global integral criteria for composition operators abstract let d denote the dirichlet type space of functions analytic on the unit disk u and q the conformal invariant version of this space any analytic self map of u induces a composition operator c acting on d respectively q by c f f where f d respectively f q the aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of

compact composition operators on weighted hilbert spaces of analytic functions we characterize the compactness of composition operators acting on a large family of hilbert spaces of analytic functions which lie between bergman and dirichlet spaces our characterization is given in terms of generalized nevanlinna counting functions

solitons and vertex operators in twisted affine toda field theories affine toda field theories in two dimensions constitute families of integrable relativistically invariant field theories in correspondence with the affine kac moody algebras the particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary here these results are extended from the untwisted to the twisted algebras the new soliton solutions and their masses are found by a folding procedure which can be applied to the affine kac moody algebras themselves to provide new insights into their structures the relevant foldings are related to inner automorphisms of the associated finite dimensional lie group which are calculated explicitly and related to what is known as the twisted coxeter element the fact that the twisted affine kac moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions

hausdorff measures capacities and compact composition operators it is shown that there exist analytic self maps of the unit disc inducing compact composition operators on the hardy space 1 p such that the hausdorff dimension of the set is one sharpening a classical result due to schwartz moreover the same holds in the weighted dirichlet spaces with 0 1 as a consequence we deduce that there exist symbols inducing compact composition operators on such that the capacity of e is positive which is no longer true for those just inducing hilbert schmidt composition operators on

composition operators with univalent symbol in schatten classes abstract we study composition operators induced by a sub domain of the unit disc whose boundary intersects the unit circle at 1 and which has in a neighborhood of 1 a polar equation 1 r see fig 1 we obtain an explicit characterization for the membership in schatten p classes in terms of

glm equations tau function and scattering data the direct and the inverse scattering problem for affine toda mkdv systems is addressed and is found to develop non standard features within the framework of the inverse scattering method a solution scheme based on the tau function formalism is described the inverse problem is shown to be equivalent to a set of decoupled scalar gelfand levitan marchenko type equations the fredholm grothendieck determinants of the latter are shown to define tau functions in the sense of the kyoto school in particular a simple monodromy formula allows the derivation of trace identities

generalized drinfel d sokolov hierarchies ii the hamiltonian structures in this paper we examine the bi hamiltonian structure of the generalized kdv hierarchies we verify that both hamiltonian structures take the form of kirillov brackets on the kac moody algebra and that they define a coordinated system classical extended conformal algebras are obtained from the second poisson bracket in particular we construct the w n l algebras first discussed for the case n 3 and l 2 by polyakov and bershadsky

asymptotic behavior of eigenvalues of toeplitz operators on the weighted analytic spaces abstract in this paper we consider toeplitz operators on a class of analytic function spaces including standard bergman spaces on the unit disc and bargman fock spaces we give a complete characterization of those compact operators whose eigenvalues have a prescribed slow decay to zero

contact points and schatten composition operators we study composition operators on hardy and dirichlet spaces belonging to schatten classes we give some new examples and analyse the size of contact set of the symbol of such operators

denominator assignment invariants and canonical forms under dynamic feedback compensation in linear multivariable systems a result originally reported by hammer for linear time invariant lti single input single output systems and concerning an invariant and a canonical form of the transfer function matrix of the closed loop system under dynamic feedback compensation is generalized for lti multivariable systems based on this result we characterize the class of transfer function matrices that are obtainable from an open loop transfer function matrix via the use of proper dynamic feedback compensators and show that if the closed loop transfer function matrix pc s has a desired denominator polynomial matrix which satisfies a certain sufficient condition then there exists a proper compensator giving rise to an internally stable closed loop system whose transfer function matrix is pc s

exceptional sets and hilbert schmidt composition operators it is shown that an analytic map phi of the unit disk into itself inducing a hilbert schmidt composition operator on the dirichlet space has the property that the set e phi e i0 is an element ofpartial derivatived phi e 10 1 has zero logarithmic capacity we also show that this is no longer true for compact composition operators on the dirichlet space moreover such a condition is not even satisfied by hilbert schmidt composition operators on the hardy space

composition operators acting on weighted dirichlet spaces abstract we study composition operators acting on weighted dirichlet spaces we obtain estimates for the essential norm describe the membership in schatten von neumann ideals and characterize the composition operators with closed range

trace ideal criteria for toeplitz operators for a complex measure on the open unit disk u define an operator t on a hilbert space h of analytic functions with reproducing kernel k z w by u for a certain scale of hilbert spaces h 1 which includes the hardy space h2 and weighted bergman spaces a2 conditions are obtained which imply t belongs to a schatten ideal sp if is a positive measure then these conditions are necessary and sufficient application to composition operators and restriction operators are indicated

vertex operator representation of the soliton tau functions in the an 1 toda models by dressing transformations we study the relation between the group algebraic approach and the dressing symmetry one to the soliton solutions of the an 1 toda field theory in 1 1 dimensions originally solitons in the affine toda models were found by olive turok and underwood single solitons are created by exponentials of elements which ad diagonalize the principal heisenberg subalgebra alternatively babelon and bernard exploited the dressing symmetry to reproduce the known expressions for the fundamental tau functions in the sine gordon model in this paper we show the equivalence between these two methods to construct solitons in the an 1 toda models

level sets and composition operators on the dirichlet space abstract we consider composition operators in the dirichlet space of the unit disc in the plane various criteria on boundedness compactness and hilbert schmidt class membership are established some of these criteria are shown to be optimal

an 1 toda solitons and the dressing symmetry we present an elementary derivation of the solitonlike solutions in the an 1 toda models which is an alternative to the previously used hirota method the solutions of the underlying linear problem corresponding to the n solitons are calculated this enables us to obtain explicit expression for the element which by dressing group action produces a generic soliton solution in the particular example of monosolitons we suggest a relation to the vertex operator formalism previously used by olive turok and underwood our results can also be considered as generalization of the approach to the sine gordon solitons proposed by babelon and bernard

nevanlinna counting function and carleson function of analytic maps we show that the maximal nevanlinna counting function and the carleson function of analytic self maps of the unit disk are equivalent up to constants

compact composition operators on the dirichlet space and capacity of sets of contact points we prove several results about composition operators on the dirichlet space d for every compact set k d of logarithmic capacity capk 0 there exists a schur function both in the disk algebra a d and in d such that the composition operator c is in all schatten classes sp d p 0 and for which k eit eit 1 eit eit 1 for every bounded composition operator c on d and every d the logarithmic capacity of eit eit is 0 every compact composition operator c on d is compact on b 2 and on h 2 in particular c is in every schatten class sp p 0 both on h2 and on b2 there exists a schur function such that c is compact on h 2 but which is not even bounded on d there exists a schur function such that c is compact on d but in no schatten class sp d

toeplitz operators on harmonically weighted bergman spaces and applications to composition operators on dirichlet spaces abstract in this paper we consider toeplitz operators acting on bergman spaces associated with harmonic weights we give a complete description of those operators which belong to p schatten classes we also obtain an upper estimate for the reproducing kernel of these spaces applications to composition operators on dirichlet spaces are given

some examples of compact composition operators on h2 abstract we construct in an essentially explicit way various composition operators on h 2 and study their compactness or their membership in the schatten classes we construct non compact composition operators on h 2 whose symbols have the same modulus on the boundary of d as symbols whose composition operators are in various schatten classes s p with p 2 compact composition operators which are in no schatten class but whose symbols have the same modulus on the boundary of d as symbols whose associated composition operators are in s p for every p 2

tau functions and dressing transformations for zero curvature affine integrable equations the solutions of a large class of hierarchies of zero curvature equations that includes toda and kdv type hierarchies are investigated all these hierarchies are constructed from affine twisted or untwisted kac moody algebras g their common feature is that they have some special vacuum solutions corresponding to lax operators lying in some abelian up to the central term subalgebra of g in some interesting cases such subalgebras are of the heisenberg type using the dressing transformation method the solutions in the orbit of those vacuum solutions are constructed in a uniform way then the generalized tau functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest weight representations of g such definition of tau functions applies for any level of the representation and it is independent of its realization vertex operator or not the particular important cases of generalized mkdv and kdv hierarc