
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
The Editors

343346
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P. Kevei

347349
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Abstract. In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can be arbitrarily large. For every $k > 1$, we construct a planar lattice $L$ generated by $4$ atoms such that $L$ has more than $k$ elements.
DOI: 10.14232/actasm0203637
AMS Subject Classification
(1991): 06B05
Keyword(s):
planar,
atomgenerated
received 13.1.2019, revised 3.9.2020, accepted 4.9.2020 (Registered under 113/2020.)
Abstract. In 1991, Lawson introduced three partial orders on reduced $U$semiabundant semigroups. Their definitions are formally similar to recently discovered characteristics of the diamond, left star and right star orders respectively on Rickart *rings; lattice properties of these orders have been studied by several authors. Motivated by these similarities, we turn to the lattice structure of $U$semiabundant semigroups and rings under Lawson's orders. In this paper, we deal with his order $\les _l$ on (a version of) right $U$semiabundant semigroups and rings. In particular, existence of meets is investigated, it is shown that (under some natural assumptions) every initial section of such a ring is an orthomodular lattice, and explicit descriptions of the corresponding lattice operations are given.
DOI: 10.14232/actasm0194263
AMS Subject Classification
(1991): 20M10; 06A06, 06C15, 20M25, 16U99, 16W99
Keyword(s):
Baer semigroup,
Dsemigroup,
Dring,
generalized orthomodular poset,
orthomodular lattice,
relatively orthocomplemented poset,
Rickart ring,
right normal band,
right star order,
$U$semiabundant semigroup
received 26.9.2019, revised 22.5.2020, accepted 25.5.2020. (Registered under 926/2019.)
Gábor Czédli,
Lillian Oluoch

405448

Abstract. Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$element set has a fourelement generating set. In 1983, L. Zádori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $\gnu n$ of fourelement generating sets of Part$(n)$. We also present a computerassisted statistical approach to $\gnu n$ for small values of $n$. In his 1983 paper, L. Zádori also proved that for $n\geq 7$, the lattice Part$(n)$ has a fourelement generating set that is not an antichain. He left the problem whether such a generating set for $n\in \set {5,6}$ exists open. Here we solve this problem in negative for $n=5$ and in affirmative for $n=6$. Finally, the main theorem asserts that the direct product of some powers of partition lattices is fourgenerated. In particular, by the first part of this theorem, $\Part {n_1}\times \Part {n_2}$ is fourgenerated for any two distinct integers $n_1$ and $n_2$ that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product $\Part {n}\times \Part {n+1}\times \dots \times \Part {3n14}$ is fourgenerated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$th the direct power of the direct product $\Part {n}\times \Part {n+1}\times \dots \times \Part {n+u1}$ is fourgenerated for all but finitely many $n$. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the $ 10^{127}$th direct power of $\Part {1011}\times \Part {1012}\times \dots \times \Part {2020}$ is fourgenerated.
DOI: 10.14232/actasm0201267
AMS Subject Classification
(1991): 06B99, 06C10
Keyword(s):
equivalence lattice,
partition lattice,
fourelement generating set,
sublattice,
statistics,
computer algebra,
computer program,
direct product of lattices,
generating partition lattices,
semimodular lattice,
geometric lattice
received 26.6.2020, revised 22.9.2020, accepted 27.9.2020. (Registered under 626/2020.)
Arup Chattopadhyay,
Debmalya Sain,
Tanusri Senapati

449466

Abstract. We introduce the concept of nonpositive operators with respect to a fixed operator defined between two real normed linear spaces. Significantly, we observe that, in certain cases, it is possible to study such type of operators from a geometric point of view. As an immediate application of our study, we explicitly characterize certain classes of nonpositive operators between particular pairs of real normed linear spaces. Furthermore, we present a complete characterization of smooth and strictly convex Radon planes in connection with nonpositive operators.
DOI: 10.14232/actasm019554z
AMS Subject Classification
(1991): 47L05, 46B20
Keyword(s):
BirkhoffJames orthogonality,
semiinnerproduct,
nonpositive operator,
polyhedral Banach spaces
received 23.10.2019, revised 5.8.2020, accepted 11.8.2020. (Registered under 54/2019.)
Abstract. The proofs of generalized Hardy, Copson, Bennett, Leindlertype, and Levinson integral inequalities are revisited. It is contemplated to establish new proof of these classical inequalities using probability density function. New integral inequalities of Hardytype involving the $r^{th}$ order \emph {Generalized RiemannLiouville}, \emph {Generalized Weyl}, \emph {ErdélyiKober}, \emph {$(k, \nu )$RiemannLiouville}, and \emph {$(k, \nu )$Weyl fractional integrals} are established through a probabilistic approach. The KullbackLeibler inequality has been applied to compute the best possible constant factor associated with each of these inequalities.
DOI: 10.14232/actasm0197507
AMS Subject Classification
(1991): 26D15; 26D10, 26A33
Keyword(s):
Hardy's inequality for integrals,
probability density function,
RiemannLiouville integral,
Weyl integral,
scale distribution,
KullbackLeibler inequality
received 19.12.2019, revised 16.4.2020, accepted 18.4.2020. (Registered under 250/2019.)
Hirokazu Oka,
Takeshi Miura,
SinEi Takahasi

493502

Abstract. Let $\mathbf R_+$ be the space of positive real numbers with the ordinary topology and let $\star $ be an arbitrary cancellative continuous semigroup operation on $\mathbf R_+$ or some special noncancellative continuous semigroup operation on $\mathbf R_+$. We characterize the set $\mathcal {D}_{\star }^{1}(\mathbf R_+)$ of all cancellative continuous semigroup operations on $\mathbf R_+$ which are distributive over $\star $ in terms of homeomorphism. As a consequence, it is shown that if $\star $ is homeomorphically isomorphic to the ordinary addition $+$ on $\mathbf R_+$, any element of $\mathcal {D}_{\star }^{1}(\mathbf R_+)$ is homeomorphically isomorphic to the ordinary multiplication on $\mathbf R_+$, and that if $\star $ is cancellative and not homeomorphically isomorphic to $+$, then $\mathcal {D}_{\star }^{1}(\mathbf R_+)$ is empty. Moreover, if $\star $ is homeomorphically isomorphic to some special noncancellative continuous semigroup operation on $\mathbf R_+$, $\mathcal {D}_{\star }^{1}(\mathbf R_+)$ is also shown to be empty.
DOI: 10.14232/actasm0201161
AMS Subject Classification
(1991): 22A15; 06F05
Keyword(s):
cancellative semigroup operation,
distributive law,
homeomorphic isomorphism,
noncancellative semigroup operation
received 16.1.2020, revised 16.8.2020, accepted 21.8.2020. (Registered under 116/2020.)
Abstract. In [BJKP] the concept of pluquasisimilarity was introduced, which is a relation between two operators, implemented by systems of intertwining mappings. In this note we carry out a detailed analysis of this and related connections, compare them, explore their properties, and pose some relevant questions.
DOI: 10.14232/actasm0209734
AMS Subject Classification
(1991): 47A15, 47A45, 47A65, 47B20
Keyword(s):
intertwining relations,
quasisimilarity,
hyperinvariant subspace,
quasinormal operator,
isometry,
unilateral shift
received 23.2.2020, revised 30.4.2020, accepted 30.4.2020. (Registered under 223/2020.)
Canay Aykol,
Javanshir J. Hasanov

521547

Abstract. In this paper we consider the generalized shift operator associated to the LaplaceBessel differential operator $\Delta _{B}$ and investigate Bmaximal commutators, commutators of BRiesz potentials and commutators of Bsingular integral operators associated to the generalized shift operator. The boundedness of the $B$maximal commutator $M_{b,\gamma }$ and the commutator $[b,A_{\gamma }]$ of the $B$singular integral operator on the modified $B$Morrey spaces $\widetilde {L}_{p,\lambda ,\gamma }(\Rnk )$ for all $1 < p < \infty $ when $b \in BMO_\gamma ({\Rnk })$ are proved. In addition, we obtain that the commutator $[b,I_{\alpha ,\gamma }]$ of the $B$Riesz potential $I_{\alpha ,\gamma }$ is bounded from the modified $B$Morrey space $\widetilde {L}_{p,\lambda ,\gamma }(\Rnk )$ to $\widetilde {L}_{q,\lambda ,\gamma }(\Rnk )$, $1<p<\frac {n+\gamma \lambda }{\alpha }$, $\frac {\alpha }{n+\gamma } \le \frac 1p\frac 1q \le \frac {\alpha }{n+\gamma \lambda }$ and from the space $\widetilde {L}_{1,\lambda ,\gamma } (\mathbb {R} _{k,+}^{n})$ to $W\widetilde {L}_{q,\lambda ,\gamma } (\Rnk )$, $\frac {\alpha }{n+\gamma } \le 1\frac 1q \le \frac {\alpha }{n+\gamma \lambda }$.
DOI: 10.14232/actasm020224y
AMS Subject Classification
(1991): 42B20; 42B25, 42B35
Keyword(s):
commutator,
generalized shift operator,
$B$maximal function,
$B$Riesz potential,
Morrey space,
modified Morrey space,
$BMO_{\gamma }$ space
received 24.2.2020, revised 4.4.2020, accepted 18.4.2020. (Registered under 224/2020.)
Abstract. In [uch] (among other results), M. Uchiyama gave necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of normestimates of complete analytic families of eigenvectors of their adjoints. In [gam], a cyclic power bounded operator is constructed which has the requested normestimates, is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this paper, a power bounded operator is constructed which has the requested normestimates, is quasisimilar to $S$, but is not similar to $S$. The question whether the criterion for contractions to be similar to $S$ can be generalized to polynomially bounded operators remains open. Also, for every cardinal number $2\leq N\leq \infty $, a power bounded operator $T$ is constructed such that $T$ is a quasiaffine transform of $S$ and $\dim \ker T^*=N$. This is impossible for polynomially bounded operators. Moreover, the constructed operators $T$ have the requested normestimates of complete analytic families of eigenvectors of~$T^*$.
DOI: 10.14232/actasm0202831
AMS Subject Classification
(1991): 47A05; 47B99, 47B32, 30H10
Keyword(s):
power bounded operator,
unilateral shift,
similarity,
quasisimilarity,
quasiaffine transform,
analytic family of eigenvectors
received 3.3.2020, revised 1.10.2020, accepted 2.10.2020. (Registered under 33/2020.)
Abstract. We extend the FugledePutnam theorem modulo the HilbertSchmidt class to almost normal $m$tuples of operators with finite HilbertSchmidt modulus of quasitriangularity.
DOI: 10.14232/actasm0205346
AMS Subject Classification
(1991): 47B20; 47B10, 47B47
Keyword(s):
HilbertSchmidt modulus of quasitriangularity,
almost normal $m$tuples of operators,
almost hyponormal operators,
generalized FugledePutnam theorem
received 4.3.2020, revised 7.5.2020, accepted 10.5.2020. (Registered under 34/2020.)
Abstract. Let $\mathcal {A}$ be a unital semiprime, complex normed $\ast $algebra and let $f, g, h :\mathcal {A} \rightarrow \mathcal {A}$ be linear mappings such that $f$ and $g + h$ are continuous. Under certain conditions, we prove that if $f(p \circ p) = g(p) \circ p + p \circ h(p)$ holds for any projection $p$ of $\mathcal {A}$, then $f$ and $g + h$ are twosided generalized derivations, where $a \circ b = a b + ba$. We present some consequences of this result. Moreover, we show that if $\mathcal {A}$ is a semiprime algebra with the unit element $\textbf {e}$ and $n > 1$ is an integer such that the linear mappings $f, g\colon \mathcal {A} \rightarrow \mathcal {A}$ satisfy $f(x^n) = \sum _{j = 1}^{n}x^{n  j}g(x) x^{j  1}$ for all $x \in \mathcal {A}$ and further $g(\textbf {e}) \in Z(\mathcal {A})$, then $f$ and $g$ are twosided generalized derivations associated with the same derivation. Also, we show that if $\mathcal {A}$ is a unital, semiprime Banach algebra and $F, G\colon \mathcal {A} \rightarrow \mathcal {A}$ are linear mappings satisfying $F(b) =  b G(b^{1}) b$ for all invertible elements $b \in \mathcal {A}$, then $F$ and $G$ are twosided generalized derivations. Some other related results are also discussed.
DOI: 10.14232/actasm0202958
AMS Subject Classification
(1991): 47B47; 47B48, 39B05
Keyword(s):
twosided generalized derivation,
generalized derivation,
derivation,
functional equation,
normed algebra
received 5.4.2020, revised 21.7.2020, accepted 18.8.2020. (Registered under 45/2020.)
Mohamed Saad Bouh Elemine Vall,
Ahmed Ahmed

601616

Abstract. In this paper, we investigate a class of problems with Neumann boundary data in MusielakOrliczSobolev spaces $W^1L_M(\Omega )$. We prove the existence of infinitely many weak solutions under some hypotheses. We also provide some particular cases and a concrete example in order to illustrate the main results. Our results are an improvement and generalization of the relative results [1].
DOI: 10.14232/actasm0201619
AMS Subject Classification
(1991): 35A15, 58E05; 35J60
Keyword(s):
MusielakSobolev spaces,
Kirchhoff type problem,
variational methods,
critical point theory
received 11.4.2020, revised 15.6.2020, accepted 28.9.2020. (Registered under 411/2020.)
Divya Khurana,
Saikat Roy,
Debmalya Sain

617634

Abstract. We explore the relation between leftsymmetry (rightsymmetry) of elements in a real Banach space and rightsymmetry (leftsymmetry) of their supporting functionals. We obtain a complete characterization of symmetric functionals on a reflexive, strictly convex and smooth Banach space. We also prove that a bounded linear operator from a reflexive, KadetsKlee and strictly convex Banach space to any Banach space is symmetric if and only if it is the zero operator. We further characterize leftsymmetric operators from $\ell _1^n$, $n\geq 2$, to any Banach space $X$. This improves a previously obtained characterization of leftsymmetric operators from $\ell _1^n$, $n\geq 2$, to a reflexive smooth Banach space~$X$.
DOI: 10.14232/actasm0204206
AMS Subject Classification
(1991): 47L05; 46B20
Keyword(s):
BirkhoffJames orthogonality,
leftsymmetric point,
rightsymmetric point,
supporting functional,
symmetric operator
received 20.4.2020, revised 18.5.2020, accepted 20.5.2020. (Registered under 420/2020.)
Abstract. In the present paper, we suggest new proofs of many known results about the {\it relative multifractal formalism}. We provide results even at points $q$ for which the relative multifractal Hausdorff and packing functions differ. We also give some examples of two measures where the multifractal functions are different and the Hausdorff dimension of the level sets of the $\nu $local Hölder exponent is given by the Legendre transform of the multifractal Hausdorff function and their packing dimension by the Legendre transform of the multifractal packing function.
DOI: 10.14232/actasm0208018
AMS Subject Classification
(1991): 28A78, 28A80
Keyword(s):
relative multifractal analysis,
multifractal formalism,
multifractal Hausdorff measure,
multifractal packing measure,
Hausdorff dimension,
packing dimension,
Moran measures
received 1.5.2020, revised 10.9.2020, accepted 21.9.2020. (Registered under 51/2020.)
Abstract. A result due to Williams, Stampfli and Fillmore shows that an essential isometry $T$ on a Hilbert space $\mathcal H$ is a compact perturbation of an isometry if and only if ind$(T)\le 0$. A recent result of S. Chavan yields an analogous characterization of essential spherical isometries $T=(T_1,\dots ,T_n)\in \mathcal {B}(\mathcal {H})^n$ with $\dim (\bigcap _{i=1}^n\ker (T_i))\le \dim (\bigcap _{i=1}^n\ker (T_i^*))$. In the present note we show that in dimension $n>1$ the result of Chavan holds without any condition on the dimensions of the joint kernels of $T$ and $T^*$.
DOI: 10.14232/actasm0207673
AMS Subject Classification
(1991): 47L05, 46B20
Keyword(s):
BirkhoffJames orthogonality,
leftsymmetric point,
rightsymmetric point,
supporting functional,
symmetric operator
received 17.5.2020, revised 28.5.2020, accepted 28.5.2020. (Registered under 517/2020.)
Abstract. In this paper we prove that if $T\in B({\mathcal H})$ is a pure class $p$$wA(s,t)$ operator ($0 < s, t, s + t =1$ and $0 < p \leq 1$) with dense range such that $0\notin \sigma _{p}(T)$, then $T$ has a nontrivial invariant subspace if and only if its second generalized Aluthge transformation $\tilde {T}(s,t)$ has a nontrivial invariant subspace. Further, we study some conditions for class $p$$wA(s,t)$ operators to have a nontrivial invariant subspace.
DOI: 10.14232/actasm0207758
AMS Subject Classification
(1991): 47A15, 47B20
Keyword(s):
$p$hyponormal operator,
class $p$$wA(s,
t)$ operator,
invariant subspaces
received 25.5.2020, revised 25.8.2020, accepted 6.9.2020. (Registered under 525/2020.)
ChiKwong Li,
YiuTung Poon

681696

Abstract. An operator system is a unital selfadjoint subspace of bounded linear operators. It is maximal if every positive linear map from it to another operator system is completely positive. In this paper, characterizations of maximal operator systems in terms of the joint numerical range are presented. New families of maximal operator systems are identified. These results admit formulations in terms of numerical range inclusion and dilation of operators that unify and extend earlier results on the topic.
DOI: 10.14232/actasm020871y
AMS Subject Classification
(1991): 47A20, 47A12, 15A60
Keyword(s):
numerical range,
dilation,
completely positive map,
maximal operator system
received 21.6.2020, revised 23.9.2020, accepted 4.10.2020. (Registered under 621/2020.)
Abstract. It is shown that every bounded, unital linear mapping that preserves elements of square zero from a C* of real rank zero and without tracial states into a Banach algebra is a Jordan homomorphism.
DOI: 10.14232/actasm0200677
AMS Subject Classification
(1991): 47B48, 46L05, 46L30, 16W10, 17C65
Keyword(s):
C*algebras,
commutators,
nilpotents,
tracial states,
Jordan homomorphisms,
spectrally bounded operators
received 17.8.2020, revised 25.8.2020, accepted 25.8.2020. (Registered under 817/2020.)
