equivalence of hardy submodules generated by polynomials abstract in this paper we obtain a complete classification under unitary equivalence for hardy submodules on the polydisk which are generated by ideals of polynomials let i be an ideal of polynomials in n variables since i is generated by finitely many polynomials i has a greatest common divisor p so i can be uniquely written as i p l which is called the beurling form of i let i 1 p 1 l 1 i 2 p 2 l 2 we prove that i 1 and i 2 are unitarily equivalent if and only if there are polynomials q 1 and q 2 with z q 1 d n z q 2 d n such that p 1 q 1 p 2 q 2 on t n and p 1 l 1 p 1 l 2 consequently two principal submodules p 1 and p 2 are unitarily equivalent if and only if there are polynomials q 1 and q 2 with z q 1 d n z q 2 d n such that p 1 q 1 p 2 q 2 on t n furthermore we give a complete similarity classification for submodules generated by homogeneous ideals finally we point out that in the case of the hardy module on the unit ball i 1 and i 2 are unitarily equivalent if and only if they are equal if i 1 and i 2 are homogeneous ideals then i 1 and i 2 are quasi similar if and only if i 1 i 2

defect operators defect functions and defect indices for analytic submodules abstract this paper mainly concerns defect operators and defect functions of hardy submodules bergman submodules over the unit ball and hardy submodules over the polydisk the defect operator function carries key information about operator theory function theory and structure of analytic submodules the problem when a submodule has finite defect is attacked for both hardy submodules and bergman submodules our interest will be in submodules generated by polynomials the reason for choosing such submodules is to understand the interaction of operator theory function theory and algebraic geometry

the bergman space the bloch space and commutators of multiplication operators soit d le disque unite ouvert dans le plan complexe c soit h d l ensemble des fonctions analytiques bornees sur d on caracterise les fonctions feh d telles que t f t f tft f est un operateur compact ou t f est l operateur multiplication t f l a 2 l a 2 defini par t f g fg

characteristic spaces and rigidity for analytic hilbert modules abstract it is well known that a polynomial in one variable is completely determined by its zeros counting multiplicities we generalize this result to an ideal of polynomials in several variables by introducing the characteristic spaces of the ideal one finds that the ideal is completely determined by its characteristic spaces on a characteristic set in particular a primary ideal is completely determined by its characteristic space at any zero point some straightforward applications of the above results yield the algebraic reduction theorem for analytic hilbert modules in several variables also we obtain some general rigidity results for analytic hilbert modules by using the techniques of af envelopes of analytic hilbert modules

on the harris knorr correspondence in p solvable groups abstract fix a prime p and let n be a normal subgroup of a finite p solvable group g suppose that b is a p block of n with abelian defect group q and b is a p block of g covering b let b be the brauer correspondent of b in n n q and let b be the unique p block of n g q that covers b and induces b we show that there exist height preserving bijections of irr b onto irr b and of ibr b onto ibr b such that the decomposition numbers d and d are equal for all irr b and all ibr b

invariant subspaces of finite codimension in banach spaces of analytic functions abstract we give a characterization of invariant subspaces of finite codimension in banach spaces of vector valued analytic functions in several variables where invariant refers to invariance under multiplication by any polynomial we obtain very weak conditions under which our characterization applies that unifies and improves a number of previous results in the vector valued case the results are new even for one complex variable as a concrete application in several variables we consider the bergman space on a strictly pseudo convex domain and we improve previous results assuming c boundary to the case of c 2 boundary

boundedness and compactness of generalized hankel operators on bounded symmetric domains the mapb hb i p mbpfrom analytic functions on the unit diskdto the associated hankel operators on the hardy space or the weighted bergman spaces is known to be an important tool in studying the size of the functionbin terms of the size of hb moreover this map is equivariant namely it intertwines the natural actions of the mobius groupaut d on functions and operators this theory extends to some extent to the context of the open unit ballbnincn but it fails in cartan domains of rankr 1 because in this case the mapb hbtrivializes as hbis compact only if hb 0 andbis constant we study generalizations abof hankel operators in the context of weighted bergman spaces over a cartan domain of tube type with rankr 1 the mapb abis equivariant and non trivial we study also in this context generalized bloch and little bloch spaces b and b0respectively and generalized bmoa and vmoa spaces with respect to weighted volume measure the main results are that abis bounded if and only ifb b if and only ifb bmoa and abis compact if and only ifb b0if and only ifb vmoa

on the period function of x f x x 2 g x 0 abstract we study the period function t of a center o of the title s equation a sufficient condition for the monotonicity of t or for the isochronicity of o is given such a condition is also necessary when f and g are odd and analytic in this case a characterization of isochronous centers is given some classes of plane systems equivalent to such equation are considered including some kukles systems

finite rank hankel operators on the polydisk abstract we give a different proof of power s several variables generalization of the well known kronecker s result on finite rank hankel matrices this also leads to a formula for the rank of a small hankel operator on polydisk in terms of a certain degree of its rational symbol

c algebras of multiplication operators on bergman spaces soit un domaine borne dans c soit a 2 l espace de bergman de toutes les fonctions analytiques sur qui sont de carre integrable par rapport a la mesure volumetrique et soit c la c algebre engendree par les operateurs de multiplication par les fonctions coordonnee sur on etudie les proprietes de c

comment on classification of lie point symmetries for quadratic lienard type equation x f x x2 g x 0 j math phys 54 053506 2013 and its erratum j math phys 55 059901 2014 we demonstrate a simplification of some recent works on the classification of the lie symmetries for a quadratic equation of lienard type we observe that the problem could have been resolved more simply

algebraic and spectral properties of dual toeplitz operators dual toeplitz operators on the orthogonal complement of the bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement in this paper we study algebraic and spectral properties of dual toeplitz operators

products of hankel and toeplitz operators on the bergman space we consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products tftg are bounded on the bergman space we prove results analogous to those obtained by the second author 17 for such toeplitz products on the hardy space we furthermore obtain similar results for hankel products hfh g where f and g are square integrable on the unit disk and for the mixed haplitz products hftg and tgh f where f and g are square integrable on the unit disk and g is analytic

on the dirichlet space for finitely connected regions this paper is devoted to the study of the dirichlet space dir g for finitely connected regions g we are particularly interested in the algebra of bounded multiplication operators on this space results in different directions are obtained one direction deals with the structure of closed subspaces invariant under all bounded multiplication operators in particular we show that each such subspace contains a bounded function for regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by z must be invariant under all bounded multiplication operators and furthermore it is of the form p dir g where p is a polynomial with all its roots lying in g another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators typical results are if f e dir g and f is bounded away from zero then f is cyclic on the other hand if the zero set of the radial limit function of f on the boundary has positive logarithmic capacity then f is not cyclic also some other sufficient conditions for a function to be cyclic are given lastly we study transitive operator algebras containing all bounded multiplication operators we prove that they are dense in the algebra of all bounded operators in the strong operator topology

boundary representations and rigidity of submodules abstract this paper considers two problems on the fock type spaces f s 0 s 1 firstly it is shown that on the space f s 0 s 1 the identity representation of c i t 1 t n is a boundary representation for the banach subalgebra b i t 1 t n while on the space f 1 it is not secondly it is shown that all the submodules of f 1 are rigid

hilbert polynomials and arveson s curvature invariant abstract we define and study hilbert polynomials for certain holomorphic hilbert spaces we obtain several estimates for these polynomials and their coefficients our estimates inspire us to investigate the connection between the leading coefficients of hilbert polynomials for invariant subspaces of the symmetric fock space and arveson s curvature invariant for coinvariant subspaces we are able to obtain some formulas relating the curvature invariant with other invariants in particular we prove that arveson s version of the gauss bonnet chern formula is true when the invariant subspaces are generated by any polynomials

analytic hilbert spaces over the complex plane in this paper we study the structure of quasi invariant subspaces of analytic hilbert spaces over the complex plane we especially investigate when two quasi invariant subspaces are similar or unitarily equivalent for an analytic hilbert space over the complex plane

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

function theory on cartan domains and the berezin toeplitz symbol calculus cn with dv z normalized lebesgue measure on 9 we let l l2 q dv be the usual space of lebesgue square integrable complex valued functions on 9 by h2 h2 q dv we denote the bergman subspace of l2 consisting of holomorphic functions the orthogonal projection operator from l2 onto h2 is denoted by p forf in l q the space of essentially bounded measurable functions on 9 we will consider the multiplication operator

invariant subspaces quasi invariant subspaces and hankel operators abstract in this paper using the theory of hilbert modules we study invariant subspaces of the bergman spaces on bounded symmetric domains and quasi invariant subspaces of the segal bargmann spaces we completely characterize small hankel operators with finite rank on these spaces

brauer characters and the harris knorr correspondence in p solvable groups abstract if b is a p block of a normal subgroup n of a p solvable group g and b is its brauer correspondent in n n l where l is a defect group of b then harris knorr correspondents over b and b contain equal numbers of irreducible brauer characters over height zero brauer characters

comment on classification of lie point symmetries for quadratic lienard type equation ẍ f x ẋ2 g x 0 j math phys 54 053506 2013

finite codimensional invariant subspaces of bergman spaces abstract for a large class of bounded domains in c we describe thosefinite codimensional subspaces of the bergman space that are invariant undermultiplication by z using different techniques for certain domains in cn wedescribe those finite codimensional subspaces of the bergman space that areinvariant under multiplication by all the coordinate functions fix a positive integer n and let v denote lebesgue volume measure on cn sothat if n 1 then v is just area measure let q c c be a domain which asusual means that 0 is a nonempty open connected subset of c for an analyticfunction from 0 to c and 1 p oo the norm n p is defined by wfh r jn f pdv the bergman space l q is defined to be the set of analytic functions from fi toc such that n p oo our goal in this paper is to describe the closed finite codimensional subspaces oflpa 0 that are invariant under multiplication by the coordinate functions zi

the core operator and congruent submodules abstract the hardy spaces h 2 d 2 can be conveniently viewed as a module over the polynomial ring c z 1 z 2 submodules of h 2 d 2 have connections with many areas of study in operator theory a large amount of research has been carried out striving to understand the structure of submodules under certain equivalence relations unitary equivalence is a well known equivalence relation in set of submodules however the rigidity phenomenon discovered in douglas et al algebraic reduction and rigidity for hilbert modules amer j math 117 1 1995 75 92 and some other related papers suggests that unitary equivalence being extremely sensitive to perturbations of zero sets lacks the flexibility one might need for a classification of submodules in this paper we suggest an alternative equivalence relation namely congruence the idea is motivated by a symmetry and stability property that the core operator possesses the congruence relation effectively classifies the submodules with a finite rank core operator near the end of the paper we point out an essential connection of the core operator with operator model theory

invariant subspaces of h2 on a torus abstract this paper answers two questions about unitarily equivalent pairs of invariant subspaces x1 and x2 of the hardy space h2 tn these satisfy x2 x1 for some unimodular l tn i must then be a quotient of inner functions ii if g 2 g 1 is such a quotient does it follow that the inner functions g1 and g2 can be so chosen that g 1 x 1 g 2 x 2 is an invariant subspace of h2 tn not merely of l2 tn when n 1 both answers are no

invariant subspaces with finite codimension in bergman spaces let omega be a domain in the complex plane denote by l p roman a omega 1 le p infty the bergman space over omega the author presents a description of finite codimensional space e subset l p roman a omega such that ze subset e under some conditions on omega an analogous result is due to n s axler en and n p bourdon en same journal bf306 1988 no 2 805 817 mr0933319 89f 46051 for an arbitrary bounded domain in c there are described those finite codimensional subspaces of the bergman space that are invariant under multiplication by z

response to comment on classification of lie point symmetries for quadratic lienard type equation ẍ f x ẋ2 g x 0 j math phys 61 044101 2020 we respond to the comment on classification of lie point symmetries for quadratic lienard type equation f x 2 g x 0 j math phys 61 044101 2013 by iacono regarding linearizability and isochronicity we assert here that the condition for linearization of the equation f x 2 g x 0 given by us in our paper is correct with the condition g1 02 0 we present the explicit form of local and nonlocal transformations that transform the quadratic lienard equation x f 11 xx2 x 1 x 1 dx 0 into the harmonic oscillator equation for the four cases mentioned in the comment and confirm the statements given in our paper are all valid

equivalence of quotient hilbert modules ii for any open connected and bounded set c m let a be a natural function algebra consisting of functions holomorphic on let m be a hilbert module over the algebra a and m0 m be the submodule of functions vanishing to order k on a hypersurface z recently the authors have obtained an explicit complete set of unitary invariants for the quotient module q m m0 in the case of k 2 in this paper we relate these invariants to familiar notions from complex geometry we also find a complete set of unitary invariants for the general case we discuss many concrete examples in this setting as an application of our equivalence results we characterise certain homogeneous hilbert modules over the bi disc algebra one source of fascination in the study of operator theory is the wide variety of connections made with other branches of mathematics techniques from algebra topology geometry and analysis are used to understand bounded linear operators on hilbert space and in many instances the behavior and properties of the operators can be used to illustrate critical features and aspects of the other fields this is particularly true in the case of multivariate operator theory that is when several operators or an algebra of operators is studied here the setting and results from these other areas can be quite sophisticated and the techniques used to understand multivariate operator theory often require additional development such is the focus of this paper although the spectral theorem is a key tool in the study of self adjoint and normal opera tors there are large and important classes of naturally occurring operators to which this theory doesn t apply examples illustrating such phenomena can be obtained by considering multipli cation operators on spaces of holomorphic functions on some domain in c m for domains in c one is in the realm of single operator theory while it is multivariate operator theory for m 1 if one considers the unit ball b m in c m and the bergman space a 2 b m for it one obtains a

unitary invariants for hilbert modules of finite rank we associate a sheaf model to a class of hilbert modules satisfying a natural finiteness condition it is obtained as the dual to a linear system of hermitian vector spaces in the sense of grothendieck a refined notion of curvature is derived from this construction leading to a new unitary invariant for the hilbert module a division problem with bounds originating in douady s privilege is related to this framework a series of concrete computations illustrate the abstract concepts of the paper

essentially normal hilbert modules and k homology this paper mainly concerns the essential normality of graded submodules essentially all of the basic hilbert modules that have received attention over the years are p essentially normal including the d shift hilbert module the hardy and bergman modules of the unit ball arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer some positive results have been obtained by arveson and douglas however the problem has been resistant in dimensions d 2 3 this paper shows that the arveson s conjecture is true in any dimension the paper also gives an affirmative answer in the case of the graded principal submodule finally the paper is associated with k homology invariants arising from graded quotient modules by which geometry of the quotient modules and geometry of algebraic varieties are connected in dimensions d 2 3 it is shown that k homology invariants determined by graded quotients are nontrivial the paper also establishes results on p smoothness of k homology elements and gives an explicit expression for k homology invariant in dimension d 2

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

operator theory and algebraic geometry on presente des resultats en theorie des operateurs multivariable dont les demonstrations se relient a des techniques issues de la geometrie algebrique

isochronicity of analytic systems via urabe s criterion a method for studying isochronous oscillations in some systems of ode reducible to the equation is described it is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two dimensional autonomous system depending on six parameters for all isochronous systems in this family the urabe function is explicitly constructed

cyclic vectors in the dirichlet space we study the hilbert space of analytic functions with finite dinchlet integral in the open unit disc we try to identify the functions whose polynomial multiples are dense in this space theorems 1 and 2 confirm a special case of the following conjecture if ij z i ig z l at all points and if g is cyclic thenjis cyclic theorems 3 5 give a sufficient condition t is an outer function with some smoothness and the boundary zero set is at most countable and a necessary condition the radial limit can vanish only for a set of loganthmic capacity zero for a function j to be cyclic introduction in this paper we shall study the hilbert space of analytic functions in the open unit disc a in the complex plane that have a finite dirichlet integral jj if t12 dx dy g z l for some cyclic g and all z question 4 asks if f must be cyclic whenever f and l f are both in the space no examples are known where either of these questions has a negative answer in 2 we begin the study of cyclic vectors in the dirichlet space d theorems 1 and 2 give a partial answer to question 3 above for this space this section also contains 2 propositions 10 11 and 4 questions 7 10 proposition 11 says that if f and g are bounded functions in d whose product is cyclic then bothf and g must be cyclic theorem 2 gives a partial converse we require that gl be dini continuous on received by the editors july 21 1983 1980 mathematics subject classification primary 30h05 secondary 46e15 46e20 47b37

finite rank multivariable hankel forms abstract we obtain a several variables generalization of kronecker s well known result on finite rank hankel matrices

toeplitz algebras subnormal tuples and rigidity on reproducing c z1 zd modules this paper mainly considers toeplitz algebras subnormal tuples and rigidity concerning reproducing c z1 zd modules by making use of arveson s boundary representation theory we find there is more rigidity in several variables than there is in single variable we specialize our attention to reproducing c z1 zd modules with u invariant kernels by examining the spectrum and the essential spectrum of the d tuple mz1 mzd and deducing an exact sequence of c algebras associated with toeplitz algebra finally we deal with toeplitz algebras defined on arveson submodules and rigidity of arveson submodules

kernels and ranks of hankel operators on the dirichlet spaces abstract we consider hankel operators and little hankel operators on the dirichlet space and pluriharmonic dirichlet space of the unit ball we first describe kernels of hankel operators with pluriharmonic symbol next we prove that there are no non trivial finite rank hankel operators with pluriharmonic symbol finally we characterize finite rank little hankel operators with pluriharmonic symbol our results show that there are several differences between the results on the dirichlet space and pluriharmonic dirichlet space

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

star generating vectors of rudin s quotient modules abstract the purpose of this paper is to study a class of quotient modules of the hardy module h 2 d n along with the two variables quotient modules introduced by w rudin we introduce and study a large class of quotient modules namely rudin s quotient modules of h 2 d n by exploiting the structure of minimal representations we obtain an explicit co rank formula for rudin s quotient modules

duality and hankel operators on the bergman spaces of bounded symmetric domains soit un domaine symetrique borne dans c n avec dv la mesure de volume normalisee l espace de bergman l a p est le sous espace clos de l p dv constitue des fonctions holomorphes on decrit le dual et le predual de l a 1 pour tous les domaines symetriques bornes

notes on ordered reproducing hilbert spaces over the complex plane abstract let f g be entire functions if there exist m 1 m 2 0 such that f z m 1 g z whenever z m 2 we say that f g let x be a reproducing hilbert space with an orthogonal basis z n n 0 we say that x is an ordered reproducing hilbert space or x is ordered if f g and g x imply f x in this note we show that if lim inf n z n 1 z n then x is ordered if lim inf n z n 1 z n 0 then x is not ordered in the case lim inf n z n 1 z n l 0 there are examples to show that x can be of order or opposite

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