n t n f 0 a e and sup n n t n f l p

invariant means and semigroups of nonexpansive mappings on uniformly convex banach spaces let s be a semitopological semigroup i e s is a semigroup with hausdorff topology such that for each a s the mappings s sa and s as from s into s are continuous let ruc s denote the space of bounded right uniformly continuous real valued functions on s in 141 t mitchell generalizing a well known fixed point theorem of day s showed that ruc s has a right invariant mean if and only if whenever s x c 4 c is an affine jointly continuous anti action of s on a compact convex subset of a separated locally convex space then c contains a common fixed point for s furthermore an argument similar to that for the proof of theorem 1 in 14 shows that if s has the following fixed point property for dual banach space then ruc s has a right invariant mean

jordan structures in harmonic functions and fourier algebras on homogeneous spaces we study the jordan structures and geometry of bounded matrix valued harmonic functions on a homogeneous space and their analogue the harmonic functionals in the setting of fourier algebras of homogeneous spaces

fixed point properties for semigroups of nonlinear mappings and amenability abstract in this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a banach space we also study a schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space such semigroups properly include the class of extremely left amenable semitopological semigroups the free commutative semigroup on one generator and the bicyclic semigroup s 1 a b a b 1

fixed point properties for semigroups of nonlinear mappings on unbounded sets abstract a well known result of w ray asserts that if c is an unbounded convex subset of a hilbert space then there is a nonexpansive mapping t c c that has no fixed point in this paper we establish some common fixed point properties for a semitopological semigroup s of nonexpansive mappings acting on a closed convex subset c of a hilbert space assuming that there is a point c c with a bounded orbit and assuming that certain subspace of c b s has a left invariant mean left invariant mean or amenability is an important notion in harmonic analysis of semigroups and groups introduced by von neumann in 1929 28 and formalized by day in 1957 5 in our investigation we use the notion of common attractive points introduced recently by s atsushiba and w takahashi

generalised f algebras as predual algebras of jbw triples and characterisations of the fourier algebras on locally compact groups abstract we introduce the notion of a generalised f algebra as a banach algebra a whose banach space dual is a j b w triple with a certain additional condition on a distinguished character of a this generalises the well known notion of f algebras which in turn includes as special cases many important classes of banach algebras on locally compact groups and semigroups our main use of this new notion is to give some axiomatic characterisations of the fourier algebras on locally compact groups strengthening our previous work 37 but we also quickly review some well known results in f algebras that can be extended to the context of generalised f algebras

semilinear evolution equations on discrete time and maximal regularity abstract this paper deals with the existence and stability of solutions for semilinear equations on banach spaces by using recent characterizations of discrete maximal regularity as application we examine the asymptotic behavior of discrete control systems

the ritt property of subordinated operators in the group case abstract let g be a locally compact abelian group let be a regular probability measure on g let x be a banach space let g b x be a bounded strongly continuous representation consider the average or subordinated operator s g t d t x x we show that if x is a umd banach lattice and has bounded angular ratio then s is a ritt operator with a bounded h functional calculus next we show that if is the square of a symmetric probability measure and x is k convex then s is a ritt operator we further show that this assertion is false on any non k convex space x

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

finite dimensional invariant subspace property and amenability for a class of banach algebras anthony to ming lau and yong zhangabstract motivated by a result of ky fan in 1965 we establish a charac terization of a left amenable f algebra which includes the group algebra andthe fourier algebra of a locally compact group and quantum group algebras or more generally the predual algebra of a hopf von neumann algebra interms of a nite dimensional invariant subspace property this is done by rstrevealing a xed point property for the semigroup of norm one positive lin ear functionals in the algebra our result answers an open question posted intokyo in 1993 by the rst author see 25 problem 5 we also show that theleft amenability of an ideal in an f algebra may determine the left amenabilityof the algebra

fixed point properties for semigroup of nonexpansive mappings on frechet spaces abstract in this paper we shall establish a fixed point property on frechet spaces for left reversible semitopological semigroups generalizing some classical results

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

analyticity and discrete maximal regularity on lp spaces abstract in the first part of this paper we give the following interpolation result on the analyticity i e the property t i tn c n for all n n of an operator t on lp if t is powerbounded on lp and lq as well as analytic on lp then t is powerbounded and analytic on lr for all r strictly between p and q this is a discrete analogue of the well known corresponding result for analytic semigroups eta as recently shown by the author the analyticity of t is a necessary condition for the maximal regularity of the discrete time evolution equation un 1 tun fn for all n z u0 0 in the second part of this paper we establish the following two sufficient conditions for its maximal regularity t is a subpositive analytic contraction or t is an integral operator satisfying certain poisson bounds these results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u t au t f t for all t r u 0 0 due to lamberton weis coulhon and duong and hieber and pruss for the poisson bound result of coulhon and duong and hieber and pruss we give a slight improvement and a short proof

characterisations of fourier and fourier stieltjes algebras on locally compact groups abstract motivated by the beautiful work of m a rieffel 1965 and of m e walter 1974 we obtain characterisations of the fourier algebra a g of a locally compact group g in terms of the class of f algebras i e a banach algebra a such that its dual a is a w algebra whose identity is multiplicative on a for example we show that the fourier algebras are precisely those commutative semisimple f algebras that are tauberian contain a nonzero real element and possess a dual semigroup that acts transitively on their spectrums our characterisations fall into three flavours where the first one will be the basis of the other two the first flavour also implies a simple characterisation of when the predual of a hopf von neumann algebra is the fourier algebra of a locally compact group we also obtain similar characterisations of the fourier stieltjes algebras of g en route we prove some new results on the problem of when a subalgebra of a g is the whole algebra and on representations of discrete groups

the c algebra generated by operators with compact support on a locally compact group abstract let g be a locally compact group and vn g be the von neumann algebra generated by the left regular representation of g let ucb denote the c subalgebra generated by operators in vn g with compact support when g is abelian ucb corresponds to the space of bounded uniformly continuous functions on the dual group of g in this paper we prove among other things that for a large class of locally compact groups which include the heisenberg group the ax b group and the motion group the centre of the banach algebra ucb is the fourier stieltjes algebra b g

on ideals in the bidual of the fourier algebra and related algebras let g be a compact nonmetrizable topological group whose local weight b g has uncountable cofinality let h be an amenable locally compact group a g h the fourier algebra of g h and uc2 g h the space of uniformly continuous functionals in vn g h a g h we use weak factorization of operators in the group von neumann algebra vn g h to prove that there exist at least 22b g left ideals of dimensions at least 22b g in a g h and in uc2 g h we show that every nontrivial right ideal in a g h and in uc2 g h has dimension at least 22b g

primary cycles on the circle we consider cycles i e periodic orbits of continuous degree one maps of the circle one cycleaforces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second we characterize primary cycles i e those which force no other cycle of the same period

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

blow up behavior of solutions to a degenerate parabolic parabolic keller segel system qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic parabolic keller segel system the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one m c 2 n 1 n in both cases it is shown that only type ii blow up is possible that is blow up at the same rate as backward self similar solutions never occurs further information on the generation of singularities induced by mass concentration are given in the critical case

invariant submeans and semigroups of nonexpansive mappings on banach spaces with normal structure abstract in this paper we study the relation between invariant submean and normal structure in a banach space this is used to give an improvement and different proof of a fixed point theorem of lim also of belluce and kirk for commutative semigroups for left reversible semigroup of nonexpansive mappings on weakly compact convex subsets of a banach space with normal structure

amenability of hopf von neumann algebras and kac algebras abstract let m be a hopf von neumann algebra the operator predualm ofmis a completely contractive banach algebra with multiplicationm m m m we call m operator amenableif the completely contractive banach algebram isoperator amenable i e for every operatorm bimodulev every completely bounded derivation fromm into the dualm bimodulev is inner there is a weaker notion of amenability introduced by d voiculescu we say that a hopf von neumann algebra m isvoiculescu amenableif there exists a left invariant mean onm we show that if a hopf von neumann algebra m is operator amenable then it is voiculescu amenable for kac algebras there is astrong voiculescu amenability we show that for discrete kac algebras these amenabilities are all equivalent in fact if we letk m be a discrete kac algebra and letk m be its compact dual kac algebra then the following are equivalent 1 kis operator amenable 2 kis voiculescu amenable 3 the von neumann algebramis hyperfinite 4 kis strong voiculescu amenable 5 kis operator amenable 6 m has a bounded approximate identity

fourier stieltjes algebra of a topological group abstract this paper is an invitation to the study of the fourier stieltjes algebra b g the linear span of the continuous positive definite complex valued functions on a topological group g we study b g when g has a host algebra or a group c algebra the analogue of the group c algebra of a locally compact group our main challenge is that a topological group g cannot have a positive regular borel measure which is left translation invariant unless g is locally compact

2m weak amenability of group algebras a common fixed point property for semigroups is applied to show that the group algebra l1 g of a locally compact group g is 2m weakly amenable for each integer m 1

isometries between jb triples let z and w be jb triples and let t be a linear isometry from z into w for any z z with z 1 we show that