on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

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

essentially normal homogeneous quotient modules on the polydisc abstract in the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the hardy space on the unit polydisc it turns out that our approach also applies to quotients of weighted bergman modules

φ contractor and the solutions for nonlinear operator equations in fuzzy normed spaces in this paper we introduce the concept of contractor in fuzzy normed spaces and prove the existence and uniqueness theorems of solutions for set and single valued nonlinear operator equations with this kind of contractor as applications of these theorems we obtain some new fixed point theorems in fuzzy normed spaces

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

a new i vector topology generated by a fuzzy norm this paper is devoted to the research for a topological structure for i vector on the fuzzy normed linear space in the sense of felbin by using a natural method a new i vector topology on the fuzzy normed linear space is constructed we study some of its properties and prove that the fuzzy normed linear space is a locally convex i topological vector space with respect to the i topology under a proper condition we give a necessity and sufficienct condition for a fuzzy normed linear space to be hausdorff and weakly hausdorff with respect to the i vector topology respectively in addition we also discuss the relations between this i vector topology and other two known i vector topologies on the fuzzy normed linear space lastly we give some canonical examples of fuzzy normed linear spaces using this i vector topology

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

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

global aspects of light cone cuts the global behavior of light cone cuts at null infinity i e the intersection of light cones of interior point with null infinity is investigated in particular it is shown that the topological structure of these cuts for the case of an asymptotically flat space time is very simple the nature of the light cone cuts singularities is also analyzed generically only two different types of singularities occur and their local description is given

on i topology generated by fuzzy norm abstract in this paper we point out that an i topology t on the fuzzy normed linear space x min max constructed by das and das fuzzy topology generated by fuzzy norm fuzzy sets and systems 107 1999 349 354 is incompatible with the linear structure on x that is x min max is not an i topological vector space with respect to the i topology t therefore we construct a new i topology t on the fuzzy normed linear space x l r by using fuzzy norm we study some of its properties and prove that if r max then x l r is a hausdorff locally convex i topological vector space with respect to the i topology t in addition we also study the relations among three i topologies t t and where is the induced i topology of the crisp vector topology determined by fuzzy norm

essential normality of polynomial generated submodules hardy space and beyond abstract recently douglas and wang proved that for each polynomial q the submodule q of the bergman module generated by q is essentially normal 11 using improved techniques we show that the hardy space analogue of this result holds and more

a conformal boundary for space times based on light like geodesics the 3 dimensional case a new causal boundary which we will term the l boundary inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space times of any dimension m 3 proposed by one of the authors r j low the space of null geodesics and a new causal boundary lecture notes in physics 692 springer 2006 pp 35 50 is analyzed in detail for space times of dimension 3 under some natural assumptions it is shown that the completed space time becomes a smooth manifold with boundary and its relation with geroch kronheimer penrose causal boundary is discussed a number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided

free resolutions in multivariable operator theory let ad be the complex polynomial ring in d variables a contractive ad module is hilbert space h equipped with an ad action such that for any 1 2 d h z1 1 z2 zd d 2 1 2 2 2 d 2 such objects have been shown to be useful for modeling d tuples of mutually commuting operators acting on a hilbert space there is a subclass of the category of contractive ad modules whose members play the role of free objects given a contractive ad module one can construct a free resolution i e an exact sequence of partial isometries of the following form 2f1 1f0 0h 0 where fi is a free module for each i 0 the notion of a localization of a free resolution will be defined in which for each bd there is a vector space complex of linear maps derived from 3 c2 2 c1 1 c0 we shall show that the homology of this complex is isomorphic to the homology of the koszul complex of the d tuple 1 2 d of where i is the ith coordinate function of a mobius transform on bd such that 0

on local bases of fuzzy topological vector spaces abstract in this paper the relations between neighborhood base introduced by warren and quasi coincidence neighborhood base introduced by pu and liu in fuzzy topological spaces are studied the characterizations of these two kind of local bases for fuzzy topological vector spaces are further discussed moreover we prove a necessary and sufficient condition for a family of fuzzy sets in a vector space to be the q prebase of a ql type fuzzy topological vector space

a harmonic analysis approach to essential normality of principal submodules guo and the second author have shown that the closure i in the drury arveson space of a homogeneous principal ideal i in c z1 zn is essentially normal in this note the authors extend this result to the closure of any principal polynomial ideal in the bergman space in particular the commutators and cross commutators of the restrictions of the multiplication operators are shown to be in the schatten p class for p n the same is true for modules generated by polynomials with vector valued coefficients further the maximal ideal space xi of the resulting c algebra for the quotient module is shown to be contained in z i bn where z i is the zero variety for i and to contain all points in bn that are limit points of z i bn finally the techniques introduced enable one to study a certain class of weight bergman spaces on the ball

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

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

the geometry of the space of null geodesics the topology and geometry of the space of null geodesics n of a space time m are used to study the causal structure of the space time itself in particular the question of whether the topology of n is hausdorff or admits a compatible manifold structure carries information on the global structure of m and the transversality properties of the intersections of skies of points tell whether the points are conjugate points on a null geodesic

the dirac operator of a commuting d tuple abstract given a commuting d tuple t t1 td of otherwise arbitrary operators on a hilbert space there is an associated dirac operator dt significant attributes of the d tuple are best expressed in terms of dt including the taylor spectrum and the notion of fredholmness in fact all properties of t derive from its dirac operator we introduce a general notion of dirac operator in dimension d 1 2 that is appropriate for multivariable operator theory we show that every abstract dirac operator is associated with a commuting d tuple and that two dirac operators are isomorphic iff their associated operator d tuples are unitarily equivalent by relating the curvature invariant introduced in a previous paper to the index of a dirac operator we establish a stability result for the curvature invariant for pure d contractions of finite rank it is shown that for the subcategory of all such t that are a fredholm and and b graded the curvature invariant k t is stable under compact perturbations we do not know if this stability persists when t is fredholm but ungraded although there is concrete evidence that it does

some necessary and sufficient conditions for admitting a continuous sphere order representation of two dimensional space times a space time may be regarded as a partially ordered set by causal relations recently as a poset a sphere order representation of a space time is considered and some interesting results are obtained in this paper we define a weakly causally convex open set and characterize connected open subsets of the n dimensional minkowski space time which admit a continuous n sphere order representation then some necessary and sufficient conditions for admitting a continuous two sphere order representation of two dimensional space times are given as a result it is shown that any simply connected and causally simple space time of dimension two can be causally isomorphically imbedded into the two dimensional minkowski space time

on linearly topological structure and property of fuzzy normed linear space in this paper the simplified definition of fuzzy normed linear space is introduced the different structure of fuzzy normed linear space with variable right norm r is discussed in terms of topological vector space and its properties such as compactness completeness and density are studied under more general left norm l and right norm r as application the linearly topological structure of menger pn space is obtained

the curvature of a hilbert module over ℂ z1 zd a notion of curvature is introduced in multivariable operator theory the curvature invariant of a hilbert module over z1 zd is a nonnegative real number which has significant extremal properties which tends to be an integer and which is hard to compute directly it is shown that for graded hilbert modules the curvature agrees with the euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring this result is a higher dimensional operator theoretic counterpart of the gauss bonnet formula which expresses the average gaussian curvature of a compact oriented riemann surface as the alternating sum of the betti numbers of the surface and it solves the problem of calculating the curvature of graded hilbert modules the proof of that result is based on an asymptotic formula which expresses the curvature of a hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the euler characteristic

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

essential normality for quotient modules and complex dimensions abstract for analytic sets m introduced in 9 we show that the corresponding quotient module q of the bergman module is p essentially normal for all p dim c m which verifies the geometric arveson douglas conjecture in this case this result makes it possible to study the helton howe trace invariants on both q and the corresponding submodule r

the global structure of simple space times according to a standard definition of penrose a space time admitting well defined future and past null infinitiesi andi is asymptotically simple if it has no closed timelike curves and all its endless null geodesics originate fromi and terminate ati the global structure of such space times has previously been successfully investigated only in the presence of additional constraints the present paper deals with the general case it is shown thati is diffeomorphic to the complement of a point in some contractible open 3 manifold the strongly causal regioni0 ofi is diffeomorphic to mathbb s 2 times mathbb r and every compact connected spacelike 2 surface ini is contained ini0 and is a strong deformation retract of bothi0 andi moreover the space time must be globally hyperbolic with cauchy surfaces which subject to the truth of the poincare conjecture are diffeomorphic to 3

on null and causal pseudoconvex space times the condition of pseudoconvexity has important implications for geodesic structures of space time low showed that this condition guarantees the existence of a smooth structure of the space of geodesics recently the place of this property within the causality ladder is in the attention there is a conjecture which states that every strongly causal space time is causally simple if and only if it is null pseudoconvex in this paper we study the difference of causal and null pseudoconvexity in space times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity then we show that in the case of open subspaces of n minkowski space time this is causal geodesic connectedness finally we prove a refined form of the conjecture that says every strongly causal space time is causally simple if and only if it is maximal null pseudoconvex

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

finite dimensional fuzzy normed linear space abstract the concept of a fuzzy normed linear space is introduced it is proved that in a finite dimensional fuzzy normed linear space fuzzy norms are the same upto fuzzy equivalence finite dimensional fuzzy subspaces of a fuzzy normed linear space are proved to be necessarily complete fuzzy normed linear spaces

quotients of standard hilbert modules we initiate a study of hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties a standard hilbert module is a finite multiplicity version of one of these standard hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra and their quotients by submodules give rise to universal solutions of nonlinear relations essentially all of the basic hilbert modules that have received attention over the years are standard including the hilbert module of the shift the hardy and bergman modules of the unit ball modules associated with more general domains in and those associated with projective algebraic varieties we address the general problem of determining when a quotient of an essentially normal standard hilbert module is essentially normal this problem has been resistant our main result is that it can be linearized in that the nonlinear relations defining the submodule can be reduced appropriately to linear relations through an iteration procedure and we give a concrete description of linearized quotients

fuzzy topological vector spaces ii abstract some of the properties of fuzzy topological vector spaces are investigated also there are given necessary and sufficient conditions for a family of fuzzy sets in a vector space e to be the family of all neighborhoods of zero for a fuzzy linear topology

the structure of inner multipliers on spaces with complete nevanlinna pick kernels let k be the reproducing kernel for a hilbert space h k of analytic functions on bd the open unit ball in cd d 1 k is called a complete np kernel if k0 1 and if 1 1 k z is positive definite on bd bd let d be a separable hilbert space and consider h k d h k d and think of it as a space of d valued h k functions a theorem of mccullough and trent j funct anal 178 2000 226 249 partially extends the beurling lax halmos theorem for the invariant subspaces of the hardy space h2 d they show that if k is a complete np kernel and if d is a separable hilbert space then for any scalar multiplier invariant subspace m of h k d there exists an auxiliary hilbert space e and a multiplication operator h k e h k d such that is a partial isometry and m h k e such multiplication operators are called inner multiplication operators and they satisfy the orthogonal projection onto m in this paper we shall show that for many interesting complete np kernels the analogy with the beurling lax halmos theorem can be strengthened we show that for almost every z bd the nontangential limit z of the multiplier bd b e d associated with an inner multiplication operator is a partial isometry and that rank z is equal to a constant almost everywhere the result applies to certain weighted dirichlet spaces and to the space h2d which is determined by the kernel k z 11 z d z bd in particular our result implies that the curvature invariant of arveson proc natl acad sci usa96 1999 11 096 11 099 of a pure contractive hilbert module of finite rank is an integer this answers a question of w arveson proc natl acad sci usa96 1999 11096 11099

stable singularities of wave fronts in general relativity a wave front in a space time m is a family of null geodesics orthogonal to a smooth spacelike two surface in m it is of some interest to know how a wave front can fail to be a smoothly immersed surface in m in this paper we see that the space of null geodesics n of m considered as a contact manifold provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave fronts

complete nevanlinna pick kernels abstract we give a new treatment of quiggin s and mccullough s characterization of complete nevanlinna pick kernels we show that a kernel has the matrix valued nevanlinna pick property if and only if it has the vector valued nevanlinna pick property we give a representation of all complete nevanlinna pick kernels and show that they are all restrictions of a universal complete nevanlinna pick kernel

bergman inner functions and m hypercontractions abstract we characterize operator valued bergman inner functions on the unit ball as functions admitting a suitable transfer function realization thus we extend corresponding one variable results of olofsson from the case of the unit disc to the unit ball at the same time we associate with each m hypercontraction t l h n a canonical bergman inner function w t and indicate a possible definition of a characteristic function t for t

a set of new separation axioms in l fuzzy topological spaces in this paper a set of new separation axioms in l fuzzy topological spaces are defined and studied we give several characterizations of these separation axioms and discuss certain relationships among them moreover some of their basic properties are also examined

celestial spheres light cones and cuts the embedding of celestial spheres in n the space of null geodesics of a strongly causal space time m is studied with particular attention to the contact form on n the algebraic topology of the embedding is also considered light cone cuts with partial cauchy surfaces are examined by regarding n or appropriate submanifolds of it as a bundle over s and projecting down finally a means of reconstructing the original space time up to a conformal factor in the metric from the cuts of light cones with a cauchy surface is given

theory of light cone cuts of null infinity light cone cuts of null infinity are defined to be the intersection of the light cone of an interior point xa with the future null boundary of the space time i e i it is shown how from the knowledge of the set of light cone cuts of i the interior conformal metric can be reconstructed furthermore a differential equation defined only on i is proposed so that 1 the solution space the parameters defining the set of solutions is identified with or defines the space time itself and 2 the solutions themselves yield the light cone cuts which in turn give metrics conformally equivalent to vacuum solutions of the einstein equations

on bohr s inequality bohr s inequality says that if is a bounded analytic function on the closed unit disc then for 0 leq r 1 3 and that 1 3 is sharp in this paper we give an operator theoretic proof of bohr s inequality that is based on von neumann s inequality since our proof is operator theoretic our methods extend to several complex variables and to non commutative situations we obtain bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel hilbert spaces on various higher dimensional domains for the non commutative disc algebra an and for the reduced respectively full group c algebra of the free group on n generators we also include an application to banach algebras we prove that every banach algebra has an equivalent norm in which it satisfies a non unital version of von neumann s inequality 2000 mathematical subject classification 47a20 47a56

characteristic function for polynomially contractive commuting tuples abstract in this note we develop the theory of characteristic function as an invariant for n tuples of operators the operator tuple has a certain contractivity condition put on it this condition and the class of domains in c n that we consider are intimately related a typical example of such a domain is the open euclidean unit ball given a polynomial p in c z 1 z 2 z n whose constant term is zero all the coefficients are nonnegative and the coefficients of the linear terms are nonzero one can naturally associate a reinhardt domain with it which we call the p ball definition 1 1 using the reproducing kernel hilbert space h p c associated with this reinhardt domain in c n s pott constructed the dilation for a polynomially contractive commuting tuple definition 1 2 s pott standard models under polynomial positivity conditions j operator theory 41 1999 365 389 mr 2000j 47019 given any polynomially contractive commuting tuple t we define its characteristic function t which is a multiplier we construct a functional model using the characteristic function exploiting the model we show that the characteristic function is a complete unitary invariant when the tuple is pure the characteristic function gives newer and simpler proofs of a couple of known results one of them is the invariance of the curvature invariant and the other is a beurling theorem for the canonical operator tuple on h p c it is natural to study the boundary behaviour of t in the case when the domain is the euclidean unit ball we do that and here essential differences with the single operator situation are brought out

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

linking and causality in globally hyperbolic space times the classical linking number lk is defined when link components are zero homologous in 15 we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes here we apply alk to the study of causality in lorentzian manifolds

condensing operators and topological degree theory in standard fuzzy normed spaces in this paper level cut measures of noncompactness are introduced in standard fuzzy normed spaces and condensing operators are studied using the extensions of nonlinear compact operators topological degree theory with respect to condensing operators is given in standard fuzzy normed spaces as an application of degree theory a fixed point theorem for condensing operators is presented

conformally embedded spacetimes and the space of null geodesics it is shown that the space of null geodesics of a causally simple spacetime is hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime this provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones