ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. G. Czédli expressed the conjecture that there exists a lattice which is a quasi-fractal but fails to be a fractal. In the present note we show that there exists a proper class of mutually non-isomorphic lattices (in fact, chains) having the mentioned property.

AMS Subject Classification (1991): 06B9, 06F15

Keyword(s): lattice, chain, fractal lattice, quasi-fractal lattice

Received May 19, 2009, and in revised form June 1, 2009. (Registered under 68/2009.)

Abstract. We solve a restricted version of the Finite Type Problem from natural duality theory: if a finite unary algebra is dualisable, then it is dualisable via a finite set of relations.

AMS Subject Classification (1991): 08C20, 08A60

Keyword(s): natural duality, dualizable, unary algebra, Hanf number for dualizability

Received April 14, 2010, and in final form August 25, 2010. (Registered under 26/2010.)

Abstract. We completely determine all lower-modular elements of the lattice of all semigroup varieties. As a corollary, we show that a lower-modular element of this lattice is modular.

AMS Subject Classification (1991): 20M07, 08B15

Keyword(s): semigroup, variety, lattice of varieties, modular element, lower-modular element

Received February 24, 2010, and in final form May 6, 2010. (Registered under 15/2010.)

Abstract. Suppose that whenever a finite abelian group $G$ is a direct product of simulated subsets and one other subset then one of these factors must be periodic. In this paper we describe completely the class of all such finite abelian groups $G$. We then consider the same problem when cyclic subsets are admitted as well as simulated subsets. Once again a complete classification is presented.

AMS Subject Classification (1991): 20K01, 11B13

Keyword(s): factorization, Abelian group, simulated subset

Received February 2, 2009, and in revised form April 12, 2010. (Registered under 17/2009.)

Abstract. A semigroup $S$ is called a permutable semigroup if $\alpha\circ \beta =\beta\circ \alpha $ is satisfied for all congruences $\alpha $ and $\beta $ of $S$. A semigroup is called a Putcha semigroup if it is a semilattice of archimedean semigroups. In this paper we deal with finite permutable Putcha semigroups. We describe the finite permutable archimedean semigroups and finite permutable semigroups which are semilattices of a group and a nilpotent semigroup.

AMS Subject Classification (1991): 20M10

Keyword(s): permutable semigroups, Putcha semigroups, archimedean semigroups

Received April 28, 2009, and in revised form May 14, 2009. (Registered under 63/2009.)

Abstract. With a measure $\varphi $ on a $\sigma $-algebra $\Sigma $ of sets taking values in a Banach space two positive functions on $\Sigma $, called semivariations of $\varphi $, are associated. We characterize those functions as order continuous submeasures that are multiply subadditive in the sense of G. G. Lorentz (1952). In connection with some results of G. Curbera (1994) and the author (2003), we also discuss the special cases where $\varphi $ is separable and nonatomic or has relatively compact range.

AMS Subject Classification (1991): 28B05, 46G10, 28A12

Keyword(s): Banach space, vector measure, semivariation, relatively compact range, separable, nonatomic, submeasure, order continuous, multiply subadditive

Received March 23, 2009, and in revised form December 10, 2009. (Registered under 45/2009.)

Abstract. Fava's weak type $L\log L$ estimate for strong two-parameter ergodic maximal operators associated to pairs of commuting non-periodic measure-preserving transformations is shown to be sharp. Moreover, given a function $\phi $ on $[0,\infty )$ that is positive, increasing, and $o(\log(x))$ for $x \rightarrow\infty $ as well as a pair of commuting invertible non-periodic measure-preserving transformations on a space $\Omega $ of finite measure, a function $f \in L\phi(L)(\Omega )$ is constructed whose associated multiparameter ergodic averages fail to converge almost everywhere in the unrestricted sense.

AMS Subject Classification (1991): 47A25, 28D05, 28D15

Keyword(s): multiparameter ergodic averages, multiparameter ergodic maximal operators

Received June 16, 2009, and in revised form September 11, 2009. (Registered under 80/2009.)

Abstract. In this paper we discuss the spectral properties of a class of Jacobi operators defined by $\lambda_n=n^{\alpha }+c_n$ and $q_n=-2n^{\alpha }+b_n$, where $(c_n)$ and $(b_n)$ are real two-periodic sequences. From the asymptotic behavior of the solutions of the generalized eigenequation, which is in the double root case, a mixed spectrum is obtained.

AMS Subject Classification (1991): 39A10, 47B25

Keyword(s): Jacobi matrices, double root case, asymptotic behavior, subordination theory, absolutely continuous spectrum, discrete spectrum

Received July 31, 2008, and in final form March 12, 2010. (Registered under 73/2010.)

Abstract. In 2003, F. Móricz proved that the jumps of a periodic function at its simple discontinuities can be determined by its conjugate Abel--Poisson mean. Later Q. L. Shi and X. L. Shi introduced the concentration factors method of Abel--Poisson type and established a criterion for functions that satisfied a condition of Dini type. For piecewise smooth functions the convergence rate of this method is usually faster then Móricz Process. In this paper we establish a new criterion for concentration factors without the condition of Dini type.

AMS Subject Classification (1991): 42A50, 42A16

Keyword(s): jump, Abel--Poisson concentration factors, convergence rate

Received July 12, 2008, and in revised form August 30, 2010. (Registered under 74/2010.)

Abstract. The statement of Lemma 3.4 in [1] has a minor typographical error and the proof is incorrrect. Corrections are given.

AMS Subject Classification (1991): 42A55, 43A46; 43A05, 43A25

Keyword(s): associated sets, Bohr group, Fatou--Zygmund property, Hadamard sets, $I_0$ sets, Sidon sets

Received February 18, 2010. (Registered under 14/2010.)

Abstract. We show that a generalized form of the infimal convolution can be used to derive a dominated, monotone and additive extension theorem from the corresponding sandwich one on an abelian preordered semigroup with neutral element. Thus, we can put some of the results and arguments of Benno Fuchssteiner and Wolfgang Lusky into a proper perspective.

AMS Subject Classification (1991): 46A22, 06F05

Keyword(s): preordered semigroups, infimal convolutions, monotone additive functions, sandwich and extension theorems

Received June 18, 2009, and in revised form October 18, 2009. (Registered under 81/2009.)

Abstract. We investigate Banach lattices on which each semi-compact operator (resp. the second power of each semi-compact operator) is $b$-weakly compact.

AMS Subject Classification (1991): 46A40, 46B40, 46B42

Keyword(s): semi-compact operator, $b$-weakly compact operator, order continuous norm, KB-space

Received May 11, 2009, and in revised form June 26, 2009. (Registered under 67/2009.)

Abstract. In the theory of two-dimensional canonical (also called `Hamiltonian') systems, the notion of the Titchmarsh--Weyl coefficient associated to a Hamiltonian function plays a vital role. A cornerstone in the spectral theory of canonical systems is the Inverse Spectral Theorem due to Louis de Branges which states that the Hamiltonian function of a given system is (up to changes of scale) fully determined by its Titchmarsh--Weyl coefficient. Much (but not all) of this theory can be viewed and explained using the theory of entire operators due to Mark G. Kre?n. Motivated from the study of canonical systems or Sturm--Liouville equations with a singular potential, and from other developments in the indefinite world, it was a long-standing open problem to find an indefinite (Pontryagin space) analogue of the notion of canonical systems, and to prove a corresponding analogue of de Branges' Inverse Spectral Theorem. We gave a definition of an indefinite analogue of a Hamiltonian function and elaborated the operator theory of such `indefinite canoncial systems' in previous work. In the present paper we prove the corresponding version of the Inverse Spectral Theorem.

AMS Subject Classification (1991): 34A55, 46C20, 46E22, 30H05

Keyword(s): canonical system, Pontryagin space, Inverse Spectral Theorem

Received August 4, 2009. (Registered under 91/2009.)

Abstract. In this paper a new type of $*$-algebras is introduced and investigated in detail to obtain a new characterization of von Neumann algebras. The main theorem (Theorem 5) supplies a characterization of the $*$-algebras $*$-isomorphic to von Neumann algebras, not assuming any previously given analytic property. This work fits to the bundle of research performed by outstanding mathematicians devoted to similar characterization problems (see [5], [6], [8], [16]).

AMS Subject Classification (1991): 46L05, 46L10

Keyword(s): $C^*$-norm, von Neumann algebra, $GW^*$-algebra, $*$-representation

Received May 15, 2008, and in final form July 26, 2010. (Registered under 75/2010.)

Abstract. We show that an invertible operator $A$ is unitary if the numerical radius satisfies $w(A) \leqq 1$ and $w(A^{-1}) \leqq 1$.

AMS Subject Classification (1991): 47A12, 47A63

Keyword(s): numerical radius, unitary operator, numerical range

Received August 5, 2009, and in revised form August 28, 2009. (Registered under 93/2009.)

Abstract. A bounded operator $T$ in a separable Hilbert space ${\eufm H}$ is called quasi-selfadjoint if $\ker(T-T^*)\not=\{0\} $ and ${\eufm N}$-quasi-selfadjoint if ${\eufm N}\supseteq{\rm ran }(T-T^*)$, where ${\eufm N}$ is a subspace of ${\eufm H}$. An ${\eufm N}$-quasi-selfadjoint operator $T$ is called ${\eufm N}$-simple if the linear hull of $\{T^n{\eufm N}, n=0,1,\ldots\} $ is dense in ${\eufm H}$. We study the ${\eufm N}$-Weyl function $M(z)=P_{\eufm N}(T-zI_{\eufm H})^{-1}{\mathrel{|^{\kern -2pt\scriptscriptstyle\setminus }} }{\eufm N}$ of an ${\eufm N}$-quasi-selfadjoint operator and define its so-called ``Schur parameters". The main result of the paper is that any ${\eufm N}$-quasi-selfadjoint and ${\eufm N}$-simple operator is unitarily equivalent to an operator given by a special block operator Jacobi matrix constructed by means of the Schur parameters of its ${\eufm N}$-Weyl function.

AMS Subject Classification (1991): 47A45, 47A48, 47A56, 47B36

Keyword(s): quasi-selfadjoint operator, ${\eufm N}$-simple operator, the Weyl function, the Schur transformation, the Schur parameters, block Jacobi operator matrix

Received February 20, 2009. (Registered under 36/2009.)

Abstract. In this paper we consider several similarity-invariant classes of operators on a complex Hilbert space. A complete description, in terms of properties of various parts of the spectrum, is obtained for the operators in the closure and for the operators in the interior of each of these classes.

AMS Subject Classification (1991): 47A58, 47A56, 47L30

Keyword(s): algebra, polynomials, functional calculus

Received November 14, 2008, and in revised form May 10, 2010. (Registered under 55/2008.)

Abstract. It is well known that the Helly dimension of the direct sum of convex sets is the maximum of the Helly dimension of the summands. In this paper we shall investigate the Helly dimension of the $L_1$-sum of two centrally symmetric compact convex sets. In case of the $L_1$-sum, the Helly dimension is not determined by the Helly dimension of the summands. Our main result is to give sharp bounds for the Helly dimension of the $L_1$-sum depending on the Helly dimension of the summands.

AMS Subject Classification (1991): 52A35, 52B35

Keyword(s): convex sets, Helly dimension, $L_1$-norm

Received April 20, 2009, and in revised form June 22, 2009. (Registered under 50/2009.)

Abstract. We study those orbits of oblique projections under the action of the full unitary group of a Hilbert space ${\cal H}$, which are submanifolds of ${\cal B}({\cal H})$. We also consider orbits under the Schatten unitaries, and obtain a partial characterization of the submanifold condition for these orbits. Finsler metrics are introduced, and the minimality of metric geodesics is investigated.

AMS Subject Classification (1991): 22E65, 58E50, 58B20

Keyword(s): oblique projections, unitary orbits

Received June 23, 2009, and in revised form November 20, 2009. (Registered under 82/2009.)

Abstract. Fazekas and Klesov (2000) found conditions for almost sure convergence rates in the law of large numbers that effectively can be applied if maximal inequalities are available. In the spirit of Móricz (1976), we aim at using those conditions in a weakly dependent framework, and this trick is proved to be quite efficient, first in the standard law of large numbers and second in the nonparametric estimation context where rates of convergence of the density kernel estimates are also obtained.

AMS Subject Classification (1991): 60F15, 60F99, 60G10, 62G07

Keyword(s): dependence, mixing strong law of large numbers, rate of convergence, kernel estimators

Received December 11, 2008, and in revised form May 10, 2010. (Registered under 79/2008.)

Abstract. Let $p \in(0, \infty )$ be a constant and let $\{\xi_n\} \subset L^p(\Omega, {\cal F}, {\msbm P})$ be a sequence of random variables. For any integers $m, n \ge0$, denote $S_{m, n} = \sum_{k=m}^{m + n-1} \xi_k$. It is proved that, if there exist a nondecreasing function $\varphi\colon {\msbm R}_+\to{\msbm R}_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\ge2$ such that $$\sum_{n=0}^\infty\sup_{k \ge0} {\msbm E}\left|\frac{S_{k, a^n}} {\varphi(a^n)}\right|^p < \infty,\ \hbox{ then }\ \lim_{n \to\infty } {S_{0, n}\over\varphi (n)} = 0 \ \hbox{ a.s.} $$ This extends Theorem 1 in Chobanyan, Levental and Salehi [chobanyan-l-s] and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

AMS Subject Classification (1991): 60F15

Keyword(s): strong law of large numbers, moment inequality, self-similar processes, stable processes

Received May 1, 2009, and in revised form December 17, 2009. (Registered under 64/2009.)

Abstract. E. Babson and D. N. Kozlov defined the graph homomorphisms complex $\mathop{\rm Hom} (H,G)$ in [1]. This construction was introduced by Lovász to give lower bounds for the chromatic number of a graph. In this paper we prove that $\mathop{\rm Hom} (K_l,KG_{m,n})$ is homotopy equivalent to a wedge of $(m-nl)$-dimensional spheres, where $K_l$ is a complete graph and $KG_{m,n}$ is a Kneser graph. As a corollary we prove that, for a graph $G$, and positive integers $n$ and $l\ge2$, $\mathop{\rm ind} (\mathop{\rm Hom} (K_l,G))+ln\leq\chi _n(G)$.

AMS Subject Classification (1991): 55P15, 05C15

Keyword(s): graph homomorphisms complex, multichromatic number, topological lower bound

Received July 29, 2009, and in revised form January 19, 2010. (Registered under 88/2009.)