berezin toeplitz quantization and composition formulas abstract extending results in l a coburn the measure algebra of the heisenberg group j funct anal 161 1999 509 525 l a coburn on the berezin toeplitz calculus proc amer math soc 129 11 2001 3331 3338 we derive composition formulas for berezin toeplitz operators with i g unbounded symbols in the range of certain integral transforms the question whether a finite product of berezin toeplitz operators is an operator of this type again can be answered affirmatively in several cases but there are also well known counter examples we explain some consequences of such formulas to c algebras generated by toeplitz operators

quantization of edge currents for continuous magnetic operators abstract for a magnetic hamiltonian on a half plane given by the sum of the landau operator with dirichlet boundary conditions and a random potential a quantization theorem for the edge currents is proven this shows that the concept of edge channels also makes sense in presence of disorder moreover gaussian bounds on the heat kernel and its covariant derivatives are obtained

algebraic properties and the finite rank problem for toeplitz operators on the segal bargmann space abstract we study three different problems in the area of toeplitz operators on the segal bargmann space in c n extending results obtained previously by the first author and y l lee and by the second author we first determine the commutant of a given toeplitz operator with a radial symbol belonging to the class s y m 0 c n of symbols having certain growth at infinity we then provide explicit examples of zero products of non trivial toeplitz operators these examples show the essential difference between toeplitz operators on the segal bargmann space and on the bergman space over the unit ball finally we discuss the finite rank problem we show that there are no non trivial rank one toeplitz operators t f for f s y m 0 c n in all these problems the growth at infinity of the symbols plays a crucial role

gap labelling and the pressure on the boundary in quantum systems described by covariant families of 1 particle schrodinger operators on half spaces the pressure on the boundary per unit energy is topologically quantised if the fermi energy lies in a gap of the bulk spectrum its relation with the integrated density of states can be expressed in an integrated version of streda s formula this leads also to a gap labelling theorem for systems with constant magnetic field the proof uses non commutative topology

commuting toeplitz operators on the segal bargmann space abstract consider two toeplitz operators t g t f on the segal bargmann space over the complex plane let us assume that g is a radial function and both operators commute under certain growth condition at infinity of f and g we show that f must be radial as well we give a counterexample of this fact in case of bounded toeplitz operators but a fast growing radial symbol g in this case the vanishing commutator t g t f 0 does not imply the radial dependence of f finally we consider toeplitz operators on the segal bargmann space over c n and n 1 where the commuting property of toeplitz operators can be realized more easily

commutative c algebras of toeplitz operators and quantization on the unit disk abstract a family of recently discovered commutative c algebras of toeplitz operators on the unit disk can be classified as follows each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles the orthogonal trajectories to lines forming a pencil the c algebra generated by toeplitz operators with such symbols turns out to be commutative we show that these cases are the only possible ones which generate the commutative c algebras of toeplitz operators on each weighted bergman space

commuting toeplitz operators with quasi homogeneous symbols on the segal bargmann space abstract let t t z l z k with l k n 0 be a toeplitz operator with monomial symbol acting on the segal bargmann space over the complex plane we determine the symbols of polynomial growth at infinity such that t and t z l z k commute on the space of all holomorphic polynomials by using polar coordinates we represent as an infinite sum r e i j j r e i j then we are able to reduce the above problem to the case of quasi homogeneous symbols j e i j we obtain the radial part j r in terms of the inverse mellin transform of an expression which is a product of gamma functions and a trigonometric polynomial if we allow operator symbols of higher growth at infinity we point out that in some of the cases more than one toeplitz operator t j e i j exists commuting with t

boundary maps for c crossed products with r with an application to the quantum hall effect the boundary map in k theory arising from the wiener hopf extension of a crossed product algebra with r is the connes thom isomorphism in this article the wiener hopf extension is combined with the heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces and cyclic cohomol ogy it then follows directly from a non commutative stokes theorem that this map is dual w r t connes pairing of cyclic cohomology with k theory as an application we prove equality of quantized bulk and edge conductivities for the integer quantum hall effect described by continuous magnetic schrodinger operators 1 motivation and main result in a commonly used approach to study aperiodic solids particles in the bulk of the medium are described by covariant families of one particle schrodinger operators h whereis the probability space of configurations furnished with an ergodic action of space translations crossed product algebras provide a natural framework for such families be86 in particular their bounded functions are represented by elements of a c crossed product the so called bulk algebra the non commutative topology of the c algebra is a useful tool to construct topological invariants resulting from pairings between k group elements and higher traces some of these invariants may be physi cally interpreted as topologically quantised quantities the quantised hall conductivity is such an example the physics near a boundary of the solid can also be described by a c algebra the so called edge algebra the bulk algebra being essentially a crossed product of the edge algebra with r or with z in the tight binding approximation krs02 both algebras are tied together in the wiener hopf extension or respectively the toeplitz extension this extension gives rise to boundary maps between the k groups and the higher traces of the bulk and edge algebra which allow one to equate bulk and edge in variants this topological relation and its physical interpretations is our main objective we discuss one prominent physical example of this the quantum hall effect where the

an invariant volume mean value property if is harmonic and integrable over the open unit disc u then so is for every moebius transformation of u and therefore 1 u d a 0 for every conversely does this mean value property imply that is harmonic a more general question with the unit ball bn of c for arbitrary n 1 in place of the disc is investigated in the present paper the answer is found to be affirmative if n 11 negative if n 12

asymptotic behavior of bergman kernels with logarithmic weight abstract we describe singularities of weighted bergman kernels on the unit disc with respect to radial logarithmic weights

toeplitz and hankel operators with l 1 symbols on dirichlet space abstract we show that toeplitz or small hankel operators with symbols in l 1 is a generalization of the case with the harmonic symbol in c d d where d is the dirichlet space on d we also give a decomposition theorem for the sobolev space of first order on d using this result some characterizations for algebraic properties of toeplitz or small hankel operators with symbols in l 1 are given

commutants of toeplitz operators with radial symbols on the fock sobolev space abstract in the setting of the fock space over the complex plane bauer and lee have recently characterized commutants of toeplitz operators with radial symbols under the assumption that symbols have at most polynomial growth at infinity their characterization states if one of the symbols of two commuting toeplitz operators is nonconstant and radial then the other must be also radial we extend this result to the fock sobolev spaces

on algebras generated by toeplitz operators and their representations abstract we study banach and c algebras generated by toeplitz operators acting on weighted bergman spaces a 2 b 2 over the complex unit ball b 2 c 2 our key point is an orthogonal decomposition of a 2 b 2 into a countable sum of infinite dimensional spaces each one of which can be identified with a differently weighted bergman space a 2 d over the complex unit disk d moreover all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a toeplitz operator on a 2 d the symbols of the generating toeplitz operators are chosen to be suitable extensions to b 2 of families s of bounded functions on d symbol classes s that generate important classical commutative and non commutative toeplitz algebras in l a 2 d are of particular interest in this paper we discuss various examples in the case of s c d and s c d l 0 1 we characterize all irreducible representations of the resulting toeplitz operator c algebras their calkin algebras are described and index formulas are provided

commuting toeplitz operators on the hardy space of the bidisk on the hardy space of the bidisk we give a necessary and sufficient condition for a bounded symbol of a toeplitz operator that commutes with another toeplitz operator whose symbol is a certain type of bounded symbol

mellin transform monomial symbols and commuting toeplitz operators abstract in this paper we give necessary and sufficient conditions for a symbol that produces a toeplitz operator on the bergman space that commutes with another such operator whose symbol is a monomial the result is stated in terms of the mellin transform of the symbol as a corollary we show that a toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol

commuting toeplitz operators with pluriharmonic symbols let da z denote the lebesgue volume measure on the open unit ball bn f cn normalized so that the measure of bn equals 1 the 13ergman space l2 bn is the eiilbert space consisting of holomorphic functions on bn that are also in l2 bn da for f in l90 bn the toeplitz operator tf with symbol f is desned by t h p fh where p is the orthogonal projection from l2 bnnda onto ol2tbesda n called the bergman projection let dsr be the surface area measure on the unit sphere sn for 1 p 007 h s4 is the banach space of holomorphic functions on bn with the norm defined by

gleason parts and cop let h denote the banach algebra of bounded analytic functions on the open unit disc d and let m h denote the maximal ideal space of h one of the results in this paper shows that if two gleason parts in m h have distinct closures then there exists an h function which is identically zero on one part and nonzero on the other from this one obtains a corona theorem for certain subalgebras of h let cop denote the algebra of bounded analytic functions that are in the little bloch space it is shown here that there exists a cop function h and an inner function i such that i divides h in h1 but the result is not in cop

toeplitz operators defined by sesquilinear forms fock space case the classical theory of toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question a further extension of the theory was made for symbols being unbounded functions measures and compactly supported distributions all of them subject to some restrictions in the context of a reproducing kernel hilbert space we propose a certain framework for a maximally possible extension of the notion of toeplitz operators for a maximally wide class of highly singular symbols using the language of sesquilinear forms we describe a certain common pattern for a variety of analytically defined forms which besides the covering of all previously considered cases permits us to introduce a further substantial extension of a class of admissible symbols that generate bounded toeplitz operators although our approach is unified for all reproducing kernel hilbert spaces for concrete operator consideration in this paper we restrict ourselves to toeplitz operators acting on the standard fock or segal bargmann space

a class of globally controllable semilinear heat equations with superlinear terms abstract it is well known nowadays that a rather general semilinear parabolic equation governed in a bounded domain by boundary or locally distributed controls can be made globally approximately controllable in l p type spaces provided the nonlinearity is globally lipschitz however the question whether this property can possibly take place in the superlinear case in several space dimensions seems to be open the goal of this article is to describe a special class of semilinear heat equations with superlinear terms for which this property indeed holds our technique combines the asymptotic method of a y khapalov 1995 j math anal appl 194 858 882 with certain a priori estimates for the corresponding truncated spectral problem

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

toeplitz products with pluriharmonic symbols on the hardy space over the ball on the hardy space over the unit ball in cn we consider operators which have the form of a finite sum of products of several toeplitz operators we study characterizing problems of when such an operator is compact or of finite rank some of our results show higher dimensional phenomena

hankel operators and toeplitz operators on the bergman space abstract we consider in this paper the question of when the semi commutator t fg t f t g on the bergman space with bounded harmonic symbols is compact several conditions equivalent to compactness of t fg t f t g are given as a consequence we prove a conjecture of axler that for bounded analytic functions f and g on the unit disk t fg t g t f is compact iff either f or g is constant on each gleason part p m except d

algebraic properties of toeplitz operators on generalized fock spaces on cd abstract we study two problems involving algebraic properties of toeplitz operators on generalized fock spaces on c d with weights of the form z 2 s e z 2 m m 1 s 0 we determine the commutant of a given toeplitz operator with a radial symbol which satisfies certain growth conditions we also discuss the equation t f t g 0 when f or g is radial

regional observability for distributed semi linear hyperbolic systems the objective of this paper is to study the notion of regional observability of hyperbolic semilinear systems for the state reconstruction in a given subregion we give a first approach which combines the extension of the hum method and the fixed point techniques the analytical case is then explored using sectorial property of the considered dynamic operator and converted to a fixed point problem the two approaches lead to algorithms which are successfully implemented numerically and illustrated with examples and simulations

cohomology of canonical projection tilings we define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in schlottmann s formulation we prove the cohomology to be equivalent to a certain cohomology of the lattice we discuss one of its qualitative features namely that it provides a topological obstruction for a generic tiling to be substitutional we develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2 presenting explicit formulae these in turn give computations for the k theory of certain associated non commutative c algebras

constrained exact controllability of semilinear systems in the paper infinite dimensional dynamical control systems described by semilinear abstract differential equations are considered using a generalized open mapping theorem sufficient conditions for constrained exact local controllability are formulated and proved it is generally assumed that the values of admissible controls are in a convex and closed cone with vertex at zero constrained exact local controllability of semilinear abstract second order dynamical systems are also formulated and proved as an illustrative example constrained exact local controllability problem for semilinear hyperbolic type distributed parameters dynamical system is solved in details some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented

some weighted estimates for the equation and a finite rank theorem for toeplitz operators in the fock space we consider the equation in c1 in classes of functions with gaussian decay at infinity we prove that if the right hand side of the equation is majorated by exp q z 2 with some positive q together with derivatives up to some order and is orthogonal as a distribution to all analytical polynomials then there exists a solution with decays together with derivatives as exp q z 2 for any q q e this result carries over to the equation in classes of distributions again with gaussian decay at infinity in some precisely defined sense the properties of the solution are used further on to prove the finite rank theorem for toeplitz operators with distributional symbols in the fock space the symbol of such operator must be a combination of finitely many distributions and their derivatives the latter result generalizes the recent theorem on finite rank toeplitz operators with symbols functions

micro motions of a 2 d swimming model governed by multiplicative controls abstract the goal of this paper is to derive a formula for the micro motions of a simplified model of an object swimming by itself in a 2 d stokes fluid mathematics wise the model equations are described by a nonlinear coupled hybrid system of fluid equations and integro differential equations we view this swimming motion as one governed by multiplicative controls such problems are of interest in biological and engineering applications dealing with propulsion systems in fluids

local controllability for a swimming model we study the local controllability of a mathematical model of an abstract object which swims in the two dimensional 2d nonstationary stokes fluid we assume that this object consists of finitely many subsequently connected small sets thick points each of which can act upon any of the adjacent sets in a rotation fashion with the purpose of generating its fish or snake like motion we regard the magnitudes of the respective rotation forces entering the system s equations as coefficients as multiplicative or bilinear controls the structural integrity of the object is maintained by the elastic forces acting between the aforementioned adjacent sets according to hooke s law models like this are of a interest in biology and engineering applications dealing with propulsion systems in fluids

finite rank toeplitz operators some extensions of d luecking s theorem abstract the recent theorem by d luecking about finite rank bergman toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions

the finite sum of finite products of toeplitz operators on the polydisk abstract a limit theorem is established for a finite sum of finite products of toeplitz operators on the hardy space of the polydisk as a consequence we show that the product of six toeplitz operators with pluriharmonic symbols is compact iff the product equals zero iff one of these toeplitz operators equals zero

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

commuting toeplitz operators with pluriharmonic symbols on the fock space abstract in the setting of the bergman space over the disk or the ball it has been known that two toeplitz operators with bounded pluriharmonic symbols can semi commute only in the trivial cases in this paper we study the analogues on the fock space over the multi dimensional complex space as is the case in various other settings we are naturally led to the problem of characterizing a certain type of fixed points of the berezin transform for such fixed points we obtain a complete characterization by means of eigenfunctions of the laplacian we also obtain other characterizations in particular it turns out that there are many nontrivial cases on the fock space for semi commuting toeplitz operators with pluriharmonic symbols all in all our results reveal that the situation on the fock space appears to be much more complicated than that on the classical bergman space setting which partly is caused by the unboundedness of the operator symbols some of our results are restricted to the one variable case and the corresponding several variable case is left open

commuting toeplitz operators on the polydisk we obtain characterizations of essentially commuting toeplitz operators with pluriharmonic symbols on the bergman space of the polydisk we show that commuting and essential commuting properties are the same for dimensions bigger than 2 while they are not for dimensions less than or equal to 2 also the corresponding results for semi commutators are obtained

polynomials having leading terms over c2 in the fock space we characterize polynomials p having leading terms over c2 in a generalized sense as applications we study a quasi invariant subspace p generated by p in the fock space and show that a quasi invariant subspace m is similar to p if and only if m q for some polynomial q having the same leading term as p these are generalizations of guo and hou s work

the commutants of certain toeplitz operators on weighted bergman spaces abstract for 1 let a 2 be the corresponding weighted bergman space of the unit ball in c n for a bounded measurable function f let t f be the toeplitz operator with symbol f on a 2 this paper describes all the functions f for which t f commutes with a given t g where g z z 1 l 1 z n l n for strictly positive integers l 1 l n or g z z 1 s 1 z n s n h z for non negative real numbers s 1 s n and a bounded measurable function h on 0 1

exponential dichotomy rotation number and linear differential operators with bounded coefficients on considere la relation entre la dichotomie exponentielle de certains systemes differentiels lineaires ou aux differences et le spectre des operateurs differentiels lineaires auto adjoints associes l on etudie une facon systematique d enumerer les intervalles ouverts dans la resolvante de l

compact vectorial toeplitz operators on the segal bargmann space abstract we provide a sufficient condition for the compactness of a toeplitz operator acting on the segal bargmann space of vector valued functions written in terms of an associated operator valued kernel

functions invariant under the berezin transform we show that a bounded function f satisfies bf f where b is the berezin tranform on the unit disc defined in 2 below if and only if f is harmonic there is an equivalent formulation of this result s axler and cuckovic integral equations operator theory14 1991 1 12 w rudin function theory in the unit ball of cn springer verlag new york berlin 1980 if f is bounded and satifies the invariant version of the area mean value property then f is harmonic the main tool employed is fourier analysis on the lie group of all mobius transformations

the pointwise convergence of fourier series ii strong l1 case for the lacunary carleson operator abstract we prove that the lacunary carleson operator is bounded from l log l to l 1 this result is sharp the proof is based on two newly introduced concepts 1 the time frequency regularization of a measurable set and 2 the set resolution of the time frequency plane at 0 frequency these two concepts will play the central role in providing a special tile decomposition adapted to the interaction between the structure of the lacunary carleson operator and the corresponding structure of a fix measurable set another key insight of our paper is that it provides for the first time a simultaneous treatment of families of tiles with distinct mass parameters this should be regarded as a fundamental feature difficulty of the problem of the pointwise convergence of fourier series near l 1 context in which unlike the standard l p p 1 case no decay in the mass parameter is possible

micromotions and controllability of a swimming model in an incompressible fluid governed by 2 d or 3 d navier stokes equations abstract we study the local controllability properties of generic 2 d and 3 d bio mimetic swimmers employing the change of their geometric shape to propel themselves in an incompressible fluid described by the navier stokes equations it is assumed that swimmers bodies consist of finitely many parts identified with the fluid they occupy that are subsequently linked by the rotational and elastic internal forces these forces are explicitly described and serve as the means to affect the geometric configuration of swimmers bodies similar models were previously introduced and investigated in 3 4 5 6 7 8 9 10

commuting toeplitz operators on the bidisk abstract a necessary and sufficient condition is obtained for two toeplitz operators to be commuting on the hardy space of the bidisk the main tool is the berezin transform and the harmonic extension

commutants of separately radial toeplitz operators in several variables if is a bounded separately radial function on the unit ball the toeplitz operator t is diagonalizable with respect to the standard orthogonal basis of monomials on the bergman space given such a we characterize bounded functions for which t commutes with t several examples are given to illustrate our results

commuting dual toeplitz operators with pluriharmonic symbols abstract in this paper we completely characterize commuting dual toeplitz operators with bounded pluriharmonic symbols on the bergman space of the unit ball we show that for and pluriharmonic s s s on l a 2 b n only in the trivial case here the trivial case is or holomorphic

rotation numbers boundary forces and gap labelling we review the johnson moser rotation number and the k0 theoretical gap labelling of bellissard for one dimensional schr odinger operators we compare them with two further gap labels one being related to the motion of dirichlet eigenvalues the other being a k1 theoretical gap label we argue that the latter provides a natural generalization of the johnson moser rotation number to higher dimensions

commuting toeplitz operators on the dirichlet space in this paper we study the commutativity of toeplitz operators with continuous symbols on the dirichlet space first under a mild condition concerning absolute continuity we characterize semi commuting toeplitz operators this is a generalization of the case of harmonic symbols also if one of the symbol is radial or analytic we get another characterization which is different from the case on the bergman space

toeplitz operators on the segal bargmann space in this paper we give a complete characterization of those functions on 2n dimensional euclidean space for which the berezin toeplitz quantizations admit a symbol calculus modulo the compact operators the functions in question are characterized by a condition of small oscillation at infinity

unbounded toeplitz operators in the segal bargmann space ii abstract the paper mostly deals with the questions of closedness and essential selfadjointness of toeplitz operators in the segal bargmann space general criteria for their closedness are formulated examples of unclosed toeplitz operators are given explicit formulas for their adjoints are shown for various classes of symbols the problem of whether polynomials and exponents form cores for toeplitz operators is investigated the results presented here improve and extend the ones from an earlier paper

local controllability and non controllability for a 1d wave equation with bilinear control abstract we consider a linear wave equation on the interval 0 1 with bilinear control and neumann boundary conditions we study the controllability of this nonlinear control system locally around a constant reference trajectory we prove that the following results hold generically for every t 2 this system is locally controllable in h 3 h 2 in time t with controls in l 2 0 t r for t 2 this system is locally controllable up to codimension one in h 3 h 2 in time t with controls in l 2 0 t r the reachable set is locally a non flat submanifold of h 3 h 2 with codimension one for every t 2 this system is not locally controllable more precisely the reachable set with controls in l 2 0 t r is contained in a non flat submanifold of h 3 h 2 with infinite codimension the proof of these results relies on the inverse mapping theorem and second order expansions