Hilbert pairing on Lorentz formal groups

In this paper, we construct an explicit pairing in Cartier series for formal Lorentz groups of the form (X + Y + XY)/(1 + c ² XY), where c is a unit of the ring of integers of the local field. We prove the basic properties of this pairing, namely, bilinearity and invariance, which make it possible to explicitly construct the generalized Hilbert symbol for formal Lorentz groups over rings of integers of local fields with the use of the obtained pairing.     

Hilbert pairing for the polynomial formal groups

The work is devoted to a wide class of formal groups, the ones given by polynomials, and to their relation to the Hilbert pairing. For the latter an explicit formula is obtained. The basic definitions are introduced in the work and the principal results are formulated, with the brief plans of proofs given for them. The detailed proofs are going to be given in the next work.     

Norm Hilbert spaces over Krull valued fields

Norm Hilbert spaces (NHS) are defined as Banach spaces over valued fields (see 1.4) for which each closed subspace has a norm-orthogonal complement. For fields with a rank 1 valuation, these spaces were characterized already in [10, 5.13, 5.16], where it was proved that infinite-dimensional NHS exist only if the valuation of K is discrete. The first discussion of the case of (Krall) valued fields appeared in [1] and [3]. In this paper we continue and expand this work focussing on the most interesting cases, not covered before. If K is not metrizable then each NHS is finite-dimensional (Corollary 3.2.2), but otherwise there do exist infinite-dimensional NHS; they are completely described in 3.2.5. Our main result is Theorem 3.2.1, where various characterizations of NHS of different nature are presented. Typical results are that NHS are of countable type, that they have orthogonal bases, and that no subspace is linearly homeomorphic to c₀.     

Norm of the Hilbert matrix on Bergman spaces

It is well known that the Hilbert matrix operator H is a bounded operator from the Bergman space $A^p$ into $A^p$ if and only if $2<p<\infty$. In [5] it was shown that the norm of the Hilbert matrix operator H on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin ?\frac{2\pi }{p}}$, when $4\le p<\infty$, and it was also conjectured that

${6H6}_{A^p\to A^p}=\frac{\pi }{\sin ?\frac{2\pi }{p}},$

when $2<p<4$. In this paper we prove this conjecture.     

Norm equalities of analytic mappings into Hilbert spaces

LetD be a subset of a complex linear spaceL such that for everyu∈D,v∈L the setΩ(u, v) = {ζ∣u+ζv∈D} is an open connected set in the complex plane. Denote byA (D, X) the linear space of allG-analytic mappings fromD to a complex Hilbert spaceX.Theorem: LetZ be a complex linear space and letA, B be linear operators fromZ toA (D, X), A (D, Y), respectively, whereX, Y are complex Hilbert spaces. If ∥(A p)u∥ _X =∥(B p)u∥ _Y (p∈Z,u∈D) then a maximal partial isometryW:X→Y exists such that(Bp)u=W((Ap)u) (p∈Z, u∈D).     

Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra with unit over the fields or , the infimum of its norms with respect to all scalar products in this algebra (with ) is either infinite or at most . Sufficient conditions for this bound to be not less than are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).     

The Hilbert pairing for formal groups over σ-rings

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 5–58.     

Norm of a linear combination of projectors in Hilbert space

V. I. Lenin Kishinev State Institute. Institute of Mathematics and Computing, Academy of Sciences of the Moldavian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 4, pp. 85–86, October–December, 1989.     

Hilbert空间上最终范数连续半群的特征刻画

本文研究Hilbert空间上最终范数连续半群的特征条件,仅利用半群生成元的预解式,给出Hilbert空间上C0-半群最终范数连续的一个的充要条件.     

Hilbert空间中范数连续半群的特征

给出Hilbert空间中范数连续半群的一个新特征，同时极大地简化了已有结果的证明。     

Hilbert lattices with the extension property

Separable Hilbert lattices are constructed that enjoy the following property: each ortho-isomorphism between arbitrary interval sublattices [0, a] and [0, b] of height at least 3 (a, b ) extends to an ortho-automorphism of the lattice #x2112.     

广义算子半群范数连续问题研究

研究了Hilbert空间上范数连续广义算子半群的特征条件.利用广义半群的的预解式,给出了广义算子半群范数连续的充分条件.     

Degeneration of the Hilbert pairing in formal groups over local fields

For an arbitrary local field K (a finite extension of the field Q_p) and an arbitrary formal group law F over K, we consider an analog c_F of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing c_F has a certain fundamental symbol property for all Lubin–Tate formal groups, then c_F = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring O_K of integers of K and have a fixed ring O₀ of endomorphisms, where O₀ is a subring of O_K. We prove that if the symbol c_F has the above-mentioned symbol property, then c_F = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.     

Hilbert空间中广义算子半群范数连续的特征

研究了Hilbert空间上最终范数连续广义算子半群的特征条件,利用半群的生成元的预解式,给出了Hilbert空间上广义算子半群范数连续的三个特征条件.     

A problem about the weak Hilbert property

In this paper, we prove the main result: Let both (K, S) and (K ^*,S ^*) be preordered fields, and let (K ^*,S ^*) be a finitely generated extension of (K, S). IfK ^* is transcendental overK, then (K ^*,S ^*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established. Project supported by National Natural Science Foundation of China     

The Kneser relation and the Hilbert pairing in multidimensional local field

We generalize the Kneser relation to multidimensional local fields and use it to obtain an explicit formula for the Hilbert pairing of multidimensional local fields directly from the definition of the pairing. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.