ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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/actasm-020-363-7

AMS Subject Classification (1991): 06B05

Keyword(s): planar, atom-generated

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/actasm-019-426-3

AMS Subject Classification (1991): 20M10; 06A06, 06C15, 20M25, 16U99, 16W99

Keyword(s): Baer semigroup, D-semigroup, D-ring, 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.)

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 four-element 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 four-element generating sets of Part$(n)$. We also present a computer-assisted 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 four-element 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 four-generated. In particular, by the first part of this theorem, $\Part {n_1}\times \Part {n_2}$ is four-generated 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 $\Part {n}\times \Part {n+1}\times \dots \times \Part {3n-14}$ is four-generated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$-th the direct power of the direct product $\Part {n}\times \Part {n+1}\times \dots \times \Part {n+u-1}$ is four-generated 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 four-generated.

DOI: 10.14232/actasm-020-126-7

AMS Subject Classification (1991): 06B99, 06C10

Keyword(s): equivalence lattice, partition lattice, four-element 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.)

Abstract. We introduce the concept of non-positive 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 non-positive 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 non-positive operators.

DOI: 10.14232/actasm-019-554-z

AMS Subject Classification (1991): 47L05, 46B20

Keyword(s): Birkhoff--James orthogonality, semi-inner-product, non-positive 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, Leindler-type, 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 Hardy-type involving the $r^{th}$ order \emph {Generalized Riemann--Liouville}, \emph {Generalized Weyl}, \emph {Erdélyi--Kober}, \emph {$(k, \nu )$-Riemann--Liouville}, and \emph {$(k, \nu )$-Weyl fractional integrals} are established through a probabilistic approach. The Kullback--Leibler inequality has been applied to compute the best possible constant factor associated with each of these inequalities.

DOI: 10.14232/actasm-019-750-7

AMS Subject Classification (1991): 26D15; 26D10, 26A33

Keyword(s): Hardy's inequality for integrals, probability density function, Riemann--Liouville integral, Weyl integral, scale distribution, Kullback--Leibler inequality

received 19.12.2019, revised 16.4.2020, accepted 18.4.2020. (Registered under 250/2019.)

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/actasm-020-116-1

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/actasm-020-973-4

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.)

Abstract. In this paper we consider the generalized shift operator associated to the Laplace--Bessel differential operator $\Delta _{B}$ and investigate B-maximal commutators, commutators of B-Riesz potentials and commutators of B-singular 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/actasm-020-224-y

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 norm-estimates of complete analytic families of eigenvectors of their adjoints. In [gam], a cyclic power bounded operator is constructed which has the requested norm-estimates, 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 norm-estimates, 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 norm-estimates of complete analytic families of eigenvectors of~$T^*$.

DOI: 10.14232/actasm-020-283-1

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 Fuglede--Putnam theorem modulo the Hilbert--Schmidt class to almost normal $m$-tuples of operators with finite Hilbert--Schmidt modulus of quasitriangularity.

DOI: 10.14232/actasm-020-534-6

AMS Subject Classification (1991): 47B20; 47B10, 47B47

Keyword(s): Hilbert--Schmidt modulus of quasitriangularity, almost normal $m$-tuples of operators, almost hyponormal operators, generalized Fuglede--Putnam 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 two-sided 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 two-sided 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 two-sided generalized derivations. Some other related results are also discussed.

DOI: 10.14232/actasm-020-295-8

AMS Subject Classification (1991): 47B47; 47B48, 39B05

Keyword(s): two-sided 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.)

Abstract. In this paper, we investigate a class of problems with Neumann boundary data in Musielak--Orlicz--Sobolev 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/actasm-020-161-9

AMS Subject Classification (1991): 35A15, 58E05; 35J60

Keyword(s): Musielak--Sobolev 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.)

Abstract. We explore the relation between left-symmetry (right-symmetry) of elements in a real Banach space and right-symmetry (left-symmetry) 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, Kadets--Klee and strictly convex Banach space to any Banach space is symmetric if and only if it is the zero operator. We further characterize left-symmetric operators from $\ell _1^n$, $n\geq 2$, to any Banach space $X$. This improves a previously obtained characterization of left-symmetric operators from $\ell _1^n$, $n\geq 2$, to a reflexive smooth Banach space~$X$.

DOI: 10.14232/actasm-020-420-6

AMS Subject Classification (1991): 47L05; 46B20

Keyword(s): Birkhoff--James orthogonality, left-symmetric point, right-symmetric 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/actasm-020-801-8

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/actasm-020-767-3

AMS Subject Classification (1991): 47L05, 46B20

Keyword(s): Birkhoff--James orthogonality, left-symmetric point, right-symmetric 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 non-trivial invariant subspace if and only if its second generalized Aluthge transformation $\tilde {T}(s,t)$ has a non-trivial invariant subspace. Further, we study some conditions for class $p$-$wA(s,t)$ operators to have a non-trivial invariant subspace.

DOI: 10.14232/actasm-020-775-8

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.)

Abstract. An operator system is a unital self-adjoint 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/actasm-020-871-y

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/actasm-020-067-7

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.)