
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Abstract. In this paper, we consider RotaBaxter operators on involutive associative algebras. We define cohomology for RotaBaxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard FardGuo construction of the functor from the category of dendriform algebras to the category of RotaBaxter algebras restricts to the involutive case.
DOI: 10.14232/actasm0206160
AMS Subject Classification
(1991): 16E40, 16S80, 16W99
Keyword(s):
involutive algebras,
Hochschild cohomology,
RotaBaxter operators,
deformations,
dendriform algebras
received 16.6.2020, accepted 21.5.2021. (Registered under 616/2020.)
Simon M. Goberstein

367379

Abstract. Two semigroups are lattice isomorphic if the lattices of their subsemigroups are isomorphic, and a class of semigroups is lattice closed if it contains every semigroup which is lattice isomorphic to some semigroup from that class. An orthodox semigroup is a regular semigroup whose idempotents form a subsemigroup. We prove that the class of all orthodox semigroups in which every nonidempotent element has infinite order is lattice closed.
DOI: 10.14232/actasm0205587
AMS Subject Classification
(1991): 20M15, 20M18, 20M19; 08A30
Keyword(s):
torsionfree semigroups,
orthodox semigroups,
monogenic orthodox semigroups,
inverse semigroups,
monogenic inverse semigroups,
lattice isomorphisms of semigroups,
lattice determined semigroups,
lattice closed classes of semigroups
received 27.10.2020, revised 4.8.2021, accepted 13.8.2021. (Registered under 58/2020.)
Abstract. A planar (upper) semimodular lattice $L$ is \emph {slim} if the fiveelement nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) \emph {Slim rectangular lattices} as particular slim planar semimodular lattices were defined by G. Grätzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of \emph {lamps} of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the \emph {Twopendant Fourcrown Property} and the \emph {Forbidden Marriage Property}.
DOI: 10.14232/actasm021865y
AMS Subject Classification
(1991): 06C10
Keyword(s):
rectangular lattice,
slim semimodular lattice,
multifork extension,
lattice diagram,
edge of normal slope,
precipitous edge,
lattice congruence,
twopendant fourcrown property,
lamp,
congruence lattice,
forbidden marriage property
received 15.1.2021, revised 5.3.2021, accepted 11.3.2021. (Registered under 115/2021.)
Delbrin Ahmed,
Gábor Czédli

415427

Abstract. A lattice is $(1+1+2)$generated if it has a fourelement generating set such that exactly two of the four generators are comparable. We prove that the lattice $\Quo n$ of all quasiorders (also known as preorders) of an $n$element set is $(1+1+2)$generated for $n=3$ (trivially), $n=6$ (when $\Quo 6$ consists of $209\,527$ elements), $n=11$, and for every natural number $n\geq 13$. In 2017, the second author and J. Kulin proved that $\Quo n$ is $(1+1+2)$generated if either $n$ is odd and at least $13$ or $n$ is even and at least $56$. Compared to the 2017 result, this paper presents twentyfour new numbers $n$ such that $\Quo n$ is $(1+1+2)$generated. Except for $\Quo 6$, an extension of Zádori's method is used.
DOI: 10.14232/actasm0213031
AMS Subject Classification
(1991): 06B99
Keyword(s):
quasiorder lattice,
lattice of preorders,
minimumsized generating set,
fourgenerated lattice,
$(1+1+2)$generated lattice,
Zádori's method
received 3.5.2021, revised 19.5.2021, accepted 19.5.2021. (Registered under 53/2021.)
Monojit Bhattacharjee,
Kalpesh J. Haria,
Jaydeb Sarkar

429461

Abstract. A characteristic function is a special operatorvalued analytic function defined on the open unit ball of $\mathbb {C}^n$ associated with an $n$tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.
DOI: 10.14232/actasm020303x
AMS Subject Classification
(1991): 47A45, 47A20, 47A48, 47A56
Keyword(s):
characteristic functions,
analytic model,
nilpotent operators,
operatorvalued polynomials,
Gleason's problem,
factorizations
received 22.10.2020, revised 28.6.2021, accepted 3.7.2021. (Registered under 53/2020.)
H. Ezzahraoui,
M. Mbekhta,
E. H. Zerouali

463483

Abstract. We show in this paper that a Woldtype decomposition holds for the class of regular operators with regular MoorePenrose inverse. We also give several examples and investigate various properties of such class of operators.
DOI: 10.14232/actasm0203992
AMS Subject Classification
(1991): 47A15; 47B37
Keyword(s):
Woldtype decomposition,
regular and biregular operators,
MoorePenrose inverse,
Cauchy dual
received 18.11.2020, revised 10.7.2021, accepted 30.7.2021. (Registered under 149/2020.)
Kanae Hatano,
Yoshimichi Ueda

485503

Abstract. This note is a complement to PuszWoronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, selfcontained and purely Hilbert space operator explanation of their functional calculus, we show that any operator connection type operations (including any operator perspectives) are captured by their functional calculus.
DOI: 10.14232/actasm0212636
AMS Subject Classification
(1991): 47A60; 47A64
Keyword(s):
functional calculus,
operator connection,
operator perspective,
convexity
received 3.1.2021, revised 28.8.2021, accepted 31.8.2021. (Registered under 13/2021.)
Abstract. We introduce a new concept of Lebesgue points, the socalled $\omega $Lebesgue points, where $\omega >0$. As a generalization of the classical Lebesgue's theorem, we prove that the Cesàro means $\sigma _n^{a}f$ of the Fourier series of a multidimensional function $f\in L_1(\T ^d)$ converge to $f$ at each $\omega $Lebesgue point $(0<\omega <\alpha )$ as $n\to \infty $.
DOI: 10.14232/actasm0216143
AMS Subject Classification
(1991): 42B08, 42A38, 42A24, 42B25
Keyword(s):
Cesàro summability,
HardyLittlewood maximal function,
Lebesgue points
received 14.1.2021, revised 29.8.2021, accepted 31.8.2021. (Registered under 114/2021.)
Biswaranjan Behera,
Md. Nurul Molla

517539

Abstract. Let $K$ be a totally disconnected, locally compact and nondiscrete field of positive characteristic and $\D $ be its ring of integers. We characterize the Schauder basis property of the Gabor systems in $K$ in terms of $A_2$ weights on $\D \times \D $ and the Zak transform $Zg$ of the window function $g$ that generates the Gabor system. We show that the Gabor system generated by $g$ is a Schauder basis for $L^2(K)$ if and only if $Zg^2$ is an $A_2$ weight on $\D \times \D $. Some examples are given to illustrate this result. Moreover, we construct a Gabor system which is complete and minimal, but fails to be a Schauder basis for $L^2(K)$.
DOI: 10.14232/actasm0211208
AMS Subject Classification
(1991): 43A70; 42B25, 43A25
Keyword(s):
local field,
Gabor system,
Zak transform,
$A_p$weight,
Schauder basis
received 20.1.2021, accepted 20.3.2021. (Registered under 120/2021.)
Roksana Krystyna Słowik

541550

Abstract. Banded lower triangular $\mathbb N\times \mathbb N$ Toeplitz matrices $A$ are considered. A sufficient condition for the elements of $A^{1}$ to decay to $0$ fast is given. Moreover, some bounds of the norms of these inverses are also found.
DOI: 10.14232/actasm0210287
AMS Subject Classification
(1991): 15B05, 15A99
Keyword(s):
triangular Toeplitz matrices,
matrix inverse,
decay of elements,
norm of a Toeplitz matrix
received 8.2.2021, revised 31.5.2021, accepted 13.6.2021. (Registered under 28/2021.)
Abstract. Using the technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces, respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces.
DOI: 10.14232/actasm0217855
AMS Subject Classification
(1991): 46B40; 46B20
Keyword(s):
adjoining an order unit,
strictly convex space,
absolutely ordered space,
absolute order unit space,
$JB$algebra,
spin factor
received 5.3.2021, revised 19.4.2021, accepted 30.7.2021. (Registered under 35/2021.)
Abstract. In this paper, we consider positive DeschSchappacher perturbations of bicontinuous semigroups on $\mathrm {AM}$spaces with an additional property concerning the additional locally convex topology. As an example, we discuss perturbations of the lefttranslation semigroup on the space of bounded continuous functions on the real line and on the space of bounded linear operators.
DOI: 10.14232/actasm0219145
AMS Subject Classification
(1991): 47D03, 47A55, 34G10, 46A70, 46A40
Keyword(s):
bicontinuous semigroups,
positivity,
DeschSchappacher perturbation,
Gamma function
received 14.4.2021, revised 8.7.2021, accepted 10.7.2021. (Registered under 414/2021.)
Emmanuel Fricain,
Javad Mashreghi

595613

Abstract. We provide an orthogonal basis of polynomials for the local Dirichlet space $\mathcal {D}_\zeta $. These polynomials have numerous interesting features and a very unique algebraic pattern. We obtain the recurrence relation, the generating function, a simple formula for their norm, and explicit formulae for the distance and the orthogonal projection onto the subspace of polynomials of degree at most $n$. The latter implies a new polynomial approximation scheme in local Dirichlet spaces. Orthogonal polynomials in a harmonically weighted Dirichlet space, created by a finitely supported singular measure, are also studied.
DOI: 10.14232/actasm0214654
AMS Subject Classification
(1991): 30H05, 33C45, 33C47, 42B35
Keyword(s):
harmonically weighted Dirichlet spaces,
orthogonal polynomials,
polynomial approximation
received 15.7.2021, accepted 12.9.2021. (Registered under 715/2021.)
Sami Mezal Almohammad

615647

Abstract. In 2018, Edelsbrunner and IglesiasHam defined a notion of density, called first soft density, for lattice packings of congruent balls in Euclidean $3$space, which penalizes gaps and multiple overlaps. In their paper, they showed that this density is maximal in a $1$parameter family of lattices, called diagonal family, for a configuration of congruent balls whose centers are the points of a facecentered cubic lattice. In this note we extend their notion of density, which we call first soft density of weight $t$, and show that it is maximal in the diagonal family for some family of congruent balls centered at the points of a facecentered cubic lattice, for every $t \geq 1$, and at the points of a bodycentered cubic lattice for $t=0.5$.
DOI: 10.14232/actasm020483y
AMS Subject Classification
(1991): 52C17, 52A38, 52A15
Keyword(s):
packing and covering,
soft density of weight $t$,
lattice configurations,
Voronoi domains,
Brillouin zones
received 2.12.2020, revised 23.4.2021, accepted 17.8.2021. (Registered under 233/2020.)
Amenah ALNajafi,
László L. Stachó,
László Viharos

649678

Abstract. We propose a class of weighted least squares estimators for the tail index of a distribution function with a regularly varying tail. Our approach is based on the method developed by Holan and McElroy (2010) for the Parzen tail index. We prove asymptotic normality and consistency for the estimators under suitable assumptions. These and earlier estimators are compared in various models through a simulation study using the mean squared error as criterion. The results show that the weighted least squares estimator has good performance.
DOI: 10.14232/actasm0203616
AMS Subject Classification
(1991): 60F05, 62G32
Keyword(s):
tail index,
Pareto model,
weighted least squares estimators,
quantile process
received 11.6.2020, revised 31.8.2021, accepted 2.9.2021. (Registered under 611/2020.)
Takeshi Yoshimoto

679707

Abstract. We give a necessary and sufficient condition for the strong $(C,\alpha )$ law of large numbers with real order $\alpha >0$ for weighted sums of independent random variables satisfying the property $\alpha $WH analogous to, though weaker than, the Hartman's type property. In particular, if a sequence of random variables is twosided, then the strong $(C,\alpha )$ law of large numbers for the sequence can also be characterized by the ergodic Hilbert transform.
DOI: 10.14232/actasm021271y
AMS Subject Classification
(1991): 60F15; 47A35
Keyword(s):
strong law of large numbers,
weak homogeneity,
Bourgain's return time theorem,
sampling scheme,
Doob scheme,
ergodic Hilbert transform,
universal sequence of weights
received 1.2.2021, revised 15.6.2021, accepted 30.6.2021. (Registered under 21/2021.)
