Subseries convergence in the space of functions of bounded φ-variation

Let there be given two F-spaces (X, ∥ ∥*), (Y, ∥ ∥*), X ? Y. A series Σ ₁ ^∞ x_i of elements in X is said to have property O(X, Y) (in the case when X = Y, ∥ ∥ = ∥ ∥*-property O) if perfect boundedness of this series in (X, ∥ ∥) implies its perfect (subseries) convergence in ( Y, ∥ ∥*). Conditions are determined for a series of elements in the space of functions of bounded φ-variation to have property O(X, Y). These conditions are rephrased for the case when convergence with respect to the norm ∥∥ _? ^v is replaced by modular convergence generated by φ-variation.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

Jacobians and differents of projective varieties

Let X be a reduced and irreducible projective variety of dimension d. Let π:X→Y be a separable noetherian normalization of X and ? the canonical morphism Ω^d _X/k→Ω^d _L/k. From our main result: $$J_X \varphi (\pi ^* \Omega ^d _{Y/k} ) = \theta _k (X/Y)\varphi (\Omega ^d _{X/k} )$$ we deduce relations among: complementary module C(X/Y), Kähler different θ_k(X/Y), Dedekind different θ_D(X/Y), jacobian ideal J_K and ω-jacobian ideal \(\tilde J_X\) .     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

Very Strange Projective Curves

Let Y ? Pⁿ, n ≥ 3, be an integral non-degenerate very strange projective curve, i.e. assume that the general hyperplane section of Y is not in linearly general position; hence we are in characteristic p. Let π: X → Y be the normalization. Set d? deg(Y), g? Pa(X) and L? π* (Oy(1)). Here we prove that d > 2g?2 and in particular h¹(X,L) = 0 and h⁰(X,L) = d+1?g ≥ (d?1)/2 > n+1.     

Automorphiesummanden und Cousin-I-Probleme auf Faserräumen

This article deals with the classification of summands of automorphy on locally trivial fibre spaces E(X, Y), the base space X having the property H¹ (X,O)=0 and the fibre Y being a compact Kähler manifold. Further we assume that the structure group is connected and that the transgression H¹(Y)H²(X) vanishes. Then every summand of automorphy on E is equivalent to a homomorphism on B₁(E) with values in the ring of holomorphic functions on X (theorem I). As a corollary we prove a theorem on the solvability of Cousin-I-Problems on E by additive functions (theorem II).

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Meinem verehrten Lehrer KARL STEIN in Dankbarkeit zum 60. Geburtstag gewidmet     

Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

MULTIPLIERS OF MATRIX SPACES

Abstract

Suppose X and Y are FK spaces in which ? the span of the coordinate vectors (e_n) is dense. Let L(X,Y) denote the space of all matrices of the form E_i(T(e_j)) as T ranges over all continuous linear operators from X into Y; here e_i represents the i^th coordinate vector and E_i represents the i^th coordinate functional. Let M(L(X, Y)) denote the space of all matrices B such that (B(i,j)A(i,j)) is in L(X,Y) whenever A is in L(X,Y). In this paper we shall show how the summability properties of X and Y determine the extent of M(L(X,Y)) and conversely how the extent of M(L(X,Y)) determines the summability properties of both X and Y.     

上三角算子矩阵的谱

设X,y是Banach空间,对A∈B(X),B∈B(y),C∈B(Y,X),以M_C记X⊕Y上的算子(ACOB).本文给出了算子M_C的20种谱的结构表示,18种谱的填洞性质以及关于这些问题的有趣例子.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Fortsetzung Ganzzahliger Cozyklen von Kompakten Teilen

Let Y be a topological space and X a subspace of Y. We assume that X is the union of an increasing sequence of subspaces K_S such that every quasi-compact subset of X is contained in some K_S and the singular homology groups of all K_S are finitely generated. The object of this paper is to give a purely algebraic characterisation of the following subgroup of H^q (X,Z): it consists of all those elements in H^q (X,Z) whose image in each of the H^q (K_S,Z) lies in the image of the induced homomorphism H^q (Y,Z)H^q (K_S,Z), These subgroups are encountered in Runge approximation theory. Partial results were obtained in an earlier common paper with K. Stein, [1].

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Meinem verehrten Lehrer Karl Stein zum 60. Geburtstag gewidmet     

On rank-one corrections of complex symmetric matrices

Let a matrix A ∈ M_n(C) be a rank-one perturbation of a complex symmetric matrix, i.e., A = X + Y for some unknown matrices X and Y such that X = X^T and rank Y = 1. The problem of determining the matrices X and Y is solved. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 78–83.     

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     

Quotient normed cones

Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖ ^p ,p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)^*, || · ‖·‖^p,p) can be identified as a distinguished subcone of the dual cone (X ^*, || · ||_{p, u}). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.     

Adjoint subspaces in Banach spaces,with applications to ordinary differential subspaces

Summary Given two subspaces A₀ ⊂ A₁ ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A₀ ⊂ A ⊂ A₁, A ₁ ^* ⊂ A∗ ⊂ A ₀ ^* ⊂ W⁺=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary differential subspaces in W=L^q(ι) ⊕ L^r(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C( ) ⊕ C( ), where is a compact interval. Entrata in Redazione il 21 febbraio 1977. The work of EarlA. Coddington was supported in part by the National Science Foundation under NSF Grant No. MCS-76-05855.     

Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

On a stabilizing feedback attitude control

The attitude control of a rotating satellite with two control jets leads to a system of four controlled ordinary differential equations of the form (S) $$dx/dt = X(x) + u_1 Y^1 (x) + u_2 Y^2 (x),x(0) = 0.$$ Our goal is to derive feedback controlsu ₁,u ₂ which automatically stabilize the system (S), i.e., drive the solution to the (uncontrolled) rest solution zero. Let $$(ad^0 X,Y) = Y,(adX,Y) = [X,Y],$$ the Lie product of the vector fieldsX, Y, and inductively $$(ad^{k + 1} X,Y) = [X,(ad^k X,Y)].$$ It is known that, if $$dim span\left\{ {\left( {ad^j X,Y^1 } \right)\left( 0 \right),j = 0,1,...} \right\} = 4,$$ then all points in some neighborhood of zero can be controlled to zero with just the controlu ₁, i.e.,u ₂≡0. In this problem,Y ¹(0), ..., (ad ³ X, Y ¹)(0) are linearly independent. We give a formula for generating the directions (ad ⁱ X, Y ⁱ )(0) as endpoints of admissible trajectories. Our modified feedback control is then formed as follows. Given an ε>0, if the state of system (S) is measured to beq ¹ ∈ ?⁴, we write $$q^1 = \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0),$$ and choose a controlu(t,q ¹) on the interval 0≤t≤ε to drive the solution in the direction $$ - \sum\limits_{i = 1}^4 {\alpha _1 } (ad^{i - 1} X,Y^1 )(0).$$ Thus, we assume that the state is measured (say) at time intervals 0, ε, 2ε, ..., while the control depends on the measured state, but then is open loop during a time interval ε until a new state is measured; hence, the terminologymodified feedback control. Numerical results are included for both the case of one control component and the case of two control components.     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.