ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $\langle\underline k,\leq\rangle $ denote the chain on $k$ elements, $k \geq2$. We present an effective procedure to determine, given a set $F$ of isotone operations on $\langle\underline k,\leq\rangle $, if it generates the whole clone of isotone operations or not. We determine all maximal subclones of ${\rm Pol}\leq $ specified by relations of arity $2$, $(k-1)$ or $k$, and give several examples of maximal subclones. For $k \leq5$ we characterize the maximal subclones of ${\rm Pol}\leq $, and exhibit minimal generating sets.

AMS Subject Classification (1991): 08A40, 06A06

Received August 16, 1993 and in revised form March 16, 1994. (Registered under 5590/2009.)

Abstract. It was recently shown in [Re] that the set of all periodic endomorphisms of a free monoid over a finite alphabet is a finite union of semigroups each of which is an ideal extension of a rectangular group by a nilpotent semigroup of finite index. It is the purpose of this paper to investigate these basic semigroups more closely to determine, among other things, the relationship between the parameters defining them and their algebraic structure. We also establish conditions for inclusion of two basic semigroups and describe a lower semilattice of the lattice of all ideals of such a semigroup.

AMS Subject Classification (1991): 20M05

Keyword(s): periodic endomorphisms of free monoids

Received November 1, 1993. (Registered under 5591/2009.)

Abstract. A {\it cutset} of a poset is a subset which meets every maximal chain, and a {\it fibre} is a subset which meets every maximal antichain. The questions we address are: {\it When is every minimal cutset an antichain?} and the analogous {\it When is every minimal fibre a chain?} For finite posets, Lonc and Rival showed that the answer to both questions is precisely {\it when the poset is fence-free}. We extend this result to classes of infinite posets satisfying certain chain conditions.

AMS Subject Classification (1991): 06A07

Received October 27, 1993 and in revised form March 11, 1994. (Registered under 5592/2009.)

Abstract. In this paper we introduce the angular cutting for $p$-hyponormal operators on a Hilbert space and study spectral properties of a section $T_{\gamma }$ of a $p$-hyponormal operator $T$ cut by the arc $\gamma\subset $ {\bf T}$=\{z\in{\bf C}:|z|=1\}$.

AMS Subject Classification (1991): 47B20

Keyword(s): Hilbert space, $p$-hyponormal, angular cutting

Received September 14, 1993. (Registered under 5594/2009.)

Abstract. Let $(X,\|\cdot\|)$ be a normed space, $$\eqalign{C_J(X) &=\sup\{\| x+y\|\wedge\| x-y\|:x,y\in S(X)\};\cr C_S(X) &=\inf\{\| x+y\|\vee\| x-y\|:x,y\in S(X)\}.}$$ In this paper we will show that if $X$ is a real normed space with $\dim(X)>1$, then $C_J(X)C_S(X)=2$. For some classical Banach spaces we get that $$\eqalign{C_J(X) &=\sup\{\| x+y\|:\| x+y\|=\| x-y\|, x,y\in S(X)\};\cr C_S(X) &=\inf\{\| x+y\|:\| x+y\|=\| x-y\|, x,y\in S(X)\}.}$$ We also give the expression of $C_J(X)$, $C_S(X)$ in two classes of Orlicz spaces, which involves the result in $L^p$.

AMS Subject Classification (1991): 46B30

Keyword(s): James nonsquare constant, Schäffer nonsquare constant, Orlicz space, Uniformly nonsquare

Received January 4, 1994 and in revised form June 20, 1994. (Registered under 5595/2009.)

Abstract. Rudin proved that outer functions are not generators of $H^2$ on the torus generally. In this paper it is proved that outer functions are characterized as generators of some generalized Hardy spaces.

AMS Subject Classification (1991): 47A15, 32A35

Received January 25, 1994. (Registered under 5596/2009.)

Abstract. A spectral theorem for a normal operator on a real Hilbert space is proved by using the techniques of Banach algebras. This gives a unified treatment for the theory of normal operators on real, complex and quaternionic Hilbert spaces. A well known spectral theorem for a normal operator on a complex Hilbert space is deduced. An example is given to illustrate a difference between the behaviour of operators on a real and a complex Hilbert space.

AMS Subject Classification (1991): 46L89, 47B15, 46L05

Received February 28, 1994 and in revised form May 17, 1994. (Registered under 5597/2009.)

Abstract. We characterize the mappings inducing the weighted composition operators on $LV_b(X)$ and $LV_0(X)$, the weighted locally convex spaces of cross-sections with the topology generated by seminorms which are weighted analogoues of the supremum norm. A few properties of the composition operators are discussed and some examples of weighted composition operators are presented to illustrate the theory. The paper presents a broad account of the theory of these operators on weighted spaces of functions. Some of the results of [7], [10] and [11] can be derived as an application of the results presented in this paper.

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

Keyword(s): Weighted composition operators, weighted spaces of cross-sections, seminorms

Received March 8, 1994. (Registered under 5598/2009.)

Abstract. We show that the commutant lifting theorem for $n$-tuples of commuting contractions with regular dilations fails to be true. A positive answer is given for operators which ``double intertwine" given $n$-tuples of contractions.

AMS Subject Classification (1991): 47A20

Received March 29, 1994. (Registered under 5599/2009.)

Abstract. It is becoming more and more apparent that the class of well-bounded operators of type {\rm(B)} is naturally suited for the investigation of many problems in analysis. In this note a complete description is given of the structure and spectral properties of this family of operators in the class of Banach spaces called hereditarily indecomposable. The special properties of this class of spaces (intensively studied in [9]) forces such operators to have a particularly simple form.

AMS Subject Classification (1991): 47B40, 47B06

Received April 22, 1994. (Registered under 5600/2009.)

Abstract. In this article we characterize invertible operators on a complete topological vector space generalizing the classical concept on invertibility of operators on normed linear spaces and Hilbert spaces. This characterization is employed to characterize invertible composition operators on the weighted locally convex spaces of continuous functions and the weighted spaces of cross-sections. Some examples are presented to give an insight into the theory.

AMS Subject Classification (1991): 47B38, 47A05, 46E10, 46E40

Received June 21, 1994. (Registered under 5601/2009.)

Abstract. The following conjecture of J. R. Partington, raised in [8], is proved: If $T$ is a Hermitian operator in a complex Banach space and $d(\lambda,\sigma(T))$ denotes the distance of the complex number $\lambda $ to the spectrum $\sigma(T)$ of $T,$ then $$\|(\lambda I-T)^{-1}\|\le{\pi\over 2} d(\lambda,\sigma(T))^{-1}$$ for all $\lambda\in \msbm C\setminus\sigma (T).$ For the proof we compute the infimum of the norms of all functions $f\in L^1 ({\msbm R}),$ whose inverse Fourier transform extends $${\msbm R}\setminus(-1, 1)\ni x\mapstochar\rightarrow {1\over\alpha -ix},$$ $\alpha $ being a real parameter

AMS Subject Classification (1991): 47B44, 42A38

Received July 17, 1994. (Registered under 5602/2009.)

Abstract. In this paper, we define a new spectrum (singular spectrum) for a closed operator in a Banach space. The singular spectrum is contained in the classical spectrum and contains the boundary of the latter. We give several characterizations of the points of this new spectrum and show that it has several properties of the classical spectrum. In particular it satisfies the spectral mapping theorem.

AMS Subject Classification (1991): 47A53, 47A55

Keyword(s): Opérateur s-régulier, spectre singulier, conorme, métrique du gap

Received September 23, 1993 and in revised form October 6, 1994. (Registered under 5603/2009.)

Abstract. We show that operators on Hilbert space in the dual operator algebra class ${\msbm A}_{n}$ have as the compression to a semi-invariant subspace, and up to unitary equivalence, both any diagonal operator on $n$ dimensional space with eigenvalues in the open unit disk and a certain block upper triangular operator. From the latter compression follows the construction of some sequences of vectors yielding point evaluations for analytic functions of the original operator.

AMS Subject Classification (1991): 47D27, 47A20

Received November 10, 1993 and in revised form November 1, 1994. (Registered under 5604/2009.)

Abstract. We consider contraction operators on Hilbert space with finite defect index. For those in $C_{\cdot0}$ we provide some new equivalent conditions, including one based on Fredholm index, for membership in the dual operator algebra classes ${\msbm A}_n$ or ${\msbm A}_{n, \aleph_0}$. For those in $C_{11}$, we give characterizations for membership in these classes including the size of scalar that can be compressed to a semi-invariant subspace and multiplicity of the unitary piece of the minimal coisometric extension.

AMS Subject Classification (1991): 47D27, 47A20

Received November 10, 1993 and in revised form November 1, 1994. (Registered under 5605/2009.)

Abstract. An integral inequality is established for nonnegative, continuous, concave functions. This subsumes a number of results derived recently in the literature.

AMS Subject Classification (1991): 26D15, 39B72

Received September 27, 1993. (Registered under 5606/2009.)

Abstract. Necessary and sufficient conditions for the existence of fixed and common fixed points of a pair of self-mappings of compact metric spaces are given. Our results extend properly some results of Jungck [1,2,3] and Park [4].

AMS Subject Classification (1991): 54H25

Received November 23, 1993. (Registered under 5607/2009.)

Abstract. It is shown that in the algebra freely generated by $a$, $b$ and $c$ satisfying the commutation relations $[a,b]=ab-ba=c$, $[a,c]=[b,c]=0$ the identity $(abc)^n=a^nb^nc^n$ holds for every $n=0,1,\ldots $. Some applications of this assertion are also pointed out.

AMS Subject Classification (1991): 17B01, 17B35, 33C45, 34A05, 81S05

Received December 30, 1993. (Registered under 5608/2009.)

Abstract. Let $G$ be a locally compact abelian group with Haar measure $dx.$ Here we obtain a concrete dual space characterisation for the space $M(S(G),A^p(G)),$ where $S(G)$ is a Segal algebra contained in $A^p(G)$, $1< p< \infty$. Further, we define $\Lambda(A^p)$ sets for each $1< p< \infty $ and show that when $G$ is compact and $F$, a $\Lambda(A^p)$ set, every element of $L^1_F(G)$ is $L^p$-improving.

AMS Subject Classification (1991): 43A22

Received January 25, 1994 and in revised form April 12, 1994. (Registered under 5609/2009.)

Abstract. The existence of a generalized Gaussian Birkhoff quadrature formula is shown for incidence matrices of pyramidal structure which contain no odd non-bottom sequence in the interior rows. It extends a result of Dyn and Jetter in [2].

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

Keyword(s): Birkhoff interpolation, Gaussian quadrature formula

Received March 29, 1994. (Registered under 5610/2009.)

Abstract. Let $\omega(x)$ and $v(x)$ be non-negative functions on $[0,\infty )$. We find the best constant in the integral inequality $$\left(\int_0^\infty[Tg(x)]^q\omega(x)dx\right )^{1/q} \le C\left(\int_0^\infty g^p(x)v(x)dx\right )^{1/p},$$ in the class of non-negative, non-increasing or non-decreasing functions $g$ when $0< p\le q< \infty $, $0< p\le1$, and $T$ is a general integral operator of the form $$Tg(x)=\int_0^\infty k(x,y)g(y)dy, k(x,y)\ge0.$$ The similar problem is solved for the reversed inequality provided that $1\le p\le q< \infty $. A number of applications are given. In particular, some recently obtained sharp reversed Hardy type inequalities are analyzed in this connection.

AMS Subject Classification (1991): 26D10

Received June 13, 1994. (Registered under 5611/2009.)

Abstract. We continue the study of the Cesàro mean $\sigma\lambda (u,v)$ of a multiplier $\lambda(x,y)$ for $L({\msbm R}^2)$. Among others, we prove that if $\lambda(x,y)$ is even in each variable, then (i) $\sigma\lambda $ is also a multiplier for $L({\msbm R}^2)$; (ii) $\sigma\lambda $ is the Fourier transform of a function $f\in L({\msbm R}^2)$ if and only if the associated finite Borel measure $\mu $ is continuous on the axes $y=0$ and $x=0$, respectively; (iii) we give necessary and sufficient conditions in order that $\sigma\lambda $ be the Fourier transform of a function $f\in{\cal H}({\msbm R}\times{\msbm R}) \subset{\cal H}({\msbm R}^2)$. We also present analogous results in the case of even multipliers for $L({\msbm T}^2)$ involving double Fourier series.

AMS Subject Classification (1991): 42B30

Keyword(s): double Fourier transform, double Hilbert transforms, Hardy space on product domain, Hardy inequality, Cesàro mean, L({\msbm R}^2), {\cal H}({\msbm R}\times{\msbm R}), multiplier forand, double Fourier series, conjugate functions, arithmetic mean, L({\msbm T}^2), {\cal H}({\msbm T}\times{\msbm T}), multiplier forand

Received August 29, 1994. (Registered under 5612/2009.)

Abstract. A study of webs on differentiable manifolds is, roughly speaking, investigating of local invariants (under the group of diffeomorphisms) of a set of $d$ foliations in general position. A $d$-web of codimension $n$ on an $m$-dimensional manifold is usually introduced as a $d$-tuple of foliations such that their leaves are submanifolds of codimension $n$, and at any point, the corresponding tangent spaces are in general position. If $m$ is a $k$-multiple of $n$, we speak about a $(d,k,n)$-web. Here we will be concerned by the case $d=3$, $k=2$, $m=2n$. Tangent spaces to the foliations are usually described by systems of differential forms (satisfying suitable integrability conditions). In the papers of Chern as well as in works of Akivis - Goldberg school, the main and fruitfull tool is the use of Cartan methods. We will try to present here a dual approach (occuring also in [Ng 2]) making use of distributions, tensor fields, and projectors. In the first two parts, a definition of a three-web of codimension $n$ on a (real) $2n$-dimensional differentiable manifold is given in terms of distributions, and a (local) equivalence of webs is treated. Regarding a 3-web ${\cal W}$ on $M_{2n}$ as a triple of $n$-dimensional involutive distributions which are pairwise complementary, we associate with a web a set of (six) projectors $P_{\alpha }^ {\beta }$, and a triple $B_1$, $B_2$, $B_3$ of associated $(1,1)$- tensor fields. The original distributions can be regarded either as kernels (or images) of projectors, or as invariant subspaces under $B_{\gamma }$. No distribution is preferred (no one is chosen as ``horizontal'' or ``vertical'') which is an advantage of this new approach. In the set of associated projectors, as well as in the set of associated fields, all informations about the web are involved, and ${\cal W}$ can be fully described by a suitably chosen couple of them. Using this description, all linear connections are found with respect to which the distributions of the web are parallel. All objects under consideration are supposed to be smooth (of the class $C^{\infty }$).

AMS Subject Classification (1991): 53C05

Keyword(s): Distribution, projector, manifold, connection, web

Received December 21, 1992 and in revised form May 25, 1994. (Registered under 5613/2009.)

Abstract. We give in this paper a description of proper shape theories of arbitrary topological spaces. Our method is to use multi-valued functions with smaller and smaller images of points. An analogous intrinsic approach to shape theory of compact metric spaces was earlier considered by J. Sanjurjo. The author has extended it to arbitrary topological spaces and the present paper shows how this extension can be adapted to the proper case. The main result is a construction of the proper {shape} category ${\cal S}h_p$ whose objects are topological spaces and whose morphisms are proper homotopy classes of proper multi-nets. The category ${\cal S}h_p$ relates to the proper homotopy category ${\cal H}_p$ similarly as the shape category ${\cal S}h$ links to the homotopy category $\cal H$. On compact spaces the proper shape category agrees with the shape category. For locally compact metrizable spaces, we show the existence of a natural functor from our proper shape category into Ball's proper shape category ${\cal S}^1_p$ which is similarly related to the original Ball and Sher proper shape category.

AMS Subject Classification (1991): 54B25, 54F45, 54C56

Keyword(s): proper multi-valued function, {\sigma }, -{close}, {\sigma }, -{small}, {\gamma }, proper-{homotopy}, proper multi-net, proper shape theory, trivial proper shape, proper shape equivalence

Received December 22, 1993. (Registered under 5614/2009.)