ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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 order-preserving 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.)

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 Bigard--Keimel--Wolfenstein --- 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.)

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 well-known 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 non-endodualisable. The distinction between entailment in the clone-theoretic and duality-theoretic 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 four-element chain; this contains only fixpoint-free 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.)

Abstract. Idempotent-consistent semigroups are defined by the property that each idempotent in a morphic image has an idempotent pre-image. 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 e-varieties 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 e-variety, $\cal V$ and $\cal W$ are e-varieties 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.)

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.)

Abstract. Suppose that $a_0\ge a_1\ge\cdots \ge a_n>0$ and $(2k+1)a_{2k-1}\ge(2k+2)a_{2k}$ 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.)

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.)

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 two-parameter strong law of large numbers to an orthogonal double sequence of vectors in a noncommutative $L_2$-space. Our main tool is a two-parameter extension of the classical Rademacher--Menshov inequality to the noncommutative case. We also prove the extension of another strong law, even the one-parameter 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, ; Gelfand-Naimark-Segal representation theorem; bundle convergence; almost sure convergence; orthogonal double sequence of vectors in, ; Rademacher-Menshov 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 non-negative irreducible operator band.

AMS Subject Classification (1991): 47A15, 47D03, 15A30

Keyword(s): operator semigroups, idempotents, band, reducible, non-negative

Received February 2, 2000, and in revised form July 6, 2000. (Registered under 2786/2009.)

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.)

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 non-trivial 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.)

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 self-adjoint in the sense of Fuglede if and only if a suitable associated matrix operator is essentially self-adjoint as a single operator. As an application we obtain an elementary proof of Nelson's famous commutativity criterion for essentially self-adjoint operators.

AMS Subject Classification (1991): 47B25, 47B15

Received December 2, 1999. (Registered under 2789/2009.)

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.)

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.)

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 non-empty approximately compact convex subset, $B$ be a non-empty closed convex subset and $C$ be a non-empty 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.)