ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. -

DOI: 10.14232/actasm-019-028-x

AMS Subject Classification (1991): 01A70

Keyword(s): András Krámli

Received March 18, 2019. (Registered under 28/2019.)

Abstract. This versatile topic goes back to the inventions of Gauss, Markov, and Gibbs, whose ideas are incorporated in graphical models and regression graphs. Later, the geneticist S. Wright (1923--1934) and the philosopher and computer scientist J. Pearl (1986--1987) developed the tools, but their notation is too complicated to formulate the mathematical background. Here we mainly follow the up-to-date discussion of statisticians S. Lauritzen and N. Wermuth, and try to juxtapose the directed--undirected and discrete--continuous cases.

DOI: 10.14232/actasm-018-331-4

AMS Subject Classification (1991): 62H99, 68T30

Keyword(s): graphical models, log-linear models, Markov random fields, covariance selection, recursive linear regressions

Received September 10, 2018. (Registered under 81/2018.)

Abstract. In this paper, we define $z$-ideals in bounded lattices. A separation theorem for the existence of prime $z$-ideals is proved in distributive lattices. As a consequence, we prove that every $z$-ideal is the intersection of some prime $z$-ideals. Lastly, we prove a characterization of dually semi-complemented lattices.

DOI: 10.14232/actasm-016-012-2

AMS Subject Classification (1991): 06B10, 06D75

Keyword(s): $z$-ideals, Baer ideal, $0$-ideal, closed ideal, minimal prime ideal, maximal ideal, dense ideal, dually semi-complemented lattice

Received February 22, 2016 and in final form September 3, 2018. (Registered under 12/2016.)

Abstract. Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representation of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.

DOI: 10.14232/actasm-017-060-9

AMS Subject Classification (1991): 06B10, 06A06

Keyword(s): finite lattice, principal congruence, ordered set

Received October 10, 2017, and in revised form November 26, 2017. (Registered under 60/2017.)

Abstract. The goal of the paper is to transfer some order properties of star-ordered Rickart *-rings to Baer semigroups. A focal Baer semigroup $S$ is a semigroup with 0 expanded by two unary idempotent-valued operations, $\lt $ and $\rt $, such that the left (right) ideal generated by $x\lt $ (resp., $x\rt $) is the left (resp., right) annihilator of $x$. $S$ is said to be symmetric if the ranges of the two operations coincide and $p\lt = p\rt $ for every $p$ from the common range $P$. Such a semigroup is shown to be $P$-semiabundant. If it is also Lawson reduced, then $P$ is an orthomodular lattice under the standard order of idempotents, and a restricted version of Drazin star partial order can be defined on $S$. The lattice structure of $S$ under this order is shown to be similar, in several respects, to that of star-ordered Rickart *-rings.

DOI: 10.14232/actasm-017-319-5

AMS Subject Classification (1991): 20M25, 20M10, 06F99

Keyword(s): Baer semigroup, closed idempotent, orthomodular lattice, Rickart ring, Rickart *-ring, star order

Received November 6, 2017 and in final form April 14, 2018. (Registered under 69/2017.)

Abstract. A subset $B$ of a group $G$ is called a {\em difference basis} of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the {\em difference size} of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\tfrac{\Delta[G]}{\sqrt{|G|}}$ is called the {\em difference characteristic} of $G$. Using properties of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge11$, any finite Abelian $p$-group $G$ has difference characteristic $\eth[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup _{k\in\IN }\eth[C_{p^k}]< \sqrt{2}\cdot\frac {\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality less than $96$.

DOI: 10.14232/actasm-017-586-x

AMS Subject Classification (1991): 05B10, 05E15, 16L99, 16Z99, 20D60, 20K01

Keyword(s): finite group, Abelian group, difference basis, difference characteristic

Received December 28, 2017 and in final form May 20, 2018. (Registered under 86/2017.)

Abstract. -

DOI: 10.14232/actasm-018-279-1

AMS Subject Classification (1991): 11K65, 11N37, 11N64

Keyword(s): multiplicative functions, completely multiplicative functions

Received March 17, 2018 and in final form June 18, 2018. (Registered under 29/2018.)

Abstract. In this paper a concise but systematic account of the relations between Shabat polynomials and plane trees is given, including the famous equivalence theorem for plane trees attributed to Riemann--Belyi--Grothendieck--Shabat. Furthermore, some analytical features of Shabat polynomials and metric properties of Shabat trees are discussed.

DOI: 10.14232/actasm-017-821-6

AMS Subject Classification (1991): 05C05, 05C10, 30C10, 30C85, 30F99

Keyword(s): Shabat polynomials, plane trees, conformal mapping

Received November 11, 2017 and in final form February 7, 2018. (Registered under 71/2017.)

Abstract. In 2013, Jiménez--Melado and Llorens--Fuster proved that the renorming of $\ell ^2$, $|x|=\max\{\|x\|_2,p(x)\}$, where $p$ is a seminorm on $\ell ^2$ satisfying certain conditions, has the weak fixed point property. In this paper, we generalize this result for a Banach space having normal structure and Schauder basis. From this, we derive that every Banach space having normal structure and Schauder basis has an equivalent renorming that lacks asymptotic normal structure but has the weak fixed point property.

DOI: 10.14232/actasm-017-339-4

AMS Subject Classification (1991): 47H09, 47H10, 46B20

Keyword(s): nonexpansive mappings, weak fixed point property

Received December 30, 2017 and in final form April 28, 2018. (Registered under 89/2017.)

Abstract. Let $F_{\alpha,\beta }(x)=\beta E_{\beta } (x^{\beta })-\alpha E_{\alpha }(x^{\alpha })$, where $E_{\alpha }$ denotes the Mittag--Leffler function. We prove that if $\alpha, \beta\in (0,1]$, then $F_{\alpha,\beta }$ is completely monotonic on $(0,\infty )$ if and only if $\alpha\leq \beta $. This extends a result of T. Simon, who proved in 2015 that $F_{\alpha,1}$ is completely monotonic on $(0,\infty )$ if $\alpha\in (0,1]$. Moreover, we apply our monotonicity theorem to obtain some functional inequalities involving $F_{\alpha,\beta }$.

DOI: 10.14232/actasm-018-263-5

AMS Subject Classification (1991): 26A48, 33E12

Keyword(s): Mittag--Leffler function, completely monotonic, functional inequalities

Received January 17, 2018. (Registered under 13/2018.)

Abstract. A modulus for the Bishop--Phelps--Bollobás property for operators (BPBpo) is formerly introduced in the literature that characterizes whether a pair of Banach spaces enjoys the BPBpo and that also provides the best possible value of the BPBpo for a given pair of Banach spaces that enjoys it. We use it also to show that the BPBpo is hereditary to a class of complemented subspaces that strictly includes the $M$-summands. We also provide an equivalent reformulation of this modulus. Finally, the continuity properties of this modulus are also discussed.

DOI: 10.14232/actasm-018-765-5

AMS Subject Classification (1991): 47A05, 46B20

Keyword(s): Bishop--Phelps--Bollobás modulus, continuous linear operator, complemented subspace

Received January 19, 2018 and in final form April 24, 2018. (Registered under 15/2018.)

Abstract. We study iterates, $f^n$, of a fixed-point free compact holomorphic map $f\colon B\rightarrow B$ where $B$ is the open unit ball of any $JB^*$-triple of finite rank. These spaces include $L(H,K)$, $H,K$ Hilbert, dim$(H)$ arbitrary, dim$(K)< \infty $, or any classical Cartan factor or $C^*$-algebra of finite rank. Apart from the Hilbert ball, the sequence of iterates $(f^n)_n$ does not generally converge (locally uniformly on $B$) and little is known of accumulation points. We present a short proof of a Wolff theorem for $B$ and establish key properties of the resulting $f$-invariant subdomains. We define a concept of closed convex holomorphic hull, $\mathop{\rm Ch}(x)$, for $x \in\partial B$ and prove the following. There is a unique tripotent $u$ in $\partial B$ such that all constant subsequential limits of $(f^n)_n$ lie in $\mathop{\rm Ch}(u)$. As a consequence we also get a short proof of the classical Hilbert ball results.

DOI: 10.14232/actasm-018-518-z

AMS Subject Classification (1991): 47H10, 32M15; 32H50, 58C10

Keyword(s): iteration, bounded symmetric domain, Denjoy--Wolff theorem

Received February 8, 2018 and in final form March 8, 2018. (Registered under 18/2018.)

Abstract. In this paper we discuss the multipliers between range spaces of co-analytic Toeplitz operators.

DOI: 10.14232/actasm-018-769-7

AMS Subject Classification (1991): 30J05, 30H10, 46E22

Keyword(s): Hardy spaces, inner functions, model spaces, multipliers, de Branges--Rovnyak spaces

Received February 8, 2018 and in final form March 21, 2018. (Registered under 19/2018.)

Abstract. We extend in this paper the notion of Cauchy dual to operators with closed range. We then give several useful properties of Cauchy duals extending the case of left-invertible operators. As a consequence, we show that a weak concavity concept of an operator induces a corresponding weak hyponormality of its Cauchy dual.

DOI: 10.14232/actasm-018-020-2

AMS Subject Classification (1991): 47B20

Keyword(s): closed range, hyperexpansive operators, Cauchy dual, Moore--Penrose inverse

Received February 10, 2018 and in final form June 28, 2018. (Registered under 20/2018.)

Abstract. We characterize meromorphic function fields closed by partial derivatives in $n$ variables.

DOI: 10.14232/actasm-018-530-7

AMS Subject Classification (1991): 32A20; 32W50

Keyword(s): algebraic addition theorem, degenerate Abelian functions, Briot--Bouquet type partial differential equations

Received March 22, 2018 and in final form March 6, 2019. (Registered under 30/2018.)

Abstract. In this paper an algorithm based on polyphase matrices for constructing a pair of orthogonal wavelet frames is suggested, and a general form for all orthogonal tight wavelet frames on local fields of positive characteristic is described. Moreover, an investigation regarding their properties by means of the Fourier transform is carried out.

DOI: 10.14232/actasm-018-785-4

AMS Subject Classification (1991): 42C40; 42C15. 43A70, 11S85

Keyword(s): wavelet frame, orthogonality, framelet symbol, polyphase matrix, extension principle, Fourier transform, local field

Received April 9, 2018. (Registered under 35/2018.)

Abstract. We reprove a recent result of Z. Sebestyén and Zs. Tarcsay [9]: If $T^*T$ and $TT^*$ are self-adjoint, then $T$ is closed.

DOI: 10.14232/actasm-018-295-y

AMS Subject Classification (1991): 47B25; 47B65

Keyword(s): von Neumann's theorem

Received May 10, 2018. (Registered under 45/2018.)

Abstract. M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2({\msbm R}_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space ${\cal H}$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with ${\cal H}$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertible, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t$.

DOI: 10.14232/actasm-018-546-3

AMS Subject Classification (1991): 47B20, 47B37; 47A10, 46E22

Keyword(s): weighted translation semigroup, completely alternating, completely hyperexpansive, analytic, operator valued weighted shift

Received May 10, 2018 and in final form January 31, 2019. (Registered under 46/2018.)

Abstract. The question if every polynomially bounded operator is similar to a contraction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In [gam16], cyclic polynomially bounded operators which are not similar to contractions were constructed. The construction was based on a perturbation of a sequence of finite-dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, studied earlier by Pisier. In this paper, a cyclic polynomially bounded operator $T_0$ is constructed so that $T_0$ is not similar to a contraction and $\omega_a(T_0)={\msbm O}$. Here $\omega_a(z)=\exp(a\frac{z+1}{z-1})$, $z\in{\msbm D}$, $a>0$, and ${\msbm D}$ is the open unit disk. To obtain such a $T_0$, a slight modification of the construction from [gam16] is needed.

DOI: 10.14232/actasm-018-797-y

AMS Subject Classification (1991): 47A60; 47A65, 47A16, 47A20

Keyword(s): polynomially bounded operator, similarity, contraction, unilateral shift, isometry, $C_0$-contraction, $C_0$-operator

Received May 15, 2018 and in final form February 6, 2019. (Registered under 47/2018.)

Abstract. In this paper we study a conjugation on a Banach space $\x $ and show properties of operators concerning conjugation $C$ and show spectral properties of such operators. Next we show spectral properties of an $(m,C)$-symmetry (isometry) operator $T$ on a complex Banach space $\x $. We prove that, for a $C$-doubly commuting pair $(T,S)$, if $T$ is an $(m,C)$-symmetry (isometry) and $S$ is an $(n,C)$-symmetry (isometry), then $T + S$ and $TS$ are $(m + n - 1,C)$-symmetries (isometries).

DOI: 10.14232/actasm-018-801-y

AMS Subject Classification (1991): 47A05; 47B25, 47B99

Keyword(s): Banach space, linear operator, conjugation, spectrum

Received June 4, 2018 and in final form August 31, 2018. (Registered under 51/2018.)

Abstract. Let $K_0$ be a compact convex subset of the plane $\preal $, and assume that whenever $K_1\subseteq\preal $ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-Tóth. A theorem of L. Fejes-Tóth from 1967 states that the assumption above holds for $K_0$ if and only if $K_0$ is a disk. In a paper that appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-Tóth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive than the old one. Therefore, L. Fejes-Tóth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursors in combinatorics and, mainly, in lattice theory.

DOI: 10.14232/actasm-018-522-2

AMS Subject Classification (1991): 52C99; 52A01, 06C10

Keyword(s): compact convex set, circle, characterization of circles, disk, crossing, abstract convex geometry, Adaricheva-Bolat property, boundary of a compact convex set, supporting line, slide-turning, lattice

Received February 18, 2018. (Registered under 22/2018.)