Theory of (<Emphasis Type="Italic">q</Emphasis> <Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis> <Subscript>2</Subscript>)-quasimetric spaces and coincidence points

We introduce (q₁, q₂)-quasimetric spaces and examine their properties. Covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q₁, q₂)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.     

Coincidence points of multivalued mappings in (<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetric spaces

The properties of (q₁, q₂)-quasimetric spaces are examined. Multivalued covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Given two multivalued mappings between (q₁, q₂)-quasimetric spaces such that one of them is covering and the other satisfies the Lipschitz condition, sufficient conditions for these mappings to have a coincidence point are obtained. A theorem on the stability of coincidence points with respect to small perturbations in the considered mappings is proved.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

On tree characterizations of <Emphasis Type="Italic">G</Emphasis><Subscript>δ</Subscript>-embeddings and some Banach spaces

We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a G _δ-embedding if and only if every T-null tree in S _E has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained. (a) A separable Banach space E has separable dual if and only if every w*-null tree in S _E * has a branch which is a boundedly complete basic sequence. (b) A Banach space E with separable dual has the point of continuity property if and only if every w-null tree in S _E has a branch which is a boundedly complete basic sequence. We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*-null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence. The research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Uniqueness of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroups on a general locally convex vector space and an application

The main purpose is to generalize a theorem of Arendt about uniqueness of C ₀-semigroups from Banach space setting to general locally convex vector spaces. More precisely, we show that cores are the only domains of uniqueness for C ₀-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique L ¹(ℝ ^d ,dx) weak solution.     

Vector Variational Inequalities and the (S)<Subscript>+</Subscript> Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)₊ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)₊. Some applications to vector complementarity problems are given.     

On generalized <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> and <Emphasis Type="Italic">R</Emphasis><Subscript>1</Subscript> spaces

Characterizations of some properties of generalized R ₀ and R ₁ topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Uniqueness of unconditional bases in nonlocally convex c<Subscript>0</Subscript>-products

We show that the c ₀-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)₀ of a natural quasi-Banach space X with strongly absolute unconditional basis has a unique unconditional basis up to permutation. Our results apply to a wide range of cases, including most of the c ₀-products of the nonlocally convex classical quasi-Banach spaces.     

Non-low<Subscript>2</Subscript>-ness and computable Lipschitz reducibility

In this paper, we prove that if a c.e. Turing degree d is non-low₂, then there are two left-c.e. reals β ₀, β ₁ in d, such that, if β ₀ is wtt-reducible to a left-c.e. real α, then β ₁ is not computable Lipschitz (cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex (wtt-complete) left-c.e. real.     

Similarity Results for Operators of Class <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

If T is a multiplicity-free contraction of class C ₀ with minimal function m _T , then it is quasisimilar to the Jordan block S(m _T ). In case m _T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m _T ) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor \({u\in H^\infty}\) of m _T , the result also holds in the more general setting where the roots have bounded multiplicities.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...     

Cone characterization of Grothendieck spaces and Banach spaces containing <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>

In this article we study the embeddability of cones in a Banach space X. First we prove that c ₀ is embeddable in X if and only if its positive cone c₀⁺{c_0^+} is embeddable in X and we study some properties of Banach spaces containing c ₀ in the light of this result. So, unlike with the positive cone of ℓ ₁ which is embeddable in any non-reflexive space, c₀⁺{c_0^+} has the same behavior as the whole space c ₀. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Finiteness conditions and PD<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-group covers of PD<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-complexes

We show that an infinite cyclic covering space M′ of a PD _n -complex M is a PD _n-1-complex if and only if χ(M) = 0, M′ is homotopy equivalent to a complex with finite [(n−1)/2]-skeleton and π₁(M′) is finitely presentable. This is best possible in terms of minimal finiteness assumptions on the covering space. We give also a corresponding result for covering spaces M _ν with covering group a PD _r -group under a slightly stricter finiteness condition.      

A Lower Bound of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> Norms in the CLT for Strongly Mixing Random Variables

We derive a lower bound of L _p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X ₁,..., X _n with under the boundedness condition ℙ{|X _i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ _r⩾1 r ²α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.     

Estimates of Distances Between Sums of the Spaces ℓ<Stack><Subscript>n</Subscript><Superscript>p</Superscript></Stack>. II

We study classical, modified, and weak Banach-Mazur distances between sums of the spaces ℓ _n ^p . We calculate explicitly the classical and weak Banach-Mazur distances between sums of the spaces ℓ _n ^p and obtain bounds for ratios of distances between sums of the spaces ℓ _n ^p . Bibliography: 14 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 203–217.