ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. On a triangular grid $T$ a set $H$ of triangles with vertices at grid points is called a system of triangular islands if for every pair of triangles in $H$ one of them contains the other or they do not overlap at all. Let $I_{T}$ denote the ordered set of systems of triangular islands on $T$ and let $\max(I_{T})$ denote the maximal elements of $I_{T}$. With $n + 1$ grid points on each side of $T$ define $f(n)=\max\{|H|:H \in\max (I_{T})\} $. E. K. Horváth, Z. Németh, and G. Pluhár [3] proved $(n^2+3n)/5 \leq f(n) \leq(3n^2+9n+2)/14$. For $g(n)=\min\{|H|:H \in\max (I_{T})\} $ we show $g(n)=n$ and investigate extensions to triangular grids on trapezoids and parallelograms.

AMS Subject Classification (1991): 05A05, 05A16

Keyword(s): maximal systems of triangular islands, lower bound, upper bound, asymptotic behavior

Received July 8, 2008, and in revised form March 29, 2009. (Registered under 6418/2009.)

Abstract. It is well known that exactly two subvarieties of the variety of lattices cover the variety of distributive lattices. In a generalization of lattices, the weakly associative lattices, three more covering varieties are known. In this paper we consider a further generalization, weak lattices. We get this variety by omitting all identities keeping only the eight absorption laws. We shall prove that in this variety the variety of distributive lattices is covered by infinitely many subvarieties.

AMS Subject Classification (1991): 06A20

Keyword(s): weak lattice, weakly associative lattice, absorption law, variety

Received December 14, 2007, and in final form March 20, 2009. (Registered under 6419/2009.)

Abstract. Let $L_1$ be a finite lattice with an ideal $L_2$. Then the restriction map is a $\{0,1\} $-homomorphism from $\mathop{\rm Con} L_1$ into $\mathop{\rm Con} L_2$. In 1986, the present authors published the converse. If $D_1$ and $D_2$ are finite distributive lattices, and $\varphi \colon D_1 \to D_2$ is a $\{0,1\} $-homomorphism, then there are finite lattices $L_1$ and $L_2$ with an embedding $\eta$ of $L_2$ as an ideal of $L_1$, and there are isomorphisms $\varepsilon_1\colon\mathop{\rm Con} L_1 \to D_1$ and $\varepsilon_2 \colon\mathop{\rm Con} L_2 \to D_2$ such that $\varphi$ is represented as the restriction map of congruences from $L_1$ to $L_2$, up to the two isomorphisms. Let us call a lattice isoform, if for any congruence, all congruence classes are isomorphic lattices. In 2003, G. Grätzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of an isoform lattice. In this paper we combine the two results, reproving the 1986 result with isoform lattices.

AMS Subject Classification (1991): 06B10, 06B15

Keyword(s): congruence lattice, congruence-preserving extension, isoform

Received November 12, 2008, and in final form April 12, 2009. (Registered under 6420/2009.)

Abstract. We prove a representation theorem for any $d$-algebra $A$ with the point separating order dual $A^{\prime }$.

AMS Subject Classification (1991): 06F25, 47B65

Keyword(s): d, f, Arens product, -algebras, separating couple

Received July 16, 2008, and in revised form October 4, 2008. (Registered under 6421/2009.)

Abstract. The distance of an operation from being associative can be ``measured'' by its associative spectrum, an appropriate sequence of positive integers. Associative spectra were introduced in a publication by B. Csákány and T. Waldhauser in 2000 for binary operations (see [CsakanyWaldhauser]). We generalize this concept to $2 \le p$-ary operations, interpret associative spectra in terms of equational theories, and use this interpretation to find a characterization of fine spectra, to construct polynomial associative spectra, and to show that there are continuum many different spectra. Furthermore, an equivalent representation of bracketings is studied.

AMS Subject Classification (1991): 08B05, 08B15, 08A62, 05C05

Keyword(s): associative spectrum, bracketing, term operation, equational theory, tree, Catalan numbers

Received September 1, 2008. (Registered under 6422/2009.)

Abstract. If $E$ is an elliptic curve, then the Galois group of the extension generated by the $n$-torsion points acts on these points. We prove a quadratic reciprocity law involving this group action. This law is an extension of the usual quadratic reciprocity law.

AMS Subject Classification (1991): 14H52

Keyword(s): Elliptic curve, torsion, Galois group, quadratic reciprocity law

Received May 23, 2008, and in revised form August 21, 2008. (Registered under 6423/2009.)

Abstract. In this paper we study complex representations of the factorpower ${{\cal FP}^+}(G,M)$ of a finite group $G$ acting on a finite set $M$. This includes the finite monoid ${{\cal FP}^+(S_n)}$, which can be seen as a kind of ``balanced'' generalization of the symmetric group $S_n$ inside the semigroup of all binary relations. We describe all irreducible representations of ${{\cal FP}^+}(G,M)$ and relate them to irreducible representations of certain inverse semigroups. In particular, irreducible representations of ${{\cal FP}^+(S_n)}$ are related to irreducible representations of the maximal factorizable submonoid of the dual symmetric inverse monoid. We also show that in the latter cases irreducible representations lead to an interesting combinatorial problem in the representation theory of $S_n$, which, in particular, is related to Foulkes' conjecture. Finally, we show that all simple ${{\cal FP}^+}(G,M)$-modules are unitarizable and that tensor products of simple ${{\cal FP}^+}(G,M)$-modules are completely reducible.

AMS Subject Classification (1991): 20M30, 20M18, 20C30

Keyword(s): symmetric group, simple module, factorpower, Foulkes' conjecture, tensor product

Received July 10, 2008, and in revised form Spetember 19, 2008. (Registered under 6424/2009.)

Abstract. Minimal *-biideals of involution rings are described.

AMS Subject Classification (1991): 16W10

Keyword(s): involution ring, minimal biideal

Received July 14, 2008, and in revised form April 8, 2009. (Registered under 6425/2009.)

Abstract. We find conditions for the regular convergence of multiple series of independent terms in terms of the same three series as in Kolmogorov's theorem.

AMS Subject Classification (1991): 26A15

Keyword(s): multiple sums, almost sure convergence, regular convergence

Received December 11, 2008, and in revised form March 20, 2004. (Registered under 78/2008.)

Abstract. We give here some equivalent definitions of the so-called $\rho $-Carleson measures when $\rho(t)=(\log(4/t))^p(\log\log (e^4/t))^q$, $0\le p,q< \infty $. As applications, we characterize the pointwise multipliers on $LMOA({\msbm S}^n)$ and from this space to $BMOA({\msbm S}^n)$. Boundedness of the Cesàro type integral operators on $LMOA({\msbm S}^n)$ and from $LMOA({\msbm S}^n)$ to $BMOA({\msbm S}^n)$ is considered as well.

AMS Subject Classification (1991): 28A25; 30D45; 30D50; 47B38

Keyword(s): Carleson measures, Hardy spaces, Bergman spaces, Bloch spaces, BMOA, LMOA

Received September 26, 2008, and in final form January 28, 2009. (Registered under 6426/2009.)

Abstract. We characterize the generalized bi-circular projections on various Banach spaces of both scalar and vector valued analytic functions, including the Bergman, Bloch, and Hardy spaces. We also establish that the only projections in the convex hull of two isometries on a Hardy space are generalized bi-circular projection.

AMS Subject Classification (1991): 30D55; 30D05

Keyword(s): isometry, convex combination of isometries

Received July 16, 2008, and in revised form September 10, 2008. (Registered under 6427/2009.)

Abstract. The author extends some results of Z. Ditzian and S. Tikhonov in [3] to the case of functions defined on bounded and unbounded domains with inner singularities.

AMS Subject Classification (1991): 41A17; 41A10; 18E20

Keyword(s): embedding theorems, polynomial inequalities, moduli of smoothness, best approximation

Received September 2, 2008, and in revised form November 18, 2008. (Registered under 6428/2009.)

Abstract. Let ${\eufm M}$ be the set of (all) linear transformations of ${\msbm R}^N$, and let ${\eufm A}$ be an arbitrary set of positive measure, ${\eufm A} \subset{\msbm T}^N=[-\pi,\pi )^N$, $N\ge2$. We study the problem: how are the sets of convergence and divergence everywhere or almost everywhere (a.e.) of multiple trigonometric Fourier series (summed over rectangles) of the function $(f\circ{\eufm m})(x)=f({\eufm m}(x))$, if $f\in L_p({\msbm T}^N)$, $p\ge1$, $f(x)=0$ on ${\eufm A}$ and ${\eufm m}\in{\eufm M}$, changed (if changed) depending on the transformation ${\eufm m}$. In the paper we give some classes (of nonsingular linear transformations) $\Psi $, $\Psi\subset {\eufm M}$, which ``change" the sets of convergence and divergence everywhere or a.e. of the indicated Fourier series. Such classes are, in particular: {\it a}) the class of transformations consisting of ``almost all" elements of the group of rotations of ${\msbm R}^N$ about the origin; {\it b}) the class of transformations whose inverse transformations have matrices ${\msbm A}=\{a_{j m}\} _{j, m=1}^N$ satisfying the condition: there exists $k$, $1\le k \le N$, such that $\max_{1\le j\le N}|a_{j k}| < 1.$ Let us note that in the paper we consider two settings of the problem under investigation (depending on the way the Fourier series of a function $f\circ{\eufm m}$ is understood).

AMS Subject Classification (1991): 42B05

Keyword(s): multiple trigonometric Fourier series, convergence and divergence everywhere and almost everywhere, linear transformations, rotation group

Received October 26, 2008, and in revised form July 9, 2009. (Registered under 6429/2009.)

Abstract. New Wiener amalgam spaces are introduced for local Hardy spaces. It is proved that the maximal Fejér operator is bounded from the amalgam space $W(h_{p},\ell_\infty )$ to $W(L_{p},\ell_\infty )$. This implies the almost everywhere convergence of the Fejér means for all $f\in W(L_{1},\ell_\infty )\supset L_1$.

AMS Subject Classification (1991): 42B08, 46E30; 42B30, 42A38

Keyword(s): Wiener amalgam spaces, local Hardy spaces, Fejér summability, Fourier transforms, atomic decomposition

Received October 10, 2008. (Registered under 6430/2009.)

Abstract. We investigate the order of magnitude of the modulus of continuity of a function $f(x,y)$ with absolutely convergent double Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belong to one of the generalized Lipschitz classes Lip($\alpha, \beta; L$) and Lip($\alpha, \beta; 1/L$), where $0 \leq\alpha, \beta\leq 1$, $L=L(x,y)=L_1(x) L_2(y)$ is positive and $L_1(x)$ and $L_2(y)$ are non-decreasing, slowly varying functions such that $L_1(x), L_2(y) \rightarrow\infty $ as $x,y \rightarrow\infty $. These sufficient conditions are also necessary in the case of a certain subclass of Fourier coefficients.

AMS Subject Classification (1991): 42B99, 42A32, 26A15

Keyword(s): Fourier series, absolute convergence, multiplicative modulus of continuity, generalized multiplicative Lipschitz classes

Received October 7, 2008, and in revised form January 20, 2009. (Registered under 6431/2009.)

Abstract. Given a bounded positive linear operator $A$ on a Hilbert space ${\cal H}$ we consider the semi-Hilbertian space $({\cal H}, \left\langle, \right\rangle _A)$, where $\left\langle \xi, \eta\right \rangle_A= \left\langle A\xi, \eta\right \rangle $. On the other hand, we consider the operator range $R(A^{1/2})$ with its canonical Hilbertian structure, denoted by ${\bf{R}}(A^{1/2})$. In this paper we explore the relationship between different types of operators on $({\cal H}, \left\langle, \right\rangle _A)$ with classical subsets of operators on ${\bf{R}}(A^{1/2})$, like Hermitian, normal, contractions, projections, partial isometries and so on. We extend a theorem by M. G. Krein on symmetrizable operators and a result by M. Mbekhta on reduced minimum modulus.

AMS Subject Classification (1991): 46C05, 47A05, 47A30

Keyword(s): A, -operators, operator ranges

Received July 11, 2008. (Registered under 6432/2009.)

Abstract. In this note, it is proved that multiplier algebras of analytic reproducing kernel Hilbert spaces which are compatible with the action of the torus group possess Kraus' completely contractive approximation property (CCAP) and, consequently, have the Property $S_\sigma $. Our results apply in particular to the usual reproducing kernel Hilbert spaces on bounded symmetric domains.

AMS Subject Classification (1991): 46E22, 47B32, 47L45

Keyword(s): operator algebras, reproducing kernel Hilbert spaces, multipliers of reproducing kernel Hilbert spaces, approximation properties

Received July 11, 2008, and in revised form October 11, 2008. (Registered under 6433/2009.)

Abstract. In this article we extend the notions of a quasipolar, polar and Browder element in a normed algebra to a multivariable setting. A decomposition result is established for Browder tuples of bounded operators and characterisations in terms of ascent and descent are provided for some special cases. Finally we show that a multivariable spectral mapping theorem holds for a large class of Browder joint spectra.

AMS Subject Classification (1991): 47A13, 47A60, 47A10

Keyword(s): Browder spectrum, joint spectrum, spectral mapping theorem

Received September 23, 2008, and in revised form March 27, 2009. (Registered under 6434/2009.)

Abstract. Regular subspaces are tensor products of subspaces. The structure of regular subspaces that are invariant or reducing for the tensor product of a finite collection of Hilbert space operators is entirely characterized. Necessary and sufficient conditions for a multiple tensor product of operators to be a unilateral shift are established, and it is proved that a multiple tensor product of operators is a completely nonunitary contraction if and only if each factor is a contraction, one of them being completely nonunitary.

AMS Subject Classification (1991): 47A80, 47A15

Keyword(s): tensor product, Hilbert space operators, invariant subspaces

Received October 1, 2008, and in final form April 8, 2009. (Registered under 6435/2009.)

Abstract. In this paper, we study a weighted composition operator $uC_{\varphi }$ on the weighted Bergman space $L_{a}^2(dA_{\alpha })$ of the unit disc ${\msbm D}$. We estimate the essential norm of this type of operator. As a consequence of this estimate, we give a function-theoretic characterization of $u$ and $\varphi $ that induce a compact weighted composition operator on $L_{a}^2(dA_{\alpha })$.

AMS Subject Classification (1991): 47B38, 47B33

Keyword(s): weighted composition operator, Bergman space

Received July 23, 2008, and in revised form February 23, 2009. (Registered under 6436/2009.)

Abstract. The existence of common best proximity points for a finite commuting family of relatively nonexpansive mappings is proved. In a Hilbert space setting, the result is proved for an arbitrary family of commuting relatively nonexpansive mappings. Also the structure of the set consisting of all best proximity points of a relatively nonexpansive map is discussed.

AMS Subject Classification (1991): 54H25, 47H10

Keyword(s): proximal pair, proximal normal structure, relatively nonexpansive mapping, fixed point, commuting family, metric projection

Received July 4, 2008. (Registered under 28/2008.)

Abstract. We consider random fields with iid random variables and prove strong laws for delayed sums, i.e., sums over rectangular blocks of these random variables. We show that strong laws hold iff appropriate moment conditions are satisfied. These moment conditions depend on the size and shape of the blocks on which the delayed sums are based.

AMS Subject Classification (1991): 60F05

Keyword(s): strong laws, random fields, delayed sums, sums over rectangular blocks, moment conditions

Received October 2, 2008, and in final form September 14, 2009. (Registered under 6437/2009.)