one parameter groups of regular quasimultipliers abstract groups of unbounded operators are approached in the setting of the esterle quasimultiplier theory we introduce groups of regular quasimultipliers of growth or groups for short where is a continuous weight on the real line we study the relationship of groups with families of operators and homomorphisms such as regularized distribution and integrated groups holomorphic semigroups and functional calculi some convolution banach algebras of functions with derivatives to fractional order are needed which we construct using the weyl fractional calculus

a spectral multiplier theorem for non self adjoint operators we prove a spectral multiplier theorem for non self adjoint operators more precisely we consider non self adjoint operators a d a c l 2 l 2 having numerical range in a sector w of angle w and whose heat kernel satisfies a gaussian upper bound we prove that for every bounded holomorphic function f on w f a acts on l p with l p norm estimated by the behavior of a finite number of derivatives of f on the boundary of w

oscillatory integrals and lp estimates for schrodinger equations abstract this paper is concerned with schrodinger equations whose principal operators are elliptic under certain degenerate condition we show the estimate of an oscillatory integral related to the solution operator in free case and then employ fractionally integrated groups to obtain the l p estimate of solutions for the initial data belonging to a dense subspace of l p in the case of integrable potentials which improves the corresponding result in m balabane h a emami rad l p estimates for schrodinger evolution equations trans amer math soc 292 1985 357 373

duality for a class of binary operations on 0 1 abstract in this paper we study a duality relation for pairs of a class of binary operations on 0 1 from the solution of a functional equation system solved by de rham in 1956 we obtain several results finally by using a dense set in 0 1 a description of the key function in this work is obtained

minimal order precompensators for decoupling linear multivariable systems a b c e this paper is concerned with the weak inherent coupling problem of the system s a b c e using state space parameters a minimal order precompensator and a static feedback pair f g are computed to decouple the given system the compensated system is shown to be irreducible the algorithm presented can also be used for computing the infinite zero structure from the state space parameters alone moreover the design procedure is simple and applicable to general systems e 0

algebra homomorphisms defined via convoluted semigroups and cosine functions abstract transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions such families are now basic in the study of the abstract cauchy problem the framework they provide is flexible enough to encompass most of the concepts used up to now to treat cauchy problems of the first and second order in general banach spaces starting with the study of the classical laplace convolution and a cosine convolution along with associated dual transforms natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties these correspond to extensions of the cauchy functional equation for semigroups and the abstract d alembert equation for the case of cosine operator functions the algebra homomorphisms obtained provide a way to prove hille yosida type generation theorems for the operator families under consideration

integrated semigroups and linear partial differential equations with delay abstract we study existence and uniqueness of solutions for linear partial differential equations with delay in l p spaces using an approach of batkai and piazzera and a recent perturbation result for integrated semigroups we apply our result to an equation with delay in the highest order derivatives

the wave equation with wentzell robin boundary conditions on l p spaces abstract let a w 1 0 1 a x 0 b c l 0 1 and consider the differential operator a formally given by a u a u b u c u we prove in the first part that a realization of a with wentzell robin boundary conditions on l p 0 1 c 2 generates a cosine function for p 1 in particular we obtain that this realization of a generates a holomorphic c 0 semigroup of angle 2 on the space l 1 0 1 c 2 this solves an open problem by a favini g r goldstein j a goldstein s romanelli and w arendt of crucial importance is the formulation of the boundary conditions we show in the second part that if 0 1 n n 1 is the cube in r n then the laplacian with pure wentzell boundary conditions generates an times integrated cosine function on c for any n 1 2

regularized solutions for abstract volterra equations abstract we introduce a two kernel dependent family of strong continuous operators defined in a banach space which allows us to consider in an unified treatment the notions of among others c 0 semigroups of operators cosine families n times integrated semigroups resolvent families and k generalized solutions the results are applied to the study of existence and uniqueness of solutions for the volterra equation of convolution type u f a au in the case a is not necessarily densely defined examples for equations defined in l p spaces are also given

gaussian bounds for higher order elliptic differential operators with kato type potentials abstract let p d be a nonnegative homogeneous elliptic operator of order 2m with real constant coefficients on r n and v be a suitable real measurable function in this paper we are mainly devoted to establish the gaussian upper bound for schrodinger type semigroup e t h generated by h p d v with kato type perturbing potential v which naturally generalizes the classical result for schrodinger semigroup e t v as v k 2 r n the famous kato potential class our proof significantly depends on the analyticity of the free semigroup e t p d on l 1 r n as a consequence of the gaussian upper bound the l p spectral independence of h is concluded

a class of linear partial neutral functional differential equations with nondense domain abstract we consider a system of linear partial neutral functional differential equations with nondense domain a natural generalized notion of solutions is provided by the integral solutions we derive a variation of constants formula which allows us to transform the integral solutions of the neutral equation to solutions of an abstract volterra integral equation the basic existence and uniqueness results are given and the solutions are shown to generate an integrated semigroup

generation of the integrated semigroups by superelliptic differential operators abstract let a be a superelliptic differential operator of order 2m introduced by e b davies e b davies uniformly elliptic operators with measurable coefficients j funct anal 132 1995 141 169 in the case of 2 m n he obtained the upper gaussian bound of the integral kernel representing e z a z c and the estimates of the l p operator norm of the semigroup for all p 1 the purpose of the present paper is to show that i a k for some constant k 0 generates an integrated semigroup on l p weighted l p space and l p l q to prove this we need norm estimates of e z a z c on each of these spaces also we get another norm estimate of e z a z c on l p when 2 m n without using the integral kernel this norm estimate is better than that in e b davies uniformly elliptic operators with measurable coefficients j funct anal 132 1995 141 169 and gives a better times of the integration of the integrated semigroup

sharp endpoint l p lp estimates for schrodinger groups let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type with a dimension n suppose that the heat operator e tl satisfies the generalized gaussian p 0 p 0 estimates of order m for some 1 le p 0 2 in this paper we prove sharp endpoint l p sobolev bound for the schrodinger group e itl that is for every p in p 0 p 0 there exists a constant c c n p 0 independent of t such that begin aligned left i l s e itl f right p le c 1 t s vert f vert p quad t in mathbb r quad s ge n big 1 over 2 1 over p big end aligned as a consequence the above estimate holds for all 1 p infty when the heat kernel of l satisfies a gaussian upper bound this extends classical results due to feffermann and stein and miyachi for the laplacian on the euclidean spaces mathbb r n we also give an application to obtain an endpoint estimate for l p boundedness of the riesz means of the solutions of the schrodinger equations

on duality and spectral properties of a k regularized resolvents we construct a duality theory for a k regularized resolvents extending some of the known theorems for dual semigroups we present several classes of spaces which in the semigroup case correspond to the favard class and the sun dual space by duality arguments spectral inclusions theorems for regularized resolvents are also obtained

the cauchy problem for critical and subcritical semilinear parabolic equations in lr ii initial data in critical sobolev spaces abstract by presenting some time space l p l r estimates we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the cauchy data in critical sobolev spaces of negative indices our results contain the complex derivative ginzburg landau equation and the cahn hilliard equation as special cases

approximation of local convoluted semigroups abstract we extend the trotter kato theorem on c 0 semigroups to local convoluted semigroups on dual spaces and apply these results to the general banach space setting compared to known results we obtain weaker convergence assumptions on the resolvent

asymptotic behaviour of integrated semigroups abstract in this paper we investigate the asymptotic behaviour of cauchy problems which are not governed by strongly continuous semigroups we obtain interesting generalizations in terms of integrated semigroups of classical abelian and tauberian theorems the results are applied to the solution of the schrodinger equation in l p spaces

estimates for schrodinger evolution equations on demontre que pour des donnees de cauchy dans l 1 r n la solution d une equation d evolution de schrodinger a coefficients constants d ordre 2m est uniformement bornee pour t 0 de borne 1 t c ou c est un entier c n 2m 1

design of precompensators via infinite zero structure for dynamic decoupling of linear invertible systems the problem of designing precompensators necessary for decoupling invertible systems having weak inherent coupling is considered a simple algorithm which is directly applicable to systems characterized by square proper transfer function matrices t s is proposed for such design in the proposed algorithm the concept row reducedness at infinity of t s is utilized as an alternative necessary and sufficient condition for the decouplability of t s in addition to the simple design procedure of the compensator the algorithm yields the infinite zero structure of both t s and the resulting compensated system

laplace transforms and integrated regularized semigroups in locally convex spaces abstract in this paper we consider the vector valued laplace transforms r times r 0 integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces our theorems develop the corresponding results in 1 11 including the well known integrated version of the classical widder s representation theorem of laplace transforms for functions taking values in banach spaces moreover we study a class of differential operators on certain function spaces optimal conditions making them the generators of integrated or regularized semigroups are obtained we finally show some applications to abstract cauchy problems

integrated semigroups and the cauchy problem for systems in lp spaces abstract in this note we prove that under suitable hypotheses every homogeneous differential operator on lp r n n corresponding to a system which is well posed in l2 r n n generates an times integrated semigroup on lp r n n 1 n 1 2 1 p for some special systems of mathematical physics such as the wave equation or maxwell s equations this constant can be improved to be n 1 1 2 1 p

on the duality of aggregation operators and k negations in this paper we study a class of duality functions given by the solution of a system of functional equations related to the de rham system with the aid of a generalized dyadic representation system in the unit interval we study a negation n which is a duality function for pairs of operators satisfying certain boundary conditions new properties of n are investigated including its singularity and fractal dimensions for several related sets as an application we obtain an explicit expression for k negations

rank 1 perturbations of cosine functions and semigroups abstract let a be the generator of a cosine function on a banach space x in many cases for example if x is a umd space a b generates a cosine function for each b l d a 1 2 x if a is unbounded and 1 2 1 then we show that there exists a rank 1 operator b l d a x such that a b does not generate a cosine function the proof depends on a modification of a baire argument due to desch and schappacher it also allows us to prove the following if a b generates a distribution semigroup for each operator b l d a x of rank 1 then a generates a holomorphic c 0 semigroup if a b generates a c 0 semigroup for each operator b l d a x of rank 1 where 0 1 then the semigroup t generated by a is differentiable and t t o t as t 0 for any 1 this is an approximate converse of a perturbation theorem for this class of semigroups

spectra of pseudodifferential operators generating integrated semigroups consider the lp realization opp a of a pseudodifferential operator with symbol a e sp70 having constant coefficients we show that for a certain class of symbols the spectrum of opp a is independent of p this implies that opp a generates an n times integrated semigroup on lp i rn for a certain n if and only if p opp a 8 0 and the numerical range of a is contained in a left half plane our method allows us also to construct examples of operators generating integrated semigroups on lp rn if and only if p is sufficiently close to 2

smith mcmillan factorizations at infinity of rational matrix functions and their control interpretation in this paper the smith mcmillan factorizations at infinity of a transfer function are studied the non uniqueness of these factorizations is characterized via a multiplicative group of bicausal isomorphisms control interpretations are given in terms of system equivalence under some groups of transformations a characterization of the stabilizer of morse group at a b c is given for irreducible systems

schrodinger type evolution equations in lp ω abstract let ap be a strongly elliptic operator of order 2m m n in lp 1 p a bounded domain of rn with dirichlet or neumann boundary conditions of concern is the cauchy problem for schrodinger type evolution equations in lp formula by showing that iap is the generator of a c0 group on a certain interpolation space we obtain results of wellposedness for which are stronger than those derived from the regularized or integrated groups on lp as a by product it is shown that iap is the generator of a ap r regularized group 0 on lp for all formula

different interpretations of triangular norms and related operations triangular norms and triangular conorms and uni norms can be treated as semigroup two place function also as commutative semiring multiplications it can also be derived from the difference operation in a difference poset we discuss here these different interpretations and their consequences

integrated semigroups and higher order abstract equations abstract let e be a banach space b 0 b n 1 closed linear operators on e and f 0 e this paper is devoted to the study of well posedness of the cauchy problem for higher order abstract differential equations formula using the theory of integrated semigroups we obtain some concise and useful criteria

lp lq estimates for dispersive equations and related applications abstract this paper is mainly concerned with the l p l q estimates of solutions for a class of general dispersive equations under the condition h b with an index b 0 1 to the end the new pointwise decay estimates of oscillatory integrals related to the fundamental solutions are proved if b 1 then the pointwise estimates are particularly consistent with the sharp results of the nondegenerate cases moreover as an application of the l p l q estimates we also show that higher order differential operator i p d v x d generates a fractionally integrated group on some l p r n from which certain l p estimates for the solutions of generalized schrodinger equations are obtained

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

integrated semigroups and related partial differential equations abstract we investigate the relationship between abstract linear evolution equations of heat wave and schrodinger types in terms of well posedness in banach spaces more precisely we study our operators as generators of integrated semigroups and integrated cosine functions as applications we consider in a banach space context an abstract laplacian which generalizes the ordinary laplacian in thelp r n spaces we obtain optimal results when compared to the classical situation where the generation results were obtained by 17 using completely different methods our main tools are the boundary value theorem for holomorphic semigroups and the abstract weierstrass formula related partial differential equations have been considered previously by bragg and dettman

a landau kolmogorov inequality for generators of families of bounded operators abstract a landau kolmogorov type inequality for generators of a wide class of strongly continuous families of bounded and linear operators defined on a banach space is shown our approach allows us to recover in a unified way known results about uniformly bounded c 0 semigroups and cosine functions as well as to prove new results for other families of operators in particular if a is the generator of an times integrated family of bounded and linear operators arising from the well posedness of fractional differential equations of order 1 then we prove that the inequality a x 2 8 m 2 2 2 1 2 3 x a 2 x holds for all x d a 2

matrices of operators and regularized semigroups where p is a matr ix of polynomials in n variables and ia1 ia is a family of c o m m u t i n g generators of bounded linear strongly cont inuous groups of operators intuitively one wants to take the exponent ia l of the matr ix of opera to r s p a since the spec t rum of ak is real for all k this means we want the complex matr ix pi j x to be contained in a left half plane in some sense for all x in r the strongest sense is to have the numerical range of the complex matr ix conta ined in a half plane r e z 0 xer a sense that is easier to calculate or at least estimate is to have the spectrum of p d x conta ined in a left half plane for all x r the system 1 1 is then said to be petrowsky correct see 9 the intuit ion we just described is not quite true in general wi thout some