ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. A description is obtained of all pseudocomplemented semilattices $S$ for which the set of idempotent endomorphisms is closed under composition. Those $S$ for which the endomorphism monoid is orthodox are then identified.

AMS Subject Classification (1991): 06A12, 20M19; 08A35

Keyword(s): Endomorphism monoid, orthodox semigroup, regular semigroup, pseudocomplemented semilattice

Received September 4, 1995. (Registered under 5667/2009.)

Abstract. Lattices $L$ with $0$ are investigated such that each ideal of $L$ is of the form $\{x\colon \langle x,0\rangle\in \tau\} $ for some tolerance relation $\tau $. We show that $L$ has this property iff for any $b\in L$ and every unary lattice polynomial $p(x)$ with $p(0)=0$ we have $p(b)\le b$. If, in addition, $L$ is atomic then the ideal generated by any finite set of atoms in $L$ is shown to be a Boolean sublattice of $L$.

AMS Subject Classification (1991): 06B05, 06B10, 06B15

Keyword(s): Lattice, 0, tolerance, kernel, ideal, distributivity

Received March 16, 1995. (Registered under 5668/2009.)

Abstract. Gábor Czédli and György Pollák proved that if $P$ is a finite quasi ordered set in which no two incomparable elements have a common upper bound, then the coalitions form a quasi lattice. We give a short proof of this result.

AMS Subject Classification (1991): 06B99; 06A99

Keyword(s): Coalition, lattice

Received September 22, 1995, and in revised form October 11, 1995. (Registered under 5669/2009.)

Abstract. A binary relation $\rho $ on a set $U$ is strongly rigid if every universal algebra on $U$ such that $\rho $ is a subuniverse of its square is trivial. Rosenberg (1973) found a strongly rigid relation on every universe $U$ of at least 3 elements. We exhibit a new strongly rigid relation for every finite $U$ with $|U|\ge3$. We also show that, for $|U|=3$, there are only 2 strongly rigid binary relations up to isomorphism.

AMS Subject Classification (1991): 08A40

Received July 25, 1994, and in revised form November 8, 1994. (Registered under 5670/2009.)

Abstract. We present here a construction of an object which can be used as a free object for partial algebras and strong equations. Spectral algebras are useful in finding an algebraic characterization of classes of partial algebras definable by a set of strong equations.

AMS Subject Classification (1991): 08A55; 08B20

Received June 12, 1995. (Registered under 5671/2009.)

Abstract. We show that a minimal clone has a nontrivial abelian representation if and only if it is isomorphic to a minimal subclone of a finite cyclic group. As an application, we show that a minimal clone contains a Mal'cev operation if and only if it is isomorphic to the clone of idempotent operations of a group of prime order.

AMS Subject Classification (1991): 08B05

Keyword(s): Varietal product, tame congruence theory, transfer principles

Received November 16, 1994, and in revised form February 20, 1995. (Registered under 5672/2009.)

Abstract. We study arithmetical properties of sequences of polynomials defined over some field ${\msbm F}$ and satisfying various linear and nonlinear recurrence relations. Such sequences may describe the sequence of states of cellular automata. Some of the results obtained here are analogues of the corresponding ones for the case of recurrence sequences of integers. On the other hand, we also show that the function and number cases; the cases of one variable and of several variable polynomials; and the characteristic zero and positive characteristic cases for the base field ${\msbm F}$, each display substantial differences.

AMS Subject Classification (1991): 11B37, 11D61, 11D88

Received October 28, 1994, and in revised form March 16, 1995. (Registered under 5673/2009.)

Abstract. Rings satisfying various identities of the form $x_1\cdots x_n=w(x_1,\ldots,x_n)$, $|w|>n\ge2$, have been considered many times. The purpose of this paper is to give a structural theorem for rings satisfying an arbitrary identity of this form. We introduce the notion of characteristic quadruplet $(n,p,h,t)$ of such an identity, and using it we characterize rings satisfying this identity as ideal extensions of an $n$-nilpotent ring by a ring satisfying $x=x^{p+1}$ and satisfying also some additional conditions, determined by the numbers $h$ and $t$, on nilpotents, idempotents and regular elements.

AMS Subject Classification (1991): 16A30

Received March 17, 1995. (Registered under 5675/2009.)

Abstract. The polynomial interpolation problem is considered for a class of subdirectly irreducible groups $G$ with abelian monolith $H$. The results give a possibility for a function $A\rightarrow H$ with $A$ a finite subset in $G$ to decide whether it is polynomial or not. In particular, this allows the determination of the number of the unary polynomial functions for certain finite groups.

AMS Subject Classification (1991): 16Y30, 08A40

Received July 6, 1994, and in revised form May 31, 1995. (Registered under 5676/2009.)

Abstract. Partially free semigroups are a class of finite group-free commutative semigroups. This article gives the definition and first properties of these semigroups, and computes their cohomology.

AMS Subject Classification (1991): 20M14, 20M50

Received October 5, 1994. (Registered under 5677/2009.)

Abstract. It is well-known (see [4], [5]) that the set ${\cal I}_X$ of all partial one-to-one mappings on a set $X$ is an inverse semigroup, which is called {\it the symmetric inverse semigroup} on $X$, and that any inverse semigroup can be embedded up to isomorphism in ${\cal I}_X$ on a set $X$ (Preston-Vagner Representation Theorem). The purpose of this paper is to obtain a generalization of the Preston-Vagner representation for generalized inverse $*$-semigroups. Let ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}$ be the set of all partial one-to-one $\pi $-mappings on a $\pi $-set $X(\pi ';\{\sigma_{e,f} \} )$ and ${\cal M}$ the structure sandwich set determined by $X(\pi ';\{\sigma_{e,f} \} )$. Then we shall show that ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}({\cal M})$ is a generalized inverse $*$-semigroup, which is called {\it the} $\pi $-{\it symmetric generalized inverse $*$-semigroup on a} $\pi $-{\it set} $X(\pi ';\{\sigma_{e,f} \} )$ {\it with a structure sandwich set} ${\cal M}$, and that any generalized inverse $*$-semigroup can be embedded up to $*$-isomorphism in ${\cal G}{\cal I}_{X(\pi ';\{\sigma_{e,f} \} )}({\cal M})$ on a $\pi $-set $X(\pi ';\{\sigma_{e,f} \} )$.

AMS Subject Classification (1991): 20M20, 20M19, 20M17

Received January 26, 1995. (Registered under 5678/2009.)

Abstract. Denote the difference of order $n$ $$ \sum ^n_{k=0} C(n,k)(-1)^{n-k}f(u+k\delta ) $$ by $\Delta_f(n;u,\delta )$ provided $[u,u+n\delta ] \subset D(f)$ where $C(n,k)$ is the binomial coefficient ${n\choose k}$. Call a function $f$ with domain [0,1] unpredictable if $f$ is continuous and is identically zero on $[0,1/2]$ but not identically zero on any longer subinterval of $[0,1]$. We deal with the question: For $f$ unpredictable, how large are differences $\Delta_f(n;0,\delta )$ where $n\delta\in (1/2,1)$? We show that higher order differences of such a function exhibit a much more extreme asymptotic character than was previously known.

AMS Subject Classification (1991): 26E10

Received December 28, 1994. (Registered under 5679/2009.)

Abstract. The continuity of the solution map of an infinite time horizon differential inclusion problem is discussed. The main result is proved by using a stability principle for fixed point sets of set valued contractions.

AMS Subject Classification (1991): 34A60, 47H04

Received April 4, 1995, and in revised form September 5, 1995. (Registered under 5680/2009.)

Abstract. We give -- among others -- a necessary and sufficient condition for the asymptotic stability of the solutions of the differential equation $x^{\prime\prime }+h(t)x^\prime +x=0$ under the restrictions that $h(t)=2h_n>0$ on $[t_n,t_n+\tau_n]$, and $h(t)=0$ elsewhere, $0< \tau_n\le{\pi /2}$ and $t_n$ is a multiple of $\pi,n\in{\msbm N}$.

AMS Subject Classification (1991): 34D20, 34D05

Received October 24, 1994. (Registered under 5681/2009.)

Abstract. We investigate expansions according to the eigenfunctions of the Dirac differential operator, describing the motion of a particle in quantum mechanics. We get an upper estimate for the square sum of eigenfunctions and show that the expansions have convergence properties similar to those of classical Fourier series.

AMS Subject Classification (1991): 35P10, 81Q05

Received November 1, 1994. (Registered under 5682/2009.)

Abstract. We determine the continuous duals of the spaces of $\Lambda $-strongly null and $\Lambda $-strongly convergent sequences for nondecreasing exponentially bounded sequences $\Lambda $ of positive reals tending to infinity. This is a partial solution of the problem by Móricz in [2].

AMS Subject Classification (1991): 40H05

Received December 19, 1994, and in revised form February 8, 1995. (Registered under 5683/2009.)

Abstract. Answers to some extremal problems concerning polynomials are given.

AMS Subject Classification (1991): 41A05

Keyword(s): extremal problem, polynomial, L_p, -norm

Received June 20, 1995, and in revised form October 15, 1995. (Registered under 5684/2009.)

Abstract. We continue the investigation of simultaneous rational approximants $(Q_1/Q_0,\ldots,Q_m/Q_0)$, with common denominator polynomial $Q_0$, to a vector of functions $(f_1,\ldots,f_m)$ that forms a Nikishin system. Rather complete results regarding the nature of the $m$th remainder function $R_m$ and the location of its zeros for a large class of normal multi--indices have been proved in [DrSt4]. In this paper, we shall analyse the zeros of the other $(m-1)$ remainder functions $R_1,\ldots,R_{m-1}$ for a class of multi--indices that is close to diagonal. To that end, we introduce auxiliary Nikishin systems which allow us to permute the order of the functions in the original system.

AMS Subject Classification (1991): 41A21, 30E10

Keyword(s): Nikishin systems, simultaneous rational approximants, normality, orthogonal polynomials, multiple orthogonality

Received February 20, 1995. (Registered under 5685/2009.)

Abstract. Explicit formulas for Cotes numbers of the generalized Gaussian Hermite quadrature formula based on the zeros of the $n$-th Chebyshev polynomial and their asymptotic behavior as $n\to\infty $ are given. This provides a solution of a modification of Problem 26 of P. Turán [J. Approximation Theory, 29 (1980), 23-85].

AMS Subject Classification (1991): 41A55, 65D32

Received January 25, 1995. (Registered under 5686/2009.)

Abstract. We prove a new sufficient condition for a locally absolutely continuous function $F$ that vanishes at infinity to be the Fourier transform of a function $f\in L^1({\msbm R})$ (in sign: $F\in\hat L ({\msbm R}))$ or that of a function $f\in H^1 ({\msbm R})$ (in sign $F\in\hat H ({\msbm R}))$. Namely, if $F'\in\hat H^1({\msbm R})$, then (i) $F\in\hat L({\msbm R})$, (ii) $F\in\hat H({\msbm R})$ if and only if $F(0)=0$. We reformulate two preliminary results, which give also sufficient conditions for $F$ to belong to $\hat L({\msbm R})$ or $\hat H({\msbm R})$, respectively. Finally, we characterize the endomorphisms and the automorphisms of ${\cal M}_1({\msbm R})$, the Banach algebra of all $L^1({\msbm R})$ multipliers.

AMS Subject Classification (1991): 42A38, 43A22, 46J15

Keyword(s): Fourier transform, Hilbert transform, Hardy inequality, multiplier, Fourier-Stieltjes transform, Banach algebra, endomorphism, automorphism

Received July 21, 1994, and in revised form January 12, 1995. (Registered under 5687/2009.)

Abstract. Let $d\in{\msbm N}$, $\psi :[0,\infty )\to[0,\infty )$ be a nondecreasing function and $$\psi(u)=o\left(u(\log u)^{d-1}\log\log u\right ) (u\to\infty ).$$ Then there exists a function $f_1$ integrable on ${\msbm T}^d$ such that $$\int_{{\msbm T}^d}| \psi(f_1(x))| dx< \infty,$$ and the trigonometric Fourier series of the function $f$ over cubes unboundedly diverges everywhere. There exists also a function $f_2$ integrable on ${\msbm T}^d$ such that $$\int_{{\msbm T}^d}| f_2(x+h)-f_2(x)| dx= o(| h| ^{-1}/\psi(| h| ^{-1}))\qquad (h\to0),$$ and the trigonometric Fourier series of the function $f_2$ over cubes unboundedly diverges almost everywhere.

AMS Subject Classification (1991): 42B05

Received October 12, 1994, and in revised form June 16, 1995. (Registered under 5688/2009.)

Abstract. We study double cosine, double sine, and cosine-sine series whose coefficients $\{a_{jk}\} $ are such that $\sum_{j=0}^\infty\sum _{k=0}^\infty |a_{jk}|< \infty $. Then each of these series converges uniformly to some sum denoted by $f(x,y)$, $g(x,y)$, and $h(x,y)$, respectively. We give sufficient conditions for the $L^1$-integrability of the quotients $[f(x,y)- f(x,0)-f(0,y)+f(0,0)]/xy$, $g(x,y)/xy$, and $[h(x,y)-h(0,y)]/xy$. Our theorems extend those of F. Móricz from the one-dimensional to two-dimensional series.

AMS Subject Classification (1991): 42B99, 42A16

Received May 2, 1995. (Registered under 5689/2009.)

Abstract. The convergence of weighted Fourier expansions with respect to orthogonal polynomials is studied in the spaces $C(S)$ and $L^p(\pi )$, $1 \leq p < \infty $, where the support $S$ of the orthogonal measure $\pi $ is assumed to be compact. Necessary and sufficient conditions for convergence are given. The Dirichlet kernel is regarded especially in case of Jacobi polynomials. A Fejér-like kernel is introduced for a restricted class of orthogonal polynomials. Assuming the existence of a convolution structure on $C(S),$ the convergence of Fejér-like expansions is proved, particularly for Jacobi polynomials and generalized Chebyshev polynomials.

AMS Subject Classification (1991): 42C15

Received May 25, 1994, and in revised form December 2, 1994. (Registered under 5690/2009.)

Abstract. The purpose of this paper is to characterize the multipliers for the pair $(L^1(G,A),L^p(G,A))$, $1< p< \infty $, where $G$ is a locally compact abelian group and $A$, a commutative complex Banach algebra with a bounded approximate identity.

AMS Subject Classification (1991): 43A22, 47B38

Received January 3, 1995. (Registered under 5691/2009.)

Abstract. Recently, using perturbation theory, A. Dijksma and A. Gheondea proved the relation $$\kappa ^+(L^\bot )+{\dim }(L\cap H^-)=\kappa ^-(L) +{\dim }(L^\bot\cap H^+),$$ where $H^+\oplus H^-$ is a fundamental decomposition of a Kreĭn space $H$; $L$ is a closed subspace of $H$; $\kappa ^+$ and $\kappa ^-$ denote positive and negative signature respectively, and no distinction between infinite cardinals is made. We give an elementary proof of this relation.

AMS Subject Classification (1991): 46C20

Received August 11, 1994. (Registered under 5692/2009.)

Abstract. It is shown that totally normalised isotonic sublinear functionals inherit generalized versions of the Jessen, Lupaş and Hermite--Hadamard inequalities satisfied by isotonic linear functions.

AMS Subject Classification (1991): 46C50, 26D15

Received August 24, 1994. (Registered under 5693/2009.)

Abstract. In this paper a certain kind of approximate identities in a Banach algebra is introduced. Many Banach algebras possess such approximate identities. In the case of the group algebras $L^1(G)$ the existence of such approximate identities gives a new characterization of [SIN]-groups. Examples and hereditary properties are also investigated.

AMS Subject Classification (1991): 46H05, 43A20

Received January 4, 1995, and in revised form April 27, 1995. (Registered under 5694/2009.)

Abstract. We investigate Arens regularity of semisimple Banach algebras $A$ which can be continuously embedded as dense subalgebras of Arens regular semisimple Banach algebras $B$. We assume that the algebras $A$ are right weakly completely continuous or weakly completely continuous.

AMS Subject Classification (1991): 46H20, 46H10, 46H35

Received September 23, 1994, and in revised form March 2, 1995. (Registered under 5695/2009.)

Abstract. In this paper Jordan *-homomorphisms on rings of bounded operators acting on a Hilbert space and on the ring of bounded infinite sequences are studied. More precisely, we investigate the question whether the Jordan *-homomorphisms, whose ranges contain the finite rank elements, must be surjective or injective. In the first case both questions have affirmative \hbox{answers}. In the second case such homomorphisms are always surjective but not necessarily injective. For the ring of operators it is also proved that *-automorphisms can be characterized among additive mappings using the above range condition and only one one-variable identity.

AMS Subject Classification (1991): 46L40, 46K99, 46L70, 16W10, 54D35

Keyword(s): Jordan *-homomorphism, antihomomorphism, operator algebras, Stone-Čech compactification

Received December 22, 1994. (Registered under 5696/2009.)

Abstract. In this note, we give a direct self-contained proof of the Schur parametrization of all contractive intertwining liftings of an intertwining contraction. The original part of the proof resides in the use of the Möbius transform of contractions inspired by [14], Chapter 6.

AMS Subject Classification (1991): 47A20, 47A57

Received June 1, 1995. (Registered under 5697/2009.)

Abstract. Let $T\in{\cal L}({\cal H})$ be an absolutely continuous contraction such that the spectral multiplicity function of its unitary asymptote $T^a$ is at least $n (1\le n\le\aleph_0)$ almost everywhere on a subset $\gamma$ (with positive measure) of the unit circle $\bf T$. Let ${\cal G}_n$ be an $n$-dimensional Hilbert space and let $J_{n,\gamma}$ denote the canonical embedding of the Hilbert space $H^2({\cal G}_n)$ into $\chi_{\gamma}L^2({\cal G}_n)$. It is shown that, given any positive $\varepsilon$, there exists a factorization $J_{n,\gamma}=ZY$ such that the mappings $Y\in{\cal L}(H^2({\cal G}_n),{\cal H})$ and $Z\in{\cal L}({\cal H},\chi_{\gamma}L^2({\cal G}_n))$ intertwine $T$ with the operators of multiplication by the identical function on the corresponding spaces, and the product: $\|Y\| \|Z\|\le\sqrt2+\varepsilon$. As a consequence, we obtain that the unilateral shift $S_n$ of multiplicity $n$ can be completely injected into $T$ and that $T$ has the property $({\bf A}_{n,\aleph_0}(\gamma))$ introduced in the theory of dual algebras. It follows furthermore that in the special case $\gamma={\bf T}$ the contraction $T$ has an invariant subspace ${\cal H}'$ such that the restriction $T|{\cal H}'$ is similar to $S_n$.

AMS Subject Classification (1991): 47A45, 47A20, 47A15

Received April 27, 1995. (Registered under 5698/2009.)

Abstract. Considering a completely non-unitary contraction whose defect space, together to that of the adjoint, have dimension 2, we study its perturbations obtained by modifying the action on the defect spaces. we determine when the resulting contraction is also completely non-unitary. As an application, a theorem of De Branges about entire functions is recaptured in a completely different context.

AMS Subject Classification (1991): 47A45, 47A55

Received December 28, 1993, and in revised form February 20, 1995. (Registered under 5699/2009.)

Abstract. In this paper the approximation of a normal operator $A$ on a complex Hilbert space $\cal H$ by positive or self-adjoint operators or by positive contractions is discussed for the operator norm as well as for a norm $|||\cdot |||$ introduced by R.Bouldin. The dimension of the convex set of approximants is computed for both norms. Moreover, those normal operators are characterized for which the approximants in both norms are the same. This extends previously known results of R. Bouldin and of T. Sekiguchi.

AMS Subject Classification (1991): 47A58, 47B15

Received December 28, 1994, and in revised form February 10, 1995. (Registered under 5700/2009.)

Abstract. One of the natural problems on weighted composition operators $uC_\varphi $ is to determine the condition of $u$ and $\varphi $ for an operator $uC_\varphi $ to map bounded sets into sequentially compact sets, that is, for $uC_\varphi $ to become compact. In this paper, we shall characterize compact weighted composition operators on the domain of a closed $*$-derivation in $C(K)$ which is a generalization of spaces of differentiable functions.

AMS Subject Classification (1991): 47B38, 46E15, 46L57

Received June 29, 1994, and in revised form October 24, 1994. (Registered under 5701/2009.)

Abstract. Let $X$ be an infinite-dimensional Hilbert space. We prove that a surjective linear mapping $\phi\colon {\cal B}(X)\longrightarrow{\cal B}(X)$ that preserves the nilpotent operators in both directions is either of the form $\phi(T)=cATA^{-1}$ or of the form $\phi(T)=cAT^{tr}A^{-1}$, where $A$ is a bounded bijective linear operator on $X$, $c$ is a nonzero complex constant, and $T^{tr}$ denotes the transpose of $T$ relative to a fixed but arbitrary orthonormal basis. The description of all surjective linear mappings preserving nilpotent operators in both directions is slightly more complicated in the case that $X$ is an arbitrary Banach space.

AMS Subject Classification (1991): 47B49

Received April 28, 1994, and in revised form February 15, 1995. (Registered under 5702/2009.)

Abstract. Our purpose is to give a short proof of the statements that the Cesàro operator is bounded on both $H^1({\msbm R})$ and $H^1({\msbm T})$. The first statement is new in the literature, while the second one is known. The proof of the first statement is based on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm R})$ is such that its Fourier transform $\hat f(t)=0$ for $t\le0$, then $f\in H^1({\msbm R})$. The following reversed statement is also proved: If $f\in H^1({\msbm R})$, then $f$ can be represented in the form $f=f_1 + f_2$, where both $f_1$ and $f_2$ belong to $H^1({\msbm R}), \hat f_1(t)=0$ for $t\le0$, and $\hat f_2(t)=0$ for $t\ge0$. The proof of the second statement also relies on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm T})$ is such that its Fourier coefficient $\hat f(k)=0$ for $k=-1,-2,\ldots $, then $f\in H^1({\msbm T})$.

AMS Subject Classification (1991): 47D05

Keyword(s): Hilbert transform, Hardy space, Fourier transform, finite Borel measure, Fourier-Stieltjes transform, Cesàro operator, closed graph theorem, conjugate function, Fourier series, conjugate series

Received May 27, 1994, and in revised form January 4, 1995. (Registered under 5703/2009.)

Abstract. We show that each jointly quasinormal family of operators acting on a separable Hilbert space generates a reflexive algebra with property ${\msbm A}_1(1)$.

AMS Subject Classification (1991): 47D25, 47B20

Keyword(s): Reflexive operator algebra, quasinormal operator

Received June 2, 1994. (Registered under 5704/2009.)

Abstract. A random variational inequality is established which, in turn, is used to give a random best approximation theorem for a continuous set-valued mapping defined on a compact convex subset of a normed space. As an application, random fixed points are then derived. Finally, non-compact versions are also given. Our results include the corresponding results of Sehgal and Singh (1985) as special cases and our approach is different from those in the literature.

AMS Subject Classification (1991): 47H10, 49J41, 54C60, 46C05, 42A50

Keyword(s): Random variational inequality, random best approximation theorem, random fixed point theorem, nonself mapping

Received April 19, 1994, and in revised form February 21, 1995. (Registered under 5705/2009.)