
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

357357
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Abstract. Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ \emph{represents the inclusion $Q\subseteq D$ by principal congruences} if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for \emph{every} subset $Q$ containing $0$, $1$, and all joinirreducible elements of $D$, then $D$ is said to be \emph{fully (A1)representable}. We prove that every fully (A1)representable finite distributive lattice is planar and it has at most one joinreducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one joinreducible coatom is \emph{fully chainrepresentable} in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another result of the present author, it follows that every fully (A1)representable finite distributive lattice is ``fully representable'' even by principal congruences of \emph{finite lattices}. Finally, we prove that every \emph{chainrepresentable} inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.
DOI: 10.14232/actasm0175387
AMS Subject Classification
(1991): 06B10
Keyword(s):
distributive lattice,
principal lattice congruence,
congruence lattice,
chainrepresentability
Received June 1, 2017 and in final form February 8, 2018. (Registered under 38/2017.)
P. A. Azeef Muhammed

377399

Abstract. Crossconnection theory propounded by Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^\theta $ of the full transformation semigroup $(\mathscr{T}_X,\cdot )$ for an arbitrary $\theta\in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^\theta = (\mathscr{T}_X,\ast )$ with the binary operation $\alpha\ast \beta = \alpha\cdot \theta\cdot \beta $ where $\alpha, \beta\in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part ${\msbm R}eg (\mathscr{T}_X^\theta )$ of the variant of the full transformation semigroup using crossconnections. We characterize the constituent categories of ${\msbm R}eg (\mathscr{T}_X^\theta )$ and describe how they are \emph{crossconnected} by a functor induced by the sandwich transformation $\theta $. This leads us to a structure theorem for the semigroup and gives the representation of ${\msbm R}eg (\mathscr{T}_X^\theta )$ as a crossconnection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.
DOI: 10.14232/actasm017044z
AMS Subject Classification
(1991): 20M10, 20M17, 20M50
Keyword(s):
regular semigroup,
full transformation semigroup,
crossconnections,
normal category,
variant
Received June 30, 2017, and in revised form February 12, 2018. (Registered under 44/2017.)
Gary L. Peterson,
Stuart D. Scott

401411

Abstract. In this paper we shall obtain that the nearring $C_0(V)$ of congruence preserving functions that are 0preserving of a tame $N$module $V$ of a nearring $N$ is finite when $N$ is finite. As a consequence, $C_0(V)$ of an expanded group $\langle V,+,F\rangle $ is finite when the nearring of 0preserving polynomial functions $P_0(V)$ of $\langle V,+,F\rangle $ is finite. We then go on to obtain further consequences of this result.
DOI: 10.14232/actasm0172999
AMS Subject Classification
(1991): 16Y30; 08A40
Keyword(s):
nearring,
expanded group,
tame module,
polynomial functions,
congruence preserving functions,
endomorphism nearring
Received July 10, 2017 and in final form May 20, 2018. (Registered under 49/2017.)
Zsigmond Tarcsay,
Tamás Titkos

413430

Abstract. The notion of quasiunit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potentialtheoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasiunits as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of the effect algebra, we conclude the same properties for quasiunits. Our second aim is to apply these results for nonnegative sesquilinear forms. Constructing an orderpreserving bijection between operator and form segments, we provide a characterization of being extremal in the convexity sense, and we give a necessary and sufficient condition for the existence of the greatest lower bound of two forms. Closing the paper we revisit some statements by using the machinery developed by Hassi, Sebestyén, and de Snoo. It will turn out that quasiunits are exactly the closed elements with respect to the antitone Galois connection induced by parallel addition and subtraction.
DOI: 10.14232/actasm017088z
AMS Subject Classification
(1991): 47A07, 47B65
Keyword(s):
quasiunit,
orthogonal projection,
extreme points,
Galois connection
Received December 30, 2017 and in final form June 29, 2018. (Registered under 88/2017.)
Abstract. We give short proofs of two descriptions given by Šemrl of order automorphisms of the effect algebra. This sheds new light on both formulas that look quite complicated. Our proofs rely on Molnár's characterization of order automorphisms of the cone of all positive operators.
DOI: 10.14232/actasm0180087
AMS Subject Classification
(1991): 47B49
Keyword(s):
selfadjoint operator,
operator interval,
effect algebra,
order isomorphism,
operator monotone function
Received November 15, 2017, and in revised form January 13, 2018. (Registered under 8/2018.)
Abstract. We refine earlier results concerning the structure of strongly continuous oneparameter semigroups ($C_0$SGR) of holomorphic Carathéodory isometries of the unit ball in infinitedimensional reflexive TROs (ternary rings of operators) We achieve finite algebraic formulas for them in terms of joint boundary fixed points and Möbius charts.
DOI: 10.14232/actasm0187613
AMS Subject Classification
(1991): 47D03, 32H15, 46G20
Keyword(s):
Carathéodory distance,
isometry,
fixed point,
holomorphic map,
$C_0$semigroup,
infinitesimal generator,
JB*triple,
Möbius transformation,
Cartan factor,
ternary ring of operators (TRO)
Received January 16, 2018 and in final form April 26, 2018. (Registered under 11/2018.)
Abstract. We prove that any bijective map between the positive definite cones of von Neumann algebras which preserves a certain unitarily invariant norm of a particular weighted geometric mean of elements is essentially (up to twosided multiplication by an invertible positive element) equal to the restriction of a Jordan *isomorphism between the algebras.
DOI: 10.14232/actasm018514x
AMS Subject Classification
(1991): 47B49; 46L40, 47A64
Keyword(s):
positive definite cone,
operator means,
geodesic correspondence,
preservers
Received January 18, 2018 and in final form February 23, 2018. (Registered under 14/2018.)
Seppo Hassi,
Zoltán Sebestyén,
Henk de Snoo

465507

Abstract. A linear relation, i.e., a multivalued operator $T$ from a Hilbert space $\sH $ to a Hilbert space $\sK $ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of $T$, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where $T_1$ is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.
DOI: 10.14232/actasm0187570
AMS Subject Classification
(1991): 4705, 47A06, 47A65; 46N30, 47N30
Keyword(s):
regular relations,
singular relations,
(weak) Lebesgue type decompositions,
uniqueness of decompositions,
domination of relations and operators,
closability
Received January 11, 2018 and in final form April 30, 2018. (Registered under 7/2018.)
Takeshi Yoshimoto

509554

Abstract. This paper projects another affine case study in the program of analyzing multiparameter a.e. convergence, based on the Sucheston's type convergence principles. An affine semigroup is considered as a natural extension of strongly continuous semigroups of linear operators on $L_{p}$ spaces. We prove some affine extensions of multiparameter martingale theorems, multiparameter ergodic theorems, and multiparameter ergodic theorems for the socalled nonlinear sums. Moreover, an affine (nonlinear) generalization is given of BerksonBourgainGillespie's theorem concerning the connection between the ergodic Hilbert transform and the ergodic theorem for powerbounded invertible linear operators on $L_{p}$ ($1< p< \infty $) spaces. In addition, the random ergodic Hilbert transforms will be established. We improve the local ergodic theorem of McGrath concerning strongly continuous $m$parameter semigroups of positive linear operators in a more general affine setting. We shall also show that the Sucheston convergence principle is also very effective even in yielding a multiparameter generalization of Starr's theorem.The final section includes some examples.
DOI: 10.14232/actasm0165100
AMS Subject Classification
(1991): 47A35, 40H05; 40G10
Keyword(s):
affine semigroup,
compound semigroup,
ergodic Hilbert transform,
random ergodic Hilbert transform,
Cotlar's theorem,
BerksonBourgainGillespie's theorem,
Sucheston's type convergence principle,
Orlicz class,
multiparameter martingale theorem,
nonlinear sum,
ergodic theorem for affine semigroups,
Abelian ergodic theorem for affine semigroups,
Starr's theorem
Received February 15, 2016 and in final form February 26, 2018. (Registered under 10/2016.)
Pietro Aiena,
Mohammed Kachad

555571

Abstract. Property $(UW_{\Pi })$ for a bounded linear operator $T\in L(X)$ on a Banach space $X$ is a variant of Browder's theorem, and means that the points $\lambda $ of the approximate point spectrum for which $\lambda IT$ is upper semiWeyl are exactly the spectral points $\lambda $ such that $\lambda IT$ is Drazin invertible. In this paper we investigate this property, and we give several characterizations of it by using typical tools from local spectral theory. We also relate this property with some other variants of Browder's theorem (or Weyl's theorem).
DOI: 10.14232/actasm0163035
AMS Subject Classification
(1991): 47A53, 47A10, 47A11
Keyword(s):
property $(UW {\scriptstyle\Pi })$,
SVEP
Received September 26, 2016 and in final form May 15, 2018. (Registered under 53/2016.)
M. H. M. Rashid,
Muneo Ch?,
T. Prasad,
Kotaro Tanahashi,
Atsushi Uchiyama

573589

Abstract. In this paper, we study spectral properties of class $p$$wA(s,t)$ operators with $0< p\leq1$ and $0< s,t,s+t\leq1$. We show that Weyl's theorem and Putnam's inequality hold for class $p$$wA(s,t)$ operators.
DOI: 10.14232/actasm017020y
AMS Subject Classification
(1991): 47A10, 47A20, 47B20
Keyword(s):
class $p$$wA(s,
t)$,
normaloid,
reguloid,
Weyl's theorem,
Putnam's inequality
Received March 28, 2017, and in revised form November 24, 2017. (Registered under 20/2017.)
Rodney Nillsen,
Susumu Okada

591609

Abstract. Let $G$ denote the group ${\msbm R}$ or ${\msbm T}$, let $\iota $ denote the identity element of $G$, and let $s\in{{\msbm N}}$ be given. Then, a \emph{difference of order} $s$ is a function $f\in L^2(G)$ for which there are $a\in G$ and $g \in L^2({G})$ such that $f= (\delta_{\iota }\delta_{a})^s\ast g$. Let ${{\cal D}}_s(L^2(G))$ be the vector space of functions that are finite sums of differences of order $s$. It is known that if $f\in L^2({{\msbm R}})$, $f\in{{\cal D}}_s(L^2({{\msbm R}}))$ if and only if $\int_{\infty }^{\infty }{\widehat f}(x)^2x^{2s}dx< \infty $. Also, if $f\in L^2({{\msbm T}})$, $f\in{{\cal D}}_s(L^2({{\msbm T}}))$ if and only if ${\widehat f}(0)=0$. Consequently, ${{\cal D}}_s(L^2(G))$ is a Hilbert space in a (possibly) weighted $L^2$norm. It is known that every function in ${{\cal D}}_s(L^2(G))$ is a sum of $2s+1$ differences of order $s$. However, there are functions in ${{\cal D}}_s(L^2({{\msbm R}}))$ that are not a sum of $2s$ differences of order $s$, and we call the latter type of fact a \emph{sharpness result}. In ${{\cal D}}_1(L^2({{\msbm T}}))$, it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces ${{\cal D}}_s(L^2({{\msbm T}}))$ that complement the results known for ${{\msbm R}}$, but also to present new results in ${{\cal D}}_s(L^2({{\msbm T}}))$ that do not correspond to known results in ${{\cal D}}_s(L^2({{\msbm R}}))$. Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics.
DOI: 10.14232/actasm017522y
AMS Subject Classification
(1991): 42A16, 42A38
Keyword(s):
Fourier transform,
finite differences,
subspaces of $L^2({{\msbm T}})$,
combinatorial inequalities,
badly approximable vectors in ${{\msbm R}}^n$,
sharpness results,
Sobolev spaces
Received April 7, 2017 and in final form May 22, 2018. (Registered under 22/2017.)
C. Bouraya,
A. Seddik

611627

Abstract. In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (selfadjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.
DOI: 10.14232/actasm0177736
AMS Subject Classification
(1991): 47A30, 47A05, 47B15
Keyword(s):
closed range operator,
MoorePenrose inverse,
group inverse,
selfadjoint operator,
unitary operator,
normal operator,
partial isometry operator,
isometry operator,
operator inequality
Received April 8, 2017, and in final form November 25, 2017. (Registered under 23/2017.)
Anuradha Gupta,
Renu Chugh,
Jagjeet Jakhar

629641

Abstract. In this paper, we discuss the conditions under which composition operators and weighted composition operators become quasi$A(n)$ operators, quasi$*$$A(n)$ operators, $k$quasi$A(n)$ operators and $k$quasi$*$$A(n)$ operators in terms of the RadonNikodym derivative $h_n$.
DOI: 10.14232/actasm0170325
AMS Subject Classification
(1991): 47B33, 47B20; 46C05
Keyword(s):
composition operators,
weighted composition operators,
quasi$A(n)$ operators,
quasi$*$$A(n)$ operators,
$k$quasi$A(n)$ operators and $k$quasi$*$$A(n)$ operators
Received May 23, 2017, and in revised form November 27, 2017. (Registered under 32/2017.)
Abstract. In this paper we study the problem of the boundedness and compactness of the Toeplitz operator $T_{\varphi }$ on $L_{a}^{2}(\Omega )$, where $\Omega $ is a multiplyconnected domain and $\varphi $ is not bounded. We find a necessary and sufficient condition when the symbol is $\mathcal{BMO}.$ For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.
DOI: 10.14232/actasm0172830
AMS Subject Classification
(1991): 47B35; 47B38
Keyword(s):
Bergman space,
Toeplitz operator,
Berezin transform
Received May 29, 2017 and in final form September 2, 2018. (Registered under 33/2017.)
Feliz Minhós,
Infeliz Coxe

659672

Abstract. In this paper we consider the nonlinear thirdorder coupled system composed by the differential equations \[\left\{{ u^{\prime\prime\prime}(t)=f\left(t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime}(t)\right), \atop v^{\prime\prime\prime}(t) =h\left( t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime }(t)\right ),}\right. \] with $f,h\colon[0,1] \times\mathbb {R}^{6}\rightarrow\mathbb {R}$ continuous functions, and the boundary conditions \[ \left\{{ u(0)=u^{\prime }(0) =u^{\prime}(1) =0, \atop v(0)=v^{\prime}(0) =v^{\prime}(1) =0. }\right.\] We remark that the nonlinearities can depend on all derivatives of both unknown functions, which is new in the literature, as far as we know. This is due to an adequate auxiliary integral problem with a truncature, applying lower and upper solutions method with bounded perturbations. The main theorem is an existence and localization result, which provides some qualitative data on the system solution, such as, sign, variation, bounds, etc., as it can be seen in the example.
DOI: 10.14232/actasm0177850
AMS Subject Classification
(1991): 34B15, 34B27, 34L30
Keyword(s):
coupled systems,
Green functions,
Nagumotype condition,
coupled lower and upper solutions
Received May 29, 2017 and in final form June 10, 2018. (Registered under 35/2017.)
Heybetkulu Mustafayev

673685

Abstract. Let $A$ be a complex commutative semisimple Banach algebra and let $T$ be a power bounded multiplier of $A$. This paper is concerned with finding necessary and sufficient conditions for the convergence of the sequence $\left\{ T^{n}a\right\} $ $( a\in A) $ in $A.$ Some related problems are also discussed.
DOI: 10.14232/actasm0172915
AMS Subject Classification
(1991): 46HXX, 43A20, 43A22
Keyword(s):
commutative Banach algebra,
multiplier,
set of synthesis,
convergence
Received June 23, 2017, and in revised form December 6, 2017. (Registered under 41/2017.)
E. Peyghan,
C. Arcuł,
A. Baghban,
E. Sharahi

687704

Abstract. We consider the direct sum of a Lie algebroid structure with its dual space and equip this bigger space with a contact form called the generalized almost contact structure and characterize these in the sense of contact morphisms. Attaching an almost generalized complex structure, we recover properties as normality conditions in a direct way and study some aspects of metrical kind of this spaces.
DOI: 10.14232/actasm0177778
AMS Subject Classification
(1991): 53D10
Keyword(s):
generalized almost contact structure,
Lie algebroid,
metric structure,
normality conditions
Received April 24, 2017, and in revised form September 21, 2017. (Registered under 27/2017.)
B. L. S. Prakasa Rao

705711

Abstract. We obtain a characterization for probability measures on a locally compact Abelian group $X$ based on linear forms of $Q$independent random elements taking values in $X$ generalizing the earlier work of the author in [12].
DOI: 10.14232/actasm0175303
AMS Subject Classification
(1991): 60B15, 62E10
Keyword(s):
$Q$independence,
characterization,
locally compact Abelian group
Received April 27, 2017 and in final form April 18, 2018. (Registered under 30/2017.)

713716
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