ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. A precise title would be: congruences on finitely generated free commutative semigroups, such that the quotient semigroup is finite group-free. Classes modulo such a congruence run parallel to faces of the free semigroup; this permits a description which is more explicit and detailed than Rédei's. Our results may be viewed as constructing all finite group-free commutative semigroups.

AMS Subject Classification (1991): 20M14

Keyword(s): finitely generated free commutative semigroups

Received February 28, 1991 and in revised form November 6, 1991. (Registered under 5529/2009.)

Abstract. If $A$ is a finite group or a finite lattice then the lattice of subvarieties of the variety generated by $A$ is finite (see Neumann [9] and Grätzer [3]). However, this is not the case for more general algebras (Dziobiah [2]) or for semigroups (Trakhtman [14]). It is an immediate consequence of the work of Kadourek [4] that there exist finite inverse semigroups generating varieties with an infinite number of subvarieties. Here we show that there exists an eight element inverse semigroup generating a variety with infinitely many subvarieties and which is the smallest such inverse semigroup. We also identify the smallest combinatorial inverse semigroups (two of size fourteen) with the same property.

AMS Subject Classification (1991): 20M07

Received August 16, 1991, and in revised from November 5, 1992. (Registered under 5530/2009.)

Abstract. A semigroup $S$ with zero is {\it categorical at zero} or a {\it C-semigroup} if given any $a,b,c\in S$, where $ab,bc\in S\setminus\{0\} $, then it follows that $abc\in S\setminus\{0\} $. A semigroup with zero is {\it left (right) idempotent bounded} if it is the $0$-direct union of idempotent generated principal left (right) ideals and {\it idempotent bounded} if it is both left and right idempotent bounded. We abbreviate these concepts by LIB, RIB and IB respectively. The aim of this paper is to give a structure theorem for the class of LIB C-semigroups in terms of what we call {\it left blocked Rees matrix semigroups}. In a sequel, we specialize our work to describe IB C-semigroups in terms of {\it double blocked Rees matrix semigroups}. The classes of LIB C-semigroups and IB C-semigroups are extremely large, including the classes of primitive abundant semigroups and completely $0$-simple semigroups. Consideration of semigroups that are idempotent bounded on the left only, allows us to move away from structure theorems for C-semigroups satisfying conditions that are inherently two sided, such as regularity.

AMS Subject Classification (1991): 20M10

Keyword(s): left blocked Rees matrix semigroups

Received October 10,1991, and in revised form November 20,1992. (Registered under 5531/2009.)

Abstract. We describe all maximal distributive subsemilattices of a given finite distributive lattice and characterize the covering relation in the poset of finite distributive semilattices. Then we prove that there is no simultaneous representation for the poset of all distributive subsemilattices of the boolean lattice of all subsets of a four-element set by congruence semilattices of finite atomistic lattices.

AMS Subject Classification (1991): 06B10, 06D05

Received November 6, 1991. (Registered under 5532/2009.)

Abstract. A finite algebra $\underline A=(A;F)$ is called preprimal if $\underline A$ is not primal (functionally complete) but for every operation $f$ defined on $A$ which is not a term operation of $\underline A$ the algebra $\underline A'=(A;A;F\cup\{f\} )$ is primal. Some of the preprimal algebras were determined by S. V. Jablonskij in the early sixties. I. G. Rosenberg gave a complete list in 1970. In this paper all varieties generated by single preprimal algebras, their dualities and category equivalences are studied. The most surprising result is that up to category equivalence the algebra of Jablonskij's "small" list represent all preprimal algebras.

AMS Subject Classification (1991): 08A40, 08C99, 18B99

Keyword(s): Finite algebra, preprimal

Received November 20, 1991 and in revised form February 5, 1993. (Registered under 5533/2009.)

Abstract. In the category of universal algebras of the same given type that fulfil the so-called interchange laws we discover two full subcategories one of which is exponential in the other and, consequently, cartesian closed.

AMS Subject Classification (1991): 08C05, 18D15

Received August 14, 1990. (Registered under 5534/2009.)

Abstract. A clone of operations is called $n$-collapsing if it is uniquely determined by its $n$-ary operations. This paper gives internal and external characterizations of $n$-collapsing clones, for arbitrary positive integers $n$. The strongest results are obtained in the case $n=1$ which is therefore treated separately.

AMS Subject Classification (1991): 08A40, 03B50, 20M20

Received January 30, 1992, and in revised form July 13, 1993. (Registered under 5535/2009.)

Abstract. It is well known that the only simple distributive lattice is the two-element chain. In an earlier paper, we introduced the concept of a complete-simple lattice, and proved the existence of infinite complete-simple distributive lattices. In this paper we provide a new proof which is easier to visualize.

AMS Subject Classification (1991): 06B10, 06D05

Keyword(s): simple distributive lattice

Received March 9, 1992 and in revised form December 11, 1992. (Registered under 5536/2009.)

Abstract. A very general joint result of the author and A. Meir, which gives estimates for pointwise approximation of the function in question by the partial sums of a general orthogonal series, is extended here such that the partial sums are replaced by Cesaro means of negative order; and the external strong summation methods can be taken from the same large family of Hausdorff and $[J,f]$-transformation as in the original theorem.

AMS Subject Classification (1991): 41A25, 41A30, 42C15

Keyword(s): partial sum, pointwise approximation

Received August 6, 1991. (Registered under 5537/2009.)

Abstract. The Radon transform that integrates a function in ${\rm S}^n$, the $n$-dimensional sphere, over totally geodesic submanifolds with codimension 1, the great circles, and the dual Radon transform are investigated in this paper. Inversion formulas, range spaces and null spaces are given.

AMS Subject Classification (1991): 44A05

Keyword(s): Radon transform, spherical harmonics, elliptic geometry

Received 28 November, 1991. (Registered under 5538/2009.)

Abstract. This paper discusses the oscillation of nonlinear hyperbolic equation with deviating arguments of the form $${\partial ^2u\over\partial t^2}=a(t)\Delta u +\sum_{i=1}^ma_i(t)\Delta u(x,\rho_i(t)) -\sum_{j=1}^kP_j(x,t)f_j(u(x,\sigma_j(t))),$$ $(x,t)\in\Omega \times(0,\infty )$, where $\Omega\subset {\msbm R}^n$ is a bounded domain with a piecewise smooth boundary, $u=u(x,t)$ and $\Delta $ is the Laplacian in Euclidean $n$-space ${\msbm R}^n$.

AMS Subject Classification (1991): 35B05, 35R10, 34K15

Keyword(s): nonlinear hyperbolic equation, deviating argument, oscillation

Received December 3, 1991. (Registered under 5539/2009.)

Abstract. In this paper, we present and solve a two dimensional Jacobi-Bessel partial differential equation, and obtain a particular solution of it involving Fox's $H$-function.

AMS Subject Classification (1991): 33C25,33C40,35C10

Received December 5, 1991. (Registered under 5540/2009.)

Abstract. The main result of this paper can be formulated as follows: Given a sequence of real-valued random variables $\Phi = (\Phi_n)$ on a probability space $\Omega $ and given a sequence $c = (c_n)$ of reals, then there exists a subsequence $\Phi ' = (\Phi_{k_n})$ of $\Phi $ such that $$\lim_{N \to\infty } F_{W_N} (x) = {1\over\sqrt {2 \pi }}\int_{- \infty }^x e^{- t^2\over2} dt \hbox{ for } x\in{\msbm R}, W_N = {1\over C_N}\sum_{n=1}^N c_n \Phi_{k_n},$$ where $F_{W_N} (x)$ is the distribution function of the weighted sum $W_N$, provided that the following conditions are satisfied: $$C_N^2=\sum_{n=1}^N c_n^2\to\infty, c_N = o(C_N), \Phi_n\in L^2$$ with $\Phi_n\to0$ weakly in $L^1$ and $\Phi_n^2\to1$ weakly in $L^1$. Moreover, the subsequence $\Phi '$ may be chosen independently of the sequence $c$. A corresponding result on the existence of a rearrangement of $\Phi $ is shown, too. The proofs are based on moment conditions (e.g. the concept of weak multiplicativity) instead of martingale techniques, which are used e.g. by D. J. Aldous or S. D. Chatterji.

AMS Subject Classification (1991): 60F05

Keyword(s): Central limit theorem, subsequences, moment conditions

Received December 6, 1991. (Registered under 5541/2009.)

Abstract. In two previous papers we generalized four classical inequalities of Hardy and Littlewood. Now it is proved that the converses of our inequalities hold if and only if the sequence of multipliers appearing in the inequalities behaves very similar to a geometrical sequence.

AMS Subject Classification (1991): 40A05

Keyword(s): inequalities

Received February 5, 1992. (Registered under 5542/2009.)

Abstract. We prove the analogues of Bernstein's and Alexits' theorems for Fourier transform by using Hilbert transform.

AMS Subject Classification (1991): 42A38

Keyword(s): Approximation, Fourier transform

Received September 10, 1992. (Registered under 5543/2009.)

Abstract. Let $X_{1,n}\le\ldots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive integers such that $k_n\to\infty $ and $k_n/n\to\alpha $ as $n\to\infty $, where $0\le\alpha < 1$. Given a known function $f$ and known constants $d_{i,n}$, $1\le i\le n$, that are all specified by the statistician, consider the linear combinations $T_n(k,k_n)=\sum_{i=k+1}^{k_n} d_{n+1-i,n}f(X_{n+1-i,n})$ of extreme values, where $k\ge0$ is any fixed integer. We find necessary and sufficient conditions for the existence of normalizing and centering constants $A_n>0$ and $C_n$ such that the sequence $E_{n}=\{T_n(k,k_n)-C_n\} /A_n$ converges in distribution along subsequences of the integers $\{n\} $ to non-degenerate limits and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $C_n$ such that $E_n$ be asymptotically normal along a given subsequence.

AMS Subject Classification (1991): 60F05; 62E20; 62G30

Received September 14, 1992. (Registered under 5544/2009.)

Abstract. In this note we extend some results on approximation of $2\pi $-periodic functions, given in [2]--[5], to the case of strong approximation by some means of Fourier series.

AMS Subject Classification (1991): 42A10, 42A24

Received November 9, 1992, and in revised form May 5, 1993. (Registered under 5545/2009.)

Abstract. A sequence $c = (c_\nu )$ of nonnegative numbers with $$C_n := \sum ^n_{\nu =0} c_\nu >0 \hbox{ for all } n\in{\msbm N}_0 \hbox{ and } \lim_{n\to\infty }{c_n\over C_n}=0$$ generates a regular Nörlund-method. The Nörlund-transforms of a power series $\sum ^\infty_{\nu =0} a_\nu z^\nu $ with radius of convergence $1$ are given by $$\sigma_n(z):={1\over C_n}\sum ^n_{\nu =0}c_{n-\nu }\sum ^\nu_{\mu =0} a_\mu z^\mu.$$ We investigate the distribution of zeros of the polynomials $\sigma_n$. If $A_n (R)$ denotes the number of zeros of $\sigma_n$ in $| z| \le R$ we have $\limsup_{n\to\infty }{A_n(R)\over n}=1$ for all $R>1$. It is the object of the present paper to characterize those power series which satisfy $$\liminf_{n\to\infty }{A_n (R)\over n}< 1\quad\hbox{for an }\ R>1. \quad (\ast)$$ One of our main results is, that $(\ast )$ holds if and only if the power series has Ostrowski-gaps.

AMS Subject Classification (1991): 40G05, 30C15, 30B30

Received November 30, 1992. (Registered under 5546/2009.)

Abstract. We prove that for any $0<\alpha < 2$, there exists a uniformly bounded complete orthonormal system $\Phi_{\alpha }=\{\phi_{n}\} _{n=1}^\infty $ of functions on $[-1, 1]$ and a continuous function $f$, such that the orthogonal expansion of any integrable function $g$ with $|\{x : g(x)=f(x)\} |>\alpha $ by the system $\Phi_{\alpha }$ does not converge in the $L^p_{[-1, 1]}$ metric for any $p>2$. We also prove that the analogous result is true for Köthe spaces with a certain property (E). Particularly this result holds for Orlicz spaces arbitrarily near to the space $L^2$ but with norm stronger than the $L^2$ norm.

AMS Subject Classification (1991): 42C15

Keyword(s): C-strong property, Köthe spaces

Received December 30, 1992 and in revised form February 24, 1993. (Registered under 5547/2009.)

Abstract. Recently G. Bennett gave a unified approach of the results proved, among others, by E. T. Copson; M. Izumi, S. Izumi and G. M. Petersen; D. Borwein and A. Jakimowski; J. M. Cartlidge; and the author. Now we give certain converses of four theorems of G. Bennett and some slight generalizations of several known inequalities.

AMS Subject Classification (1991): 40A05

Keyword(s): inequalities

Received December 30, 1992. (Registered under 5548/2009.)

Abstract. In a preceding paper we generalized four classical results of Mulholland. Now we develop these results using the ``blocking method'' of M. Mateljevič and M. Pavlovič, and the function $x^r$ will be replaced by more general function.

AMS Subject Classification (1991): 40A05, 40A10, 40A99

Received February 2, 1993. (Registered under 5549/2009.)

Abstract. First, we consider a complex-valued function $f$ defined and absolutely continuous on the positive quadrant ${\msbm R}_+^2$ of the plane, and study the double cosine $F_c,$ double sine $F_s$, and cosine-sine Fourier transform $F_{cs}$ of $f.$ We give sufficient conditions, under which $F_c, F_s,$ and $F_{cs}$ are Lebesgue integrable on ${\msbm R}_+^2,$ respectively; and the inversion formula holds. Our basic tools are Sidon type inequalities, which we elaborate also in this paper. Second, we deduce sufficient conditions for Lebesgue integrability of double cosine, double sine, and cosine-sine series on the two-dimensional torus ${\msbm T}^2.$ Third, we extend these results to double complex Fourier transform of functions defined and absolutely continuous on the whole plane ${\msbm R}^2$ as well as to double complex trigonometric series. Fourth, as a by-product, we obtain sufficient conditions for an absolutely continuous function to be the double complex Fourier transform of a Lebesgue integrable function on ${\msbm R}^2.$

AMS Subject Classification (1991): 42B99, 42A38, 26A46

Keyword(s): double cosine, double sine, cosine-sine Fourier transforms, double complex Fourier transform, absolute continuity of function in two variables, Lebesgue integrability, inversion formula, double cosine, double sine, cosine-sine series, double complex trigonometric series, double null sequence of bounded variation, Fourier series, Hausdorff-Young inequality, Sidon type inequalities

Received February 2, 1993. (Registered under 5550/2009.)

Abstract. We give new sufficient conditions on absolutely continuous functions on ${\msbm R}^2$ to be multipliers of double Fourier transforms on $L^1({\msbm R}^2)$. Analogously, we give new sufficient conditions on double sequences of bounded variation to be multipliers of double Fourier series on $L^1({\msbm T}^2)$. We deal not only with even, but odd and even-odd functions and sequences, respectively. Our proofs rely on integrability results obtained recently.

AMS Subject Classification (1991): 42A45, 46A19

Keyword(s): Fourier transform, absolutely continuous function, Fourier series, Lebesgue integrability, sequence of bounded variation, multiplier, bounded linear operator, convolution

Received February 2, 1993. (Registered under 5551/2009.)

Abstract. In general Riemannian manifolds the shape operators and the curvature endomorphisms with respect to the normal vectors on a given hypersurface not necessarily commute. In this paper examples are presented for submanifolds of codimension one having the above commutativity property. Particularly, tubular hypersurfaces in Riemannian symmetric spaces are discussed.

AMS Subject Classification (1991): 53B25

Received February 19, 1993, and in revised form July 7, 1993. (Registered under 5552/2009.)

Abstract. Consider the class of all measurable plane sets having given horizontal and vertical projections. According to this class the plane can be divided into three subsets: the essentially common subset of all elements of the class, the essentially common subset of the complements of all elements of the class, and the remaining subset of the plane. The three sets together are called the structure of the class. In this paper a method is given by which the structure of an arbitrary class can be determined from the projections. The method is similar to the procedure applied in the case of binary matrices. First, the structure of the normalized class (having rearranged non-increasing projections from the original ones) is constructed. Then, by a measure preserving mapping the structure of the original class is derived from the structure of the normalized class. The structure can be used in the reconstruction of non-unique sets from their projections, e.g. it gives information about the shape and the position of the possible solutions and an upper bound of the measure of the difference between two solutions.

AMS Subject Classification (1991): 28A05, 28A45

Keyword(s): projections, uniqueness, rearrangements, reconstruction

Received April 1, 1993. (Registered under 5553/2009.)

Abstract. Let $\Lambda $ be an open set in ${\msbm R},$ and define $$B^1(L^1_{\Lambda }({\msbm R})) =\{f(x) \in L^1({\msbm R}) | \hat f(t) = 0\hbox{ for all }t\notin\Lambda\}.$$ We define the set $\Lambda $ to be nicely placed if this class of functions is closed with respect to almost everywhere convergence. This term is analogous to that used by Godefroy and others in the setting of compact abelian groups. Our main result is that a set of the form $\Lambda = {\msbm R}^+\setminus K$ is nicely placed if $K$ is a closed set supporting a measure $\mu $ that has density on $K$ and whose Fourier-Stieltjes transform, $\hat\mu (t),$ belongs to the class $C_0({\msbm R}).$ In particular, if $K \subseteq{\msbm R}^+$ is any closed set of positive Lebesgue measure having density, then ${\msbm R}^+ \setminus K$ is nicely placed.

AMS Subject Classification (1991): 42A38

Received July 26, 1993. (Registered under 5554/2009.)

Abstract. Let $G$ be a nondiscrete compact abelian group, and $M(p,q)$ the algebra of translation invariant operators of $L^p(G)$ to $L^q(G)$. If $1< p< q< \infty $, the closure of the range of Gelfand transform of $T$ in $M(p,q)$ coincides with the closure of range of Fourier transform of $T$.

AMS Subject Classification (1991): 42A45, 43A22

Received December 4, 1991. (Registered under 5555/2009.)

Abstract. The author considers operators $T$ acting on complex separable Banach spaces. The set $\{x,Tx,T^2x,\ldots,T^nx,\ldots\} $ is called the orbit of $x$ under $T$. If dense orbits exist $T$ is called {\it hypercyclic}. If the spectrum of a hypercyclic operator can be represented as the union of two nonvoid, compact, mutually disjoint sets then each of these sets must have nonvoid intersection with the unit circle. No nonzero reducing subspace of a hypercyclic operator $T$ reduces $T$ to a normal, respectively to a compact operator. For Hilbert space contractions $A$ if $\lambda A$ is hypercyclic for some complex $\lambda $ then $A$ is in $C_0.\setminus C_0 $.

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

Received January 31, 1992 and in revised form March 30, 1992. (Registered under 5556/2009.)

Abstract. In this paper we introduce a notion of $p$-classes in a proper $H^*$-algebra for which, in contrast with the definition due to Smith, an analogue of the duality of the $p$-classes of compact operators on a Hilbert space is valid. Furthermore, the cyclic and the irreducible $*$-representations of these classes and those of dual $B^*$-algebras are given.

AMS Subject Classification (1991): 46K15, 47B10, 46K10

Keyword(s): H^*, -algebras, centralizers, spectral theorem of compact operators, p, -classes, representations

Received March 20, 1992. (Registered under 5557/2009.)

Abstract. It is shown that every closed symmetric operator possesses a closed symmetric restriction whose square has a trivial domain.

AMS Subject Classification (1991): 47B25

Keyword(s): closed symmetric operator

Received April 8, 1992. (Registered under 5558/2009.)

Abstract. We prove that, under suitable hypothesis on the dual space ${\cal A}$, the dual space generated by ${\cal A}$ and a finite rank operator $R$ has property $({\msbm A}_{1,\chi_{0}})$ without having any property $E^r_{0,\gamma }$. We also completely discuss in terms of rank $R$, properties $({\msbm A}_{m,n})$ for our example.

AMS Subject Classification (1991): 47A55, 47D25, 47D15

Recu le 31 juillet 1992, sans forme revue le 2 février 1993. (Registered under 5559/2009.)

Abstract. The symbols of invertible Toeplitz operators from $H^p(wd\theta /2\pi )$ to $L^p(wd\theta /2\pi )/e^{-i\theta }{\bar H}^p(wd\theta /2\pi )$ are described completely where $H^p(wd\theta /2\pi )$ denotes a weighted Hardy space. The result is strongly related with a weighted norm inequality. If the weight $w$ satisfies the condition $(A_p)$ then $L^p(wd\theta /2\pi )/e^{-i\theta }{\bar H}^p(wd\theta /2\pi )=H^p(wd\theta /2\pi )$ with equivalent norms.

AMS Subject Classification (1991): 47B35

Received August 18, 1992. (Registered under 5560/2009.)

Abstract. Let $X$ be a completely regular Hausdorff space and let $E$ be a locally convex Hausdorff space. If $V$ is a system of weights, then $CV_o(X,E)$ and $CV_p(X,E)$ are weighted spaces of continuous functions with topologies derived from seminorms which are weighted analogues of the supremum norm. We characterize the self-maps of the underlying space $X$ which induce composition operators on these weighted spaces and then give a characterization of linear transformations which are composition operators on weighted spaces. Some properties of these composition operators on weighted spaces are given. Most of the results of [Si-Su2] are obtained as an application of the results of this paper.

AMS Subject Classification (1991): 47B38, 46E40

Keyword(s): weighted space, composition operators

Received October 1, 1992. (Registered under 5561/2009.)

Abstract. If $\phi $ is an analytic function taking the unit disk into itself then the composition operator $C_\phi $ can be defined on the Hardy space $H^p(D)$ by $C_\phi(f)=f\circ\phi $. In this work it is shown that if some power of $C_\phi $ is compact and $\phi $ has a nonzero derivative at its unique fixed point inside the disk, then $C_\phi $ is similar to $C_\psi $ if and only if the similarity can be induced by an invertible composition operator.

AMS Subject Classification (1991): 47B38, 47B07

Keyword(s): Composition operator

Received October 14, 1992. (Registered under 5562/2009.)

Abstract. We present an abstract form of the classical Schur Theorem from summability. Our result contains the vector subseries case and bounded multiplier convergent case for both vector-valued and operator-valued series as corollaries.

AMS Subject Classification (1991): 40B05, 40A30

Received November 3, 1992. (Registered under 5563/2009.)

Abstract. Let $w$ be a modulus function and $H(w)$ be the corresponding Hardy--Orlicz space. If $\phi $ is a self analytic map of the unit disc, then the linear transformation $C_\phi $ on $H(w)$, defined by $C_\phi f=f\circ\phi $ turns out to be continuous and is called a substitution operator. A characterization of operators which are substitution operators is presented. Quasinorm estimate for $C_\phi $ is given and it is utilized to characterize isometric substitution operators on $H(w)$.

AMS Subject Classification (1991): 47B38, 46A06

Received November 24, 1992 and in revised form June 15, 1993. (Registered under 5564/2009.)

Abstract. For $A\in{\cal L}(H)$ (the algebra of all operators on the separable complex Hilbert space $H$) we study the property of the range of $\delta_A(X\mapstochar\rightarrow AX-XA)$ if, for $T\in{\cal C}_1(H)$ (trace class operators) $AT=TA$ implies $A^*T=TA^*$. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of $\delta_A$ is closed under taking adjoints.

AMS Subject Classification (1991): 47B47

Reçu le 12 février, 1993. (Registered under 5565/2009.)