ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In this paper, we consider Rota--Baxter operators on involutive associative algebras. We define cohomology for Rota--Baxter 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 Fard--Guo construction of the functor from the category of dendriform algebras to the category of Rota--Baxter algebras restricts to the involutive case.

DOI: 10.14232/actasm-020-616-0

AMS Subject Classification (1991): 16E40, 16S80, 16W99

Keyword(s): involutive algebras, Hochschild cohomology, Rota--Baxter operators, deformations, dendriform algebras

received 16.6.2020, accepted 21.5.2021. (Registered under 616/2020.)

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/actasm-020-558-7

AMS Subject Classification (1991): 20M15, 20M18, 20M19; 08A30

Keyword(s): torsion-free 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 five-element 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 {Two-pendant Four-crown Property} and the \emph {Forbidden Marriage Property}.

DOI: 10.14232/actasm-021-865-y

AMS Subject Classification (1991): 06C10

Keyword(s): rectangular lattice, slim semimodular lattice, multifork extension, lattice diagram, edge of normal slope, precipitous edge, lattice congruence, two-pendant four-crown property, lamp, congruence lattice, forbidden marriage property

received 15.1.2021, revised 5.3.2021, accepted 11.3.2021. (Registered under 115/2021.)

Abstract. A lattice is $(1+1+2)$-generated if it has a four-element 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 twenty-four 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/actasm-021-303-1

AMS Subject Classification (1991): 06B99

Keyword(s): quasiorder lattice, lattice of preorders, minimum-sized generating set, four-generated 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.)

Abstract. A characteristic function is a special operator-valued 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/actasm-020-303-x

AMS Subject Classification (1991): 47A45, 47A20, 47A48, 47A56

Keyword(s): characteristic functions, analytic model, nilpotent operators, operator-valued polynomials, Gleason's problem, factorizations

received 22.10.2020, revised 28.6.2021, accepted 3.7.2021. (Registered under 53/2020.)

Abstract. We show in this paper that a Wold-type decomposition holds for the class of regular operators with regular Moore--Penrose inverse. We also give several examples and investigate various properties of such class of operators.

DOI: 10.14232/actasm-020-399-2

AMS Subject Classification (1991): 47A15; 47B37

Keyword(s): Wold-type decomposition, regular and bi-regular operators, Moore--Penrose inverse, Cauchy dual

received 18.11.2020, revised 10.7.2021, accepted 30.7.2021. (Registered under 149/2020.)

Abstract. This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained 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/actasm-021-263-6

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 so-called $\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 multi-dimensional 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/actasm-021-614-3

AMS Subject Classification (1991): 42B08, 42A38, 42A24, 42B25

Keyword(s): Cesàro summability, Hardy--Littlewood maximal function, Lebesgue points

received 14.1.2021, revised 29.8.2021, accepted 31.8.2021. (Registered under 114/2021.)

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/actasm-021-120-8

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.)

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/actasm-021-028-7

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/actasm-021-785-5

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 Desch--Schappacher perturbations of bi-continuous semigroups on $\mathrm {AM}$-spaces with an additional property concerning the additional locally convex topology. As an example, we discuss perturbations of the left-translation semigroup on the space of bounded continuous functions on the real line and on the space of bounded linear operators.

DOI: 10.14232/actasm-021-914-5

AMS Subject Classification (1991): 47D03, 47A55, 34G10, 46A70, 46A40

Keyword(s): bi-continuous semigroups, positivity, Desch--Schappacher perturbation, Gamma function

received 14.4.2021, revised 8.7.2021, accepted 10.7.2021. (Registered under 414/2021.)

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/actasm-021-465-4

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.)

Abstract. In 2018, Edelsbrunner and Iglesias-Ham 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 face-centered 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 face-centered cubic lattice, for every $t \geq 1$, and at the points of a body-centered cubic lattice for $t=0.5$.

DOI: 10.14232/actasm-020-483-y

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.)

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/actasm-020-361-6

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.)

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 two-sided, then the strong $(C,\alpha )$ law of large numbers for the sequence can also be characterized by the ergodic Hilbert transform.

DOI: 10.14232/actasm-021-271-y

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.)