
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
M. E. Adams,
Matthew Gould

337

Abstract. The ${\cal L}^*$ relation on a semigroup $S$ is defined as follows. For $a,b\in S$, $a{\cal L}^*b$ iff $a$ and $b$ generate the same left ideal in some semigroup $T$ of which $S$ is a subsemigroup. The relation ${\cal R}^*$ is defined dually. As introduced by Fountain, the semigroup $S$ is {\it abundant} providing each ${\cal L}^*$class and each ${\cal R}^*$class contain an idempotent. Since regularity of a semigroup may be defined analogously in terms of its ${\cal L}$ and ${\cal R}$ relations, abundancy is a generalisation of the classical concept of regularity. We determine all finite partially ordered sets $P$ for which $\mathop{\rm End}(P)$, the monoid of orderpreserving selfmaps of $P$, is abundant. Moreover, in answer to a question of V. Gould, we show that abundancy and regularity are equivalent for endomorphism monoids of semilattices.
AMS Subject Classification
(1991): 06A06, 06A12, 20M17
Keyword(s):
Poset,
semilattice,
endomorphism,
abundant semigroup,
regular semigroup
Received July 15, 1998, and in revised form November 20, 2000. (Registered under 2771/2009.)
G. Grätzer,
E. T. Schmidt

3950

Abstract. In 1990, we published the following result: {\it Let $\eufm m$ be a regular cardinal $> \aleph_0$. Every {\rm $\eufm m$algebraic lattice } $L$ can be represented as the lattice of $\eufm m$complete congruence relations of an {\rm $\eufm m$complete modular lattice } $K$.} In this note, we present a short proof of this theorem. In fact, we present a significant improvement: The lattice $K$ we construct is $2$distributive.
AMS Subject Classification
(1991): 06B10, 06D05
Keyword(s):
Complete lattice,
2,
distributive lattice,
complete congruence,
congruence lattice
Received March 3, 2000. (Registered under 2772/2009.)
Abstract. The aim of this paper is to present a forbidden sublattice characterization of strongness for semimodular lattices. We give a generalization of some results obtained by Stern [1] for lattices of finite length to lower continuous strongly coatomic lattices.
AMS Subject Classification
(1991): 06C05, 06C10
Received May 3, 2000, and in revised form July 12, 2000. (Registered under 2773/2009.)
Abstract. One of the central results of the early theory of lattice ordered groups is a theorem due to Ján Jakubik, telling that a maximal convex chain containing the identity element is a direct factor. In this paper it is shown  along the lines of BigardKeimelWolfenstein  how far this result carries over to divisibility semigroups and divisibility semiloops.
AMS Subject Classification
(1991): 06F
Keyword(s):
lattice monoids,
lattice loops,
divisibility,
decomposition
Received February 3, 1999, and in final form May 8, 2000. (Registered under 2774/2009.)
B. A. Davey,
M. Haviar,
H. A. Priestley

77103

Abstract. A finite endodualisable algebra is always endoprimal, and this fact has led to the discovery of many endoprimal algebras. Recent investigations by the authors have shown that the finite endoprimal algebras in various wellknown quasivarieties of algebras having distributive lattice reducts are exactly the finite endodualisable algebras. In the context of semilattices, B. A. Davey and J. G. Pitkethly found examples of finite algebras which are endoprimal but nonendodualisable. The distinction between entailment in the clonetheoretic and dualitytheoretic senses was first revealed in the variety $\cal K$ of Kleene algebras. This paper considers the subquasivariety $\cal L$ of $\cal K$ generated by the fourelement chain; this contains only fixpointfree Kleene algebras. The classes of finite endoprimal and endodualisable algebras in $\cal L$ are found. These do not coincide and the way in which this happens leads to a better understanding of the relationship between the two entailment concepts.
AMS Subject Classification
(1991): 08A35, 08A40, 08C05, 18A40
Keyword(s):
Kleene algebra,
natural duality,
entailment,
clone,
endodualisable algebra,
endoprimal algebra
Received November 12, 1999, and in revised form December 6, 2000. (Registered under 2775/2009.)
Phillip M. Edwards,
Peter M. Higgins,
Samuel L. Kopamu

105120

Abstract. Idempotentconsistent semigroups are defined by the property that each idempotent in a morphic image has an idempotent preimage. We show that this class is strictly contained in the class of $E$inversive semigroups while strictly containing a certain class of coextensions of eventually regular semigroups.
AMS Subject Classification
(1991): 20M05, 20M17
Received October 28, 1998, and in final form July 12, 2000. (Registered under 2776/2009.)
Abstract. We examine the problem of determining when a finite Rees quotient of a free monoid has a finite basis for its identities. In [bibjac] and [bibosa] there is shown to be many difficulties associated with this problem but the main examples and theorems there concern the Rees quotients of free monoids on small numbers of generators. Here we extend these results to arbitrary finite generating sets and provide some considerably more general conditions on when a finite Rees quotient of a free monoid is not finitely based. We also introduce the notion of a strongly not finitely based word and construct some examples.
AMS Subject Classification
(1991): 20M07, 08B05
Received January 3, 2000. (Registered under 2777/2009.)
Abstract. Let $T$ be a locally $\cal R$unipotent semigroup acting on a semigroup $S$ by endomorphisms on the left. A kind of semidirect product of $S$ by $T$ is defined, which leads to a regular (locally $\cal R$unipotent) semigroup if $S$ is regular (locally $\cal R$unipotent). On the level of evarieties we naturally obtain a binary operation ``$*$'' which satisfies $({\cal U}*{\cal V})*{\cal W}={\cal U}*({\cal V}*{\cal W})$ whenever $\cal U$ is an evariety, $\cal V$ and $\cal W$ are evarieties of locally $\cal R$unipotent semigroups and ${\cal U}*({\cal V}*{\cal W})$ is generated by $\{A*(B*C)  A\in{\cal U}, B\in{\cal V}, C\in{\cal W}\} $. This generalizes recent results found in [1], [4], [9].
AMS Subject Classification
(1991): 20M10, 20M17
Received March 25, 1998, and in final form August 4, 2000. (Registered under 2778/2009.)
David Bleecker,
George Csordas

177196

Abstract. Turán's results concerning the distribution of zeros of Hermite expansions of entire functions are generalized. Estimates for the location of zeros of entire functions under the action of certain linear operators are established.
AMS Subject Classification
(1991): 30C15, 30D15, 26C10, 33C25
Keyword(s):
Hermite expansions,
zeros,
entire functions,
differential operators
Received November 30, 1999, and in revised form May 5, 2000. (Registered under 2779/2009.)
Abstract. We prove that each closed and connected set $C \subseteq{\msbm R}^n$ is congruent to the $\omega $limit set of a solution $x\colon[0,\infty ) \to{\msbm R}^{n+2}$ of $x'=f(x)$ for some bounded $f\in C^\infty({\msbm R}^{n+2},{\msbm R}^{n+2})$.
AMS Subject Classification
(1991): 34C35, 34D45
Received June 8, 2000, and in revised form December 8, 2000. (Registered under 2780/2009.)
Abstract. A pluriparabolic problem which combines a classical and a nonlocal constraint is considered. The existence and uniqueness of a generalized solution are proved. We use a functional analysis method based on a two a priori estimates and on the density of the range of the operator generated by the considered problem.
AMS Subject Classification
(1991): 35K70
Keyword(s):
pluriparabolic equation,
Integral condition,
generalized solution
Received March 30, 2000, and in revised form July 8, 2000. (Registered under 2781/2009.)
Gavin Brown,
Qinghe Yin

221247

Abstract. Suppose that $a_0\ge a_1\ge\cdots \ge a_n>0$ and $(2k+1)a_{2k1}\ge(2k+2)a_{2k}$ for $k\ge1$. Then for $0< x< \pi $ one has $$ \sum_{k=0}^na_k\cos kx>0. $$ This is a further generalisation of a result of the first named author and Hewitt, which generalises a theorem of Vietoris.
AMS Subject Classification
(1991): 42A32
Received March 2, 1999. (Registered under 2782/2009.)
Torben Maack Bisgaard

249271

Abstract. We characterize measurable semigroups of moment functions on abelian semigroups with zero and involution.
AMS Subject Classification
(1991): 43A35, 44A60
Received March 1, 2000, and in revised form June 20, 2000. (Registered under 2783/2009.)
Barthélemy Le Gac,
Ferenc Móricz

273298

Abstract. The notion of bundle convergence for single (ordinary) sequences in von Neumann algebras and their $L_2$spaces was introduced by Hensz, Jajte, and Paszkiewicz in 1996. We adopt this notion for double sequences. Bundle convergence is stronger than almost sure convergence, and it enjoys the property of additivity. We prove an extension of the classical twoparameter strong law of large numbers to an orthogonal double sequence of vectors in a noncommutative $L_2$space. Our main tool is a twoparameter extension of the classical RademacherMenshov inequality to the noncommutative case. We also prove the extension of another strong law, even the oneparameter version of which seems to be new.
AMS Subject Classification
(1991): 46L10, 46L53, 60B12
Keyword(s):
\phi,
L_2({\eufm A},
von Neumann algebra {\eufm A},
faithful and normal state,
completion,
; GelfandNaimarkSegal representation theorem; bundle convergence; almost sure convergence; orthogonal double sequence of vectors in,
; RademacherMenshov inequality; strong law of large numbers,
\phi )L_2
Received August 12, 1999, and in revised form October 17, 2000. (Registered under 2784/2009.)
Abstract. Operator matrices in Banach and Hilbert spaces are considered. Invertibility conditions and bounds for the spectrum are established. Besides, the Gershgorin types bounds are improved for operator matrices, which are close to triangular ones. An application to matrix differential operators is discussed.
AMS Subject Classification
(1991): 47A10, 47A55
Keyword(s):
operator matrices,
invertibility conditions,
spectrum perturbations,
differential operators
Received June 14, 1999. (Registered under 2785/2009.)
Abstract. A new method for constructing irreducible operator bands is provided. Using this, we are able to answer some open problems concerning bands. In particular, there exists a nonnegative irreducible operator band.
AMS Subject Classification
(1991): 47A15, 47D03, 15A30
Keyword(s):
operator semigroups,
idempotents,
band,
reducible,
nonnegative
Received February 2, 2000, and in revised form July 6, 2000. (Registered under 2786/2009.)
Gustavo Corach,
Alejandra Maestripieri,
Demetrio Stojanoff

337356

Abstract. Let ${\cal H}$ be a Hilbert space, $L({\cal H} )$ the algebra of all bounded linear operators on ${\cal H}$ and $ \langle, \rangle_A \colon{\cal H} \times{\cal H} \to{\msbm C}$ the bounded sesquilinear form induced by a selfadjoint $A\in L({\cal H} ) $, $ \langle\xi, \eta\rangle _A = \langle A \xi, \eta\rangle, \xi, \eta\in {\cal H}.$ Given $T\in L({\cal H} )$, $T$ is $A$selfadjoint if $AT = T^*A$. If ${\cal S} \subseteq{\cal H}$ is a closed subspace, we study the set of $A$selfadjoint projections onto ${\cal S} $, $$ {\cal P}(A, {\cal S} ) = \{Q \in L({\cal H}) : Q^2 = Q, R(Q) = {\cal S}, AQ = Q^*A\} $$ for different choices of $A$, mainly under the hypothesis that $A\ge0$. There is a closed relationship between the $A$selfadjoint projections onto ${\cal S} $ and the shorted operator (also called Schur complement) of $A$ to ${\cal S} ^\perp $. Using this relation we find several conditions which are equivalent to the fact that ${\cal P}(A, {\cal S} ) \not =\emptyset $, in particular in the case of $A\ge0$ with $A$ injective or with $R(A)$ closed. If $A$ is itself a projection, we relate the set ${\cal P}(A, {\cal S} ) $ with the existence of a projection with fixed kernel and range and we determine its norm.
AMS Subject Classification
(1991): 47A64, 47A07, 46C99
Received May 2, 2000, and in revised form September 27, 2000. (Registered under 2787/2009.)
Shizuo Miyajima,
Isao Saito

357371

Abstract. This article deals with operators that are $p$hyponormal for every $p>0$ under the name of $\infty $hyponormal operators. A characterization of such operators is given, and their properties are investigated. Special attention is paid to the spectral property of $\infty $hyponormal operators having dense ranges and no nontrivial reducing subspaces.
AMS Subject Classification
(1991): 47B20, 47A63
Keyword(s):
p,
hyponormal operator,
spectral resolution,
polar decomposition,
spectrum,
\infty,
hyponormal operator
Received March 1, 2000, and in revised form September 12, 2000. (Registered under 2788/2009.)
Jörg Eschmeier,
FlorianHoria Vasilescu

373386

Abstract. Let $S = (S_1,\ldots, S_n)$ be a tuple of symmetric operators $S_j\colon D(S_j)\subset H\rightarrow H$ on a Hilbert space $H$. We show that $S$ is jointly essentially selfadjoint in the sense of Fuglede if and only if a suitable associated matrix operator is essentially selfadjoint as a single operator. As an application we obtain an elementary proof of Nelson's famous commutativity criterion for essentially selfadjoint operators.
AMS Subject Classification
(1991): 47B25, 47B15
Received December 2, 1999. (Registered under 2789/2009.)
Paul S. Bourdon,
Dylan Q. Retsek

387394

Abstract. Motivated by a question raised by C. C. Cowen and B. D. MacCluer, we investigate when the norm of a composition operator $C_\phi $ on the classical Hardy space $H^2$ of the unit disk is determined by the action of $C_\phi $ or its adjoint $C_\phi ^*$ on the set $S$ of normalized reproducing kernels of $H^2$. Our results suggest that the action of $C_\phi ^*$ on $S$ rarely determines $\C_\phi\$. We show, for example, that when $C_\phi $ is compact then the action of $C_\phi ^*$ on $S$ determines the norm of $C_\phi $ if and only if $\phi(0) = 0$ or $\phi $ has the form $z\mapstochar\rightarrow sz + t$ for some constants $s$ and $t$ satisfying $s + t < 1$. We also show that when $\phi $ is inner and $\phi(0)\not=0$, then the action of $C_\phi ^*$ on $S$ determines $\C_\phi\$ if and only if $\phi $ is an automorphism of the unit disk.
AMS Subject Classification
(1991): 47B33
Received August 30, 2000, and in revised form January 12, 2001. (Registered under 2790/2009.)
Ludmila N. Nikolskaia

395409

Abstract. Let $ W$ be the Wiener algebra of absolutely convergent Fourier series on the unit circle $ {\msbm T}$, and let $ T_{\varphi }\colon W_{+}\longrightarrow W_{+}$, $ T_{\varphi }f=: P_{+}(\varphi f)$ ($ \varphi\in W$) be an invertible Toeplitz operator, that is, $ \delta(T_{\varphi })= \inf_{\msbm T} \varphi  =\inf\{ \lambda  :\lambda\in \sigma(T_{\varphi })\} >0$ and $ \mathop{\rm Ind}(\varphi )=0$. The problems of estimates of inverses $ \T_{\varphi }^{1}\$ and of finite section inverses $ \(P_{n}T_{\varphi } {\cal P}_{n})^{1}\$, in terms of the lower spectral bound $ \delta(T_{\varphi })$, are considered. It is shown that for $ \delta(T_{\varphi })\geq\delta \T_{\varphi }\$ and $ \delta > 1/{\sqrt{2}} $ there exists a constant $ c(\delta )$ such that $ \T_{\varphi }^{1}\$, $ \(P_{n}T_{\varphi } {\cal P}_{n})^{1}\\leq\T_{\varphi }\^{1}c(\delta )$ for every $ n\geq0$, and that for $ 0< \delta\leq 1/2$ such an estimate is impossible: $ \sup\{\T_{\varphi }^{1}\: \delta(T_{\varphi })\geq\delta, \mathop{\rm Ind}(\varphi )=0, \T_{\varphi }\\leq1\} =\infty $ and $ \sup\{\overline{\lim }_{n}\(P_{n}T_{\varphi } {\cal P}_{n})^{1}\: \delta(T_{\varphi })\geq\delta, \mathop{\rm Ind}(\varphi )=0, \T_{\varphi }\\leq1\} =\infty $. For analytic symbols $ \varphi\in W_{+}$, the similar constant $ c_{+}(\delta )$ is finite iff $ 1/2< \delta\leq 1$. Similar results are obtained for Fredholm regularizers of operators $ T$ from the Toeplitz algebra $ \mathop{\rm Alg}(T_{W})$.
AMS Subject Classification
(1991): 47B35, 46J15, 65J10
Received April 19, 1999, and in revised form March 14, 2000. (Registered under 2791/2009.)
Petros Galanopoulos

411420

Abstract. In this paper we show that the Cesàro operator is bounded on the weighted Dirichlet spaces that lie between the classical Dirichlet spaces and the Hardy space $H^2$.
AMS Subject Classification
(1991): 47B38, 46E20
Received March 20, 2000, and in revised form October 31, 2000. (Registered under 2792/2009.)
Abstract. Let $A$ be a nonempty approximately compact convex subset, $B$ be a nonempty closed convex subset and $C$ be a nonempty convex subset of a normed linear space $E$. Given a multifunction $T_1\colon A \longrightarrow2^C$ with open fibres, a Kakutani factorizable multifunction $T_2\colon C\longrightarrow2^B$ and a single valued function $g\colon A \longrightarrow A$, best proximity pair theorems furnishing the sufficient conditions for the existence of an element $x_\circ\in A$ such that $$ d(gx_{\circ }, T_2 T_1 x_{\circ }) = d(A, B) $$ are explored. As a consequence, a generalization of Ko and Tan's coincidence theorem is obtained.
AMS Subject Classification
(1991): 47H10, 54H25
Keyword(s):
Best proximity pairs,
Kakutani factorizable multifunctions,
Multifunctions with open fibres,
Proper map,
Quasi affine map
Received March 9, 2000. (Registered under 2793/2009.)

431460
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