Hyperbolic Hilbert space

The hyperbolic complex space is one class of non-Euclidean spaces with continuous singular points. It corresponds with Minkowski space, and it has the characteristic that the space-time direction is different in nature. Regard the hyperbolic complex space as original spaces. We can abstract a class of the hyperbolic inner product space and the hyperbolic Hilbert space.     

The product of a Baire space with a hereditarily Baire metric space is Baire

In this paper we prove that the product of a Baire space with a metrizable hereditarily Baire space is again a Baire space. This answers a recent question of J. Chaber and R. Pol.

     

On the tangent space to the BMO-Teichmüller space

In a previous paper [31] we established the complex analytic manifold theory of the BMO-Teichmüller space. In this paper we identify the function space which is the tangent space to the BMO-Teichmüller space.     

Fock space on C∞ and Bose-Fock space

In this paper, we introduce the Fock space on ${\mathbb{C}}^{\infty }$ and obtain an isomorphism between the Fock space on ${\mathbb{C}}^{\infty }$ and Bose-Fock space. Based on this isomorphism, we obtain representations of some operators on the Bose-Fock space and answer a question in [2]. As a physical application, we study the Gibbs state.     

The product of a Gâteaux differentiability space and a separable space is a Gâteaux differentiability space

This paper shows that the product of a Gâteaux differentiability space and a separable Banach space is again a Gâteaux differentiability space.

     

Beurling type theorem on the Bergman space via the Hardy space of the bidisk

In this paper,by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk,we give a proof of the Beurling type theorem on the Bergman space of Aleman,Richter and Sundberg(1996) via the Hardy space of the bidisk.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

Composition operators on the Newton space

We investigate properties of composition operators C? on the Newton space (the Hilbert space of analytic functions which have the Newton polynomials as an orthonormal basis). We derive a formula for the entries of the matrix of C? with respect to the basis of Newton polynomials in terms of the value of the symbol ? at the non-negative integers. We also establish conditions on the symbol ? for boundedness, compactness, and self-adjointness of the induced composition operator C?. A key technique in obtaining these results is use of an isomorphism between the Newton space and the Hardy space via the Binomial Theorem.     

Embedding of Sobolev space in Orlicz space for a domain with irregular boundary

In this paper, we establish the embedding of a weighted Sobolev space in an Orlicz space for a domain with irregular boundary. We find an estimate of the order of growth of the N-function (defining the Orlicz space) and show that, under certain additional constraints on the weights, this estimate is sharp. We also establish the embedding in the space of continuous functions.     

Characterization for Beurling-Björck space and Schwartz space

We give an elementary proof of the equivalence of the original definition of Schwartz and our characterization for the Schwartz space . The new proof is based on the Landau inequality concerning the estimates of derivatives. Applying the same method, as an application, we give a better symmetric characterization of the Beurling-Björck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives.

     

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.     

Essential norms of weighted composition operators from weighted Bergman space to mixed-norm space on the unit ball

In this paper, we characterize the boundedness and compactness of the weighted composition operators from the weighted Bergman space to the standard mixed-norm space or the mixed-norm space with normal weight on the unit ball and estimate the essential norms of the weighted composition operators.     

Cohomogeneity one de Sitter space S n 1

In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.     

Bounded and compact Toeplitz operators on the weighted Bergman space over the bidisk

Forα1,let dvαdenote the weighted Lebesgue measure on the bidisk andμa complex measure satisfying some Carleson-type conditions.In this paper,we show a sufcient and necessary condition for the Toeplitz operatorTαˉμto be bounded or compact on weighted Bergman spaceL1a(dvα).     

Operators on the orthogonal complement of the Dirichlet space

In this paper we prove that a dual Hankel operator is zero if and only if its symbol is orthogonal to the Dirichlet space in the Sobolev space, and characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space.     

On the Natanzon-Turaev compactification of the Hurwitz space

Natanzon and Turaev have constructed by topological methods a compactification of the Hurwitz space, that is, the space of simple branched covers of the two-sphere. Here we show that this compactification is homeomorphic to a compactification mentioned by Diaz and Edidin (in 1996) that was constructed by algebraic methods. Using this we are able to show by example that the Natanzon-Turaev compactification can be singular, that is, not a manifold.

     

The bicompletion of an asymmetric normed linear space

A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖^s) is a Banach space, where ‖x‖^s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖_Y). Furthermore the space (Y,‖·‖_Y) is unique (up to isometric isomorphism). This revised version was published online in August 2006 with corrections to the Cover Date.     

A Gâteaux differentiability space that is not weak Asplund

In this paper we construct a Gâteaux differentiability space that is not a weak Asplund space. Thus we answer a question raised by David Larman and Robert Phelps from 1979.

     

The Urysohn space embeds in Banach spaces in just one way

In a Master's thesis in 1985 and a subsequent paper published in 1992, the author discovered that the universal separable metric space (up to isometry) discovered by Urysohn in 1925 has a uniquely determined linear closure (up to linear isometry) when isometrically embedded in a Banach space so as to include the zero of the Banach space. The proof of this result is given in this note and the current status of some related questions is discussed.