ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We study upper-bounded lattices and join-semilattices that have antitone involutions in all intervals $[x,1]$ (sectionally antitone involutions). On such a (semi)lattice we introduce a binary operation the properties of which characterize the original order and sectionally antitone involutions. It turns out that bounded lattices with sectionally antitone involutions satisfying a simple additional condition are distributive and correspond one-to-one to well-known $MV$-algebras.

AMS Subject Classification (1991): MSC: 06D05, 06D35, 08B05

Keyword(s): Distributive lattice, MV, -algebra, sectionally antitone involution, chain

Received September 30, 2003, and in revised form October 4, 2004. (Registered under 5854/2009.)

Abstract. In [1] we proved that a semigroup is a right commutative T1 semigroup if and only if it is isomorphic to a full $\Delta $-overact of a null semigroup by a commutative nil $\Delta $-semigroup with an identity adjoined (see Theorem 2.1 of [1]). Such overacts are defined by means of full $\Delta $-acts (see Construction 2.1 of [1]), and these acts are examined in the third section of [1] and are shown to be obtained by Construction 3.1 of [1]. In this note, we prove a simpler version of Theorem 2.1 of [1]: a semigroup is a right commutative T1 semigroup if and only if it is isomorphic either to a commutative nil $\Delta $-semigroup with an identity adjoined or to a semigroup $S=\{0,e,a\} $ obtained by adjoining to a null semigroup $A=\{0,a\} $ an idempotent $e$ that is both a right identity element of $S$ and a left annihilator element for $A$. This new fact shows that the constructions mentioned above provide only two very special types of semigroups and, actually, we do not need the notion of a full $\Delta $-overact and a full $\Delta $-act in order to describe right commutative T1 semigroups.

AMS Subject Classification (1991): 20M35

Received February 20, 2004, and in revised form September 16, 2004. (Registered under 5855/2009.)

Abstract. We say that a semigroup $S$ is a permutable semigroup if, for all congruences $\alpha $ and $\beta $ of $S$, $\alpha\circ \beta = \beta\circ \alpha $. In this paper we show that every permutable semigroup satisfying a permutation identity $x_1x_2\ldots x_n=x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(n)}, \sigma(1)\not=1, \sigma(n)\not=n$ is commutative. We also prove that every permutable semigroup satisfying an arbitrary non-trivial permutation identity is medial or an ideal extension of a rectangular band by a non-trivial commutative nil semigroup. The following problem is unsolved: Is every permutable semigroup satisfying a non-trivial permutation identity medial?

AMS Subject Classification (1991): 20M35

Received April 13, 2004, and in final form October 6, 2004. (Registered under 5856/2009.)

Abstract. All finite Moufang loops have the Lagrange property.

AMS Subject Classification (1991): 20N05, 20D06

Keyword(s): Moufang loop, Lagrange's Theorem, group with triality

Received October 12, 2004, and in revised form January 31, 2005. (Registered under 5857/2009.)

Abstract. For various Fourier expansions by orthogonal systems a Hörmander-type multiplier condition implies a Littlewood--Paley theory and from it a sharp Marchaud-type inequality is deduced for the $K$-functional related to the operator generating the orthogonal system and for related moduli of smoothness.

AMS Subject Classification (1991): 26A15, 41A27, 42B25

Keyword(s): Littlewood--Paley type inequality, Hörmander type multiplier theorems, Sharp Marchaud inequality

Received June 16, 2004, and in revised form November 5, 2004. (Registered under 5858/2009.)

Abstract. The definition of the class of sequences of rest bounded variation is extended in order to get a wider class of sequences. The new class is not anymore a subclass of the class of quasi monotone sequences. Namely a sequence of the new class may have infinitely many zero and negative terms. Two embedding theorems proved in [2] are generalized in such a way that the new results maintain the former embedding relations but now the coefficients of the sine series belong to the new class.

AMS Subject Classification (1991): 26A15, 42A10

Received January 21, 2004. (Registered under 5859/2009.)

Abstract. Generalizing the notions of Jensen and Wright convexity, we introduce the concept of $(M;p,q)$-convexity. Our main results show that, under continuity assumptions, $(M;p,q)$-convexity is equivalent to convexity.

AMS Subject Classification (1991): 26B25, 26D15, 39B62

Received September 6, 2004. (Registered under 5860/2009.)

Abstract. We investigate local properties of the Green function of the complement of a compact set $E\subset[0,1]$ with respect to the extended complex plane. We extend results of V. Andrievskii, L. Carleson and V. Totik which claim that the Green function satisfies the $1/2$-Hölder condition locally at the origin if and only if the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function and extend the results to the case $E\subset[-1,1]$.

AMS Subject Classification (1991): 30C10, 30C15, 41A10

Keyword(s): Green function, Equilibrium measure, Conformal invariants, Compact sets

Received September 27, 2004, and in revised form November 19, 2004. (Registered under 5861/2009.)

Abstract. In this paper we will present a definition for lacunary $P$-convergent double sequences analogues to the definition presented by Freedman, Sember and Raphael in 1978. This definition shall be used to establish a connection between the strongly double Cesaro summable and strongly double lacunary convergent sequences, and the connection strongly double almost convergence and strongly double lacunary convergent sequence. Also, we investigate inclusion relations among these spaces.

AMS Subject Classification (1991): 40A05, 40G05

Keyword(s): Double sequence, Lacunary sequence, Almost convergence, Cesaro summable.

Received January 5, 2004, and in final form September 15, 2004. (Registered under 5862/2009.)

Abstract. We extend the concept and basic results on statistical limit of measurable functions of one variable to those of two variables. We discuss the close connection between strong Cesàro summability and the existence of statistical limit. As an application, we prove that if $f\in L^1\cap L_{\rm loc}^\infty({\msbm R}^2)$, then the Dirichlet integral $s_\nu(f, x_1, x_2)$ of $f$ has statistical limit as $\nu\to \infty $ at every Lebesgue point $(x_1, x_2)$ of $f$ of order 2, that is, at almost every point of ${\msbm R}^2$. This means that a function $f$ in $L^1\cap L^\infty_{\rm loc}({\msbm R}^2)$ can be reconstructed by means of its Fourier transform in terms of statistical limit.

AMS Subject Classification (1991): 40C10, 42B10

Received June 29, 2004, and in final form February 15, 2005. (Registered under 5863/2009.)

Abstract. We establish embedding relations concerning the generalized Lipschitz-classes defined by L. Leindler.

AMS Subject Classification (1991): 42A10

Received February 4, 2004, and in revised form June 3, 2004. (Registered under 5864/2009.)

Abstract. It is known that the Fejér means -- with respect to the Walsh and bounded Vilenkin systems -- of a continuous function converge everywhere to the function, and this convergence is uniform. That is, the celebrated result of Lipót Fejér is valid also for these systems. In this work we discuss problems concerning the above written on similar, not necessarily Abelian, totally disconnected groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case completely differs from the commutative case. Even the theorem of Fejér fails to hold. We prove the existence of a $\gamma >0$, and $f\in{\rm Lip} (\gamma )$ Lipschitz function, such that $\sup |\sigma_nf|=+\infty $ on a dense set.

AMS Subject Classification (1991): 42C10

Keyword(s): noncommutative Vilenkin groups, character system, Fejér means, divergence, Lipschitz functions, Mathieu group

Received October 10, 2002, and in final form September 8, 2004. (Registered under 5865/2009.)

Abstract. We establish some general theorems for a wide class of sequences of superlinear operators $\{T_n : L^1(I^2) \to L^0(I^2),n = 1,2,\ldots\} $ about existence of a function $g$ from a certain class $L\phi(L)$ such that the sequence of functions $\{T_{n}(g), n=1,2,\ldots\} $ is not bounded in measure. These theorems imply several results about divergence for classical operators with respect to general and classical orthonormal systems.

AMS Subject Classification (1991): 42C15, 42C10

Received April 2, 2004, and in revised form December 15, 2004. (Registered under 5866/2009.)

Abstract. The compactness of the integral of an operator-valued function is proven in an elementary way, generalizing an earlier result of Karl H. Hofmann, related to the Peter--Weyl theory of continuous representations of compact groups.

AMS Subject Classification (1991): 43A50, 43A77, 43A65

Received March 9, 2004, and in revised form September 9, 2004. (Registered under 5867/2009.)

Abstract. A de Branges space is a reproducing kernel Hilbert space of entire functions which satisfies additional axioms. We consider such de Branges spaces whose elements possess a certain growth behaviour (e.g. are all of exponential type) and investigate the interplay of Hilbert space structure and growth behaviour. We characterize the presence of certain growth behaviour, prove the existence of spaces with prescribed growth and investigate the structure of subspaces defined by growth restrictions.

AMS Subject Classification (1991): 46E20, 30D15, 46E22

Keyword(s): de Branges space, growth function, exponential type

Received March 23, 2004, and in revised form March 7, 2005. (Registered under 5868/2009.)

Abstract. We introduce and study the essential numerical range for Banach space operators. This generalizes the corresponding well-known concept for Hilbert space operators.

AMS Subject Classification (1991): 47A12, 46H05

Keyword(s): essential numerical range, measure of non-compactness, Asplund spaces

Received November 5, 2004. (Registered under 5869/2009.)

Abstract. Extending a result of S. I. Ansari and P. S. Bourdon, it is proved that supercyclic, bounded representations of discrete abelian semigroups are stable. It is shown that there are supercyclic representations consisting entirely of non-supercyclic operators. Furthermore, we obtain as a consequence that supercyclicity implies stability for strongly continuous, bounded representations of ${\msbm R}_+$, too.

AMS Subject Classification (1991): 47A16, 47A13, 47A67, 47D06

Received April 13, 2004, and in final form October 11, 2004. (Registered under 5870/2009.)

Abstract. A particular case of the Kalman--Yakubovich--Popov inequality is studied, namely when the state operator and the feed through operator of the underlying linear time-invariant system are both zero. In this particular case, the system theory problems reduce to problems about representations of a bounded linear Hilbert space operator $K$ as a product $K=CB$, where $B\colon{\cal U}\rightarrow{\cal X}$ and $C\colon{\cal X}\rightarrow{\cal Y}$ are bounded linear Hilbert space operators too. These problems, which are inspired by [2], are of independent interest. We present an example of two pseudo-similar multiplicative representations of $K$, $K=C_1B_1$ and $K=C_2B_2$, such that the first is minimal, i.e, $\mathop{\rm Ker} B_1^*$ and $\mathop{\rm Ker} C_1$ are both equal to $\{0\} $, but the other is not minimal. Given a minimal multiplicative representation $K=CB$ of a contractive operator $K$, we describe the set of all (possibly unbounded) positive selfadjoint operators $H$ on $X$ such that $B_H=H^{1/2}B$ and $C_H =CH^{-1/2}$ define contractive operators which give a minimal multiplicative representation $K=C_HB_H$. The existence of minimal and maximal elements in this partially ordered set of positive operators $H$ is proved. Moreover we show that if $H$ is one of these two extremal elements, then the range of $B$ is a core for $H^{1/2}$. Other properties of this set are derived too.

AMS Subject Classification (1991): 47A45, 93B28, 47A48

Keyword(s): contractions, linear systems, factorization

Received June 8, 2004, and in revised form January 12, 2005. (Registered under 5871/2009.)

Abstract. A Hilbert space operator $T\in{\cal B}[{\cal H}]$ is said to be totally hereditarily normaloid, or $\cal THN$, if for every $T$-invariant subspace ${\cal M}\subseteq{\cal H}$ the restriction $T|_{\cal M}$ of $T$ to ${\cal M}$ is normaloid and, whenever $T|_{\cal M}\in{\cal B}[{\cal M}]$ is invertible, the inverse $(T|_{\cal M})^{-1}$ is normaloid as well. In this paper we explore the structure of $\cal THN$ contractions, and conclude some properties which follow from such a structure, specially for $\cal THN$ contractions with either compact or Hilbert--Schmidt defect operators.

AMS Subject Classification (1991): 47A45, 47B20

Keyword(s): Hereditarily normaloid, contractions, defect operator, decompositions

Received August 15, 2003, and in final form December 28, 2004. (Registered under 5872/2009.)

Abstract. The abstract analogue of classical two variables Toeplitz operators is presented. The existence of symbol operators is shown. Properties of the doubly commuting minimal isometric dilation of doubly commuting contractions are used.

AMS Subject Classification (1991): 47A62, 47A20,47B35, 47B38, 47B20

Keyword(s): Contraction, isometric dilation, doubly commuting, Toeplitz operator, symbol operator

Received June 26, 2003, and in final form October 4, 2004. (Registered under 5873/2009.)

Abstract. It is known that any positive integer power of an $\omega $-hyponormal operator is $\omega $-hyponormal. In this note we show that, for any $0< p\leq1$, there exists an invertible operator whose integer powers are all $p-\omega $-hyponormal. We also show that there exists a ${1\over2}-\omega $-hyponormal operator $T$ such that $T^3$ is not ${1\over2}-\omega $-hyponormal.

AMS Subject Classification (1991): 47B20, 47A63

Keyword(s): p-$\omega$, $\omega$-hyponormal, Furuta inequality

Received June 26, 2003. (Registered under 5874/2009.)

Abstract. We investigate Carleson measures $\mu $ on $\overline{\msbm D}$ where ${\msbm D}$ is the open unit disk in ${\msbm C}$, along with functional analytic properties of the formal identity of the Hardy space $H^p({\msbm D})$ into the Lebesgue space $L^q(\mu )$, for any previously fixed $0< p,q< \infty $. Our corresponding characterizations do not only extend the classical results for measures concentrated on ${\msbm D}$ but also provide different proofs for the latter ones. Among the applications are generalizations to formal identities as above of several results which have been known for composition operators only.

AMS Subject Classification (1991): 47B38, 46E5, 30D55, 47B33

Keyword(s): Carleson measures, Carleson embeddings, Hardy spaces, compactness properties

Received April 8, 2004. (Registered under 5875/2009.)

Abstract. The Dirac operators are important differential operators. Bismut got a local index theorem for the modified Dirac operators over the Spin manifolds. In this paper, we obtained a similar theorem over the Spin$^c$-manifolds and used the heat equation to prove this theorem.

AMS Subject Classification (1991): 53C27

Keyword(s): ^c, spin-manifold, Dirac operator, local index, MP Parametrix, Chern root

Received March 18, 2004, and in revised form April 16, 2005. (Registered under 5876/2009.)

Abstract. A fully probabilistic proof is given for Kruglov's basic characterization of the existence of generalized moments of infinitely divisible random variables.

AMS Subject Classification (1991): 60E07

Received October 4, 2004. (Registered under 5877/2009.)

Abstract. We propose a new test to check the null hypothesis that the distribution of the inter-event times of a point process belongs to a parametric family of distributions. The test is based on the joint asymptotic normality of the total time on test and partial sum processes.

AMS Subject Classification (1991): 62N05, 62G10

Received August 23, 2004, and in revised form September 13, 2004. (Registered under 5878/2009.)