Remarks on regular (LF)-spaces

Developing ideas found in a recent paper of Gilsdorf to arbitrary topological vector spaces (tvs) one shows that a Hausdorff (LB) _tv -spaceE is regular provided every null-sequence inE has a series convergent subsequence inE. Received by the editors 1980Mathematics Subject Classification (1985 Revision) 46A12.     

When a Compact (Countably Compact) set is Closed

We give necessary and sufficient conditions for a compact (countably compact) set to be closed in S ₂ (Fréchet, S ₂) and in normal (Fréchet, normal) spaces. Sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets, in such spaces. Countable compactness of the closure of a countably compact set in Fréchet, S ₂-spaces, and related results are also obtained.     

Universal classes of nuclear Köthe spaces with a continuous norm

We study the existence of universal and quotient universal spaces in the class of nuclear Köthe spaces with a continuous norm. It is shown that no countable set of these spaces has all L_f(a, r) spaces, ?∞ < r ? 0, as subspaces or those with 0 < r ? ∞ as quotients. On the other hand, the quotients and subspaces of (s) with basis constitute a universal and a quotient universal class, respectively.     

On the zeros ofζ (1)(s) — a (on the zeros of a class of a generalized Dirichlet series — XVII)

Some very precise results (see Theorems 4 and 5) are proved about thea-values of thelth derivative of a class of generalized Dirichlet series, forl≥l _o =l _o(a) (l _o being a large constant). In particular for the precise results on the zeros ofζ ⁽¹⁾ (s) —a (a any complex constant andl≥l _o) see Theorems 1 and 2 of the introduction.     

On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ₁ is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.     

Convolution on L p -spaces of a locally compact group

We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ L ^p (G). Here, we study the existence of f * g for all f, g ∈ L ^p (G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L ^p (G) * L ^p (G) to be contained in certain function spaces on G.     

Measure data problems,lower-order terms and interpolation effects

We deal with the solutions to nonlinear elliptic equations of the form $$-{\rm div}\, a(x, Du) + g(x, u)=f$$ , with f being just a summable function, under standard growth conditions on g and a. We prove general local decay estimates for level sets of the gradient of solutions in turn implying very general estimates in rearrangement and non-rearrangement function spaces, up to Lorentz–Morrey spaces. The results obtained are in clear accordance with the classical Gagliardo–Nirenberg interpolation theory.     

(DFS)-spaces of holomorphic functions invariant under differentiation

Let be a bounded convex domain, A^−∞(G) be the (DFS)-space of all holomorphic functions of polynomial growth on G and A^∞(G) be the Fréchet space of C^∞-functions on closure of G which are holomorphic on G. With the help of the Laplace transform we describe the strong dual of A^−∞(G) and prove that A^−∞(G) is the unique (DFS)-space H such that the space A^∞(G) is contained in H, H is embedded continuously in A^−∞(G) and H is invariant under differentiation.     

H p -spaces of martingales, 0<p≦1

For 0 <p≦1 the dual Banach space to the space of H _p -martingales is characterized as a Lipschitz class of order α=l/p?1. When p=1 and α=0 this reduces to BMO being the dual of H ₁ . In the course of the investigation a hierarchy of martingale spaces _a L _p is introduced with the property that the index p is in the range ?∞ <1/p<∞. For l≦b≦q<∞ the space L _q is a subspace of _b L _q (equality when b=q) while for q<0 the space _b L _q corresponds to a Lipschitz class of order α=?1/q.     

Uniqueness of linear combinations (mod p)

In an earlier paper [1] one of us examined the question: Given a prime p what is the maximal number n=f(p) in any set of residues {a₁, a₂, …, a_n} (mod p) so that at least one of the differences a_i−a_j is incongruent to all other differences a_i′−a_j′?It was determined that the answer is of the order of magnitude log p. That is, there exist constants 0<c₁<c₂ so that c₁ log p<f(p)<c₂ log p for all p.In this note we examine a natural generalization, where instead of differences we examine the set of linear expressions with prescribed coefficients.     

A remark on CD 0(K)-spaces

A representation of the CD ₀(K)-space is given in [1, 2] for a compact Hausdorff space K without isolated points. We generalize this to an arbitrary countably compact space K without any assumption on isolated points.     

Characterizations of stratifiable μ-spaces

We show that the following classes of topological spaces coincide: (1) stratifiable μ-spaces, (2) stratifiable spaces with an M-structure, (3) regularly stratifiable spaces, and (4) strongly regularly stratifiable spaces.     

The core operator and congruent submodules

The Hardy spaces H²(D²) can be conveniently viewed as a module over the polynomial ring C[z₁,z₂]. Submodules of H²(D²) 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.     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

Some results concerning Lp(μ)-spaces

It is proved that L^p(μ) does not have an unconditional basis if the cardinality of L^p(μ) is sufficiently large and μ is a finite measure. It is also shown that L^p(μ) has a weaker kind of basis for arbitrary μ and 1 < p < ∞. A new truncation lemma concerning sequences in L^p equivalent to the usual l^p-basis is given. This lemma is used in solving the problem of when l^p(Γ) imbeds in L^r(μ) for uncountable sets Γ and finite measures μ. It may also be used to give a nonprobabilistic proof of the fact (due to Schwartz-Kwapien) that there exist non-q-absolutely summing operators from L^∞to L^qfor 2 < q < ∞. It is again used in proving that basic sequences in L^p equivalent to the usual l^p-basis admit subsequences with a complemented linear span. Other applications of the techniques introduced are also given.     

Approximation by spherical waves inL p -spaces

The paper is devoted to the following problem. Consider the set of all radial functions with centers at the points of a closed surface inR ⁿ . Are such functions complete in the spaceL ^q (R ⁿ )? It is shown that the answer is positive if and only ifq is not less than 2n/(n + 1). A similar question is also answered for Riemannian symmetric spaces of rank 1. Relations of this problem with the wave and heat equations are also discussed.     

On multiplicative perturbation of integral resolvent families

In this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a Banach space X. Assuming that a(t) is a creep function which satisfies a(⁰⁺)>0, we prove that if (A,a) generates an integral resolvent, then (A(I+B),a) also generates an integral resolvent for all B∈B(X,Z), where Z belongs to a class of admissible Banach spaces. In special instances of a(t) the space Z is proved to be characterized by an extended class of Favard spaces.