ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ \emph{represents the inclusion $Q\subseteq D$ by principal congruences} if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for \emph{every} subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be \emph{fully (A1)-representable}. We prove that every fully (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is \emph{fully chain-representable} in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another result of the present author, it follows that every fully (A1)-representable finite distributive lattice is ``fully representable'' even by principal congruences of \emph{finite lattices}. Finally, we prove that every \emph{chain-representable} inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.

DOI: 10.14232/actasm-017-538-7

AMS Subject Classification (1991): 06B10

Keyword(s): distributive lattice, principal lattice congruence, congruence lattice, chain-representability

Received June 1, 2017 and in final form February 8, 2018. (Registered under 38/2017.)

Abstract. Cross-connection theory propounded by Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^\theta $ of the full transformation semigroup $(\mathscr{T}_X,\cdot )$ for an arbitrary $\theta\in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^\theta = (\mathscr{T}_X,\ast )$ with the binary operation $\alpha\ast \beta = \alpha\cdot \theta\cdot \beta $ where $\alpha, \beta\in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part ${\msbm R}eg (\mathscr{T}_X^\theta )$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of ${\msbm R}eg (\mathscr{T}_X^\theta )$ and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation $\theta $. This leads us to a structure theorem for the semigroup and gives the representation of ${\msbm R}eg (\mathscr{T}_X^\theta )$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.

DOI: 10.14232/actasm-017-044-z

AMS Subject Classification (1991): 20M10, 20M17, 20M50

Keyword(s): regular semigroup, full transformation semigroup, cross-connections, normal category, variant

Received June 30, 2017, and in revised form February 12, 2018. (Registered under 44/2017.)

Abstract. In this paper we shall obtain that the nearring $C_0(V)$ of congruence preserving functions that are 0-preserving of a tame $N$-module $V$ of a nearring $N$ is finite when $N$ is finite. As a consequence, $C_0(V)$ of an expanded group $\langle V,+,F\rangle $ is finite when the nearring of 0-preserving polynomial functions $P_0(V)$ of $\langle V,+,F\rangle $ is finite. We then go on to obtain further consequences of this result.

DOI: 10.14232/actasm-017-299-9

AMS Subject Classification (1991): 16Y30; 08A40

Keyword(s): nearring, expanded group, tame module, polynomial functions, congruence preserving functions, endomorphism nearring

Received July 10, 2017 and in final form May 20, 2018. (Registered under 49/2017.)

Abstract. The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential-theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasi-units as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of the effect algebra, we conclude the same properties for quasi-units. Our second aim is to apply these results for nonnegative sesquilinear forms. Constructing an order-preserving bijection between operator and form segments, we provide a characterization of being extremal in the convexity sense, and we give a necessary and sufficient condition for the existence of the greatest lower bound of two forms. Closing the paper we revisit some statements by using the machinery developed by Hassi, Sebestyén, and de Snoo. It will turn out that quasi-units are exactly the closed elements with respect to the antitone Galois connection induced by parallel addition and subtraction.

DOI: 10.14232/actasm-017-088-z

AMS Subject Classification (1991): 47A07, 47B65

Keyword(s): quasi-unit, orthogonal projection, extreme points, Galois connection

Received December 30, 2017 and in final form June 29, 2018. (Registered under 88/2017.)

Abstract. We give short proofs of two descriptions given by Šemrl of order automorphisms of the effect algebra. This sheds new light on both formulas that look quite complicated. Our proofs rely on Molnár's characterization of order automorphisms of the cone of all positive operators.

DOI: 10.14232/actasm-018-008-7

AMS Subject Classification (1991): 47B49

Keyword(s): self-adjoint operator, operator interval, effect algebra, order isomorphism, operator monotone function

Received November 15, 2017, and in revised form January 13, 2018. (Registered under 8/2018.)

Abstract. We refine earlier results concerning the structure of strongly continuous one-parameter semigroups ($C_0$-SGR) of holomorphic Carathéodory isometries of the unit ball in infinite-dimensional reflexive TROs (ternary rings of operators) We achieve finite algebraic formulas for them in terms of joint boundary fixed points and Möbius charts.

DOI: 10.14232/actasm-018-761-3

AMS Subject Classification (1991): 47D03, 32H15, 46G20

Keyword(s): Carathéodory distance, isometry, fixed point, holomorphic map, $C_0$-semigroup, infinitesimal generator, JB*-triple, Möbius transformation, Cartan factor, ternary ring of operators (TRO)

Received January 16, 2018 and in final form April 26, 2018. (Registered under 11/2018.)

Abstract. We prove that any bijective map between the positive definite cones of von Neumann algebras which preserves a certain unitarily invariant norm of a particular weighted geometric mean of elements is essentially (up to two-sided multiplication by an invertible positive element) equal to the restriction of a Jordan *-isomorphism between the algebras.

DOI: 10.14232/actasm-018-514-x

AMS Subject Classification (1991): 47B49; 46L40, 47A64

Keyword(s): positive definite cone, operator means, geodesic correspondence, preservers

Received January 18, 2018 and in final form February 23, 2018. (Registered under 14/2018.)

Abstract. A linear relation, i.e., a multivalued operator $T$ from a Hilbert space $\sH $ to a Hilbert space $\sK $ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of $T$, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where $T_1$ is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.

DOI: 10.14232/actasm-018-757-0

AMS Subject Classification (1991): 4705, 47A06, 47A65; 46N30, 47N30

Keyword(s): regular relations, singular relations, (weak) Lebesgue type decompositions, uniqueness of decompositions, domination of relations and operators, closability

Received January 11, 2018 and in final form April 30, 2018. (Registered under 7/2018.)

Abstract. This paper projects another affine case study in the program of analyzing multiparameter a.e. convergence, based on the Sucheston's type convergence principles. An affine semigroup is considered as a natural extension of strongly continuous semigroups of linear operators on $L_{p}$ spaces. We prove some affine extensions of multiparameter martingale theorems, multiparameter ergodic theorems, and multiparameter ergodic theorems for the so-called nonlinear sums. Moreover, an affine (nonlinear) generalization is given of Berkson--Bourgain--Gillespie's theorem concerning the connection between the ergodic Hilbert transform and the ergodic theorem for power-bounded invertible linear operators on $L_{p}$ ($1< p< \infty $) spaces. In addition, the random ergodic Hilbert transforms will be established. We improve the local ergodic theorem of McGrath concerning strongly continuous $m$-parameter semigroups of positive linear operators in a more general affine setting. We shall also show that the Sucheston convergence principle is also very effective even in yielding a multiparameter generalization of Starr's theorem.The final section includes some examples.

DOI: 10.14232/actasm-016-510-0

AMS Subject Classification (1991): 47A35, 40H05; 40G10

Keyword(s): affine semigroup, compound semigroup, ergodic Hilbert transform, random ergodic Hilbert transform, Cotlar's theorem, Berkson-Bourgain-Gillespie's theorem, Sucheston's type convergence principle, Orlicz class, multiparameter martingale theorem, nonlinear sum, ergodic theorem for affine semigroups, Abelian ergodic theorem for affine semigroups, Starr's theorem

Received February 15, 2016 and in final form February 26, 2018. (Registered under 10/2016.)

Abstract. Property $(UW_{\Pi })$ for a bounded linear operator $T\in L(X)$ on a Banach space $X$ is a variant of Browder's theorem, and means that the points $\lambda $ of the approximate point spectrum for which $\lambda I-T$ is upper semi-Weyl are exactly the spectral points $\lambda $ such that $\lambda I-T$ is Drazin invertible. In this paper we investigate this property, and we give several characterizations of it by using typical tools from local spectral theory. We also relate this property with some other variants of Browder's theorem (or Weyl's theorem).

DOI: 10.14232/actasm-016-303-5

AMS Subject Classification (1991): 47A53, 47A10, 47A11

Keyword(s): property $(UW {\scriptstyle\Pi })$, SVEP

Received September 26, 2016 and in final form May 15, 2018. (Registered under 53/2016.)

Abstract. In this paper, we study spectral properties of class $p$-$wA(s,t)$ operators with $0< p\leq1$ and $0< s,t,s+t\leq1$. We show that Weyl's theorem and Putnam's inequality hold for class $p$-$wA(s,t)$ operators.

DOI: 10.14232/actasm-017-020-y

AMS Subject Classification (1991): 47A10, 47A20, 47B20

Keyword(s): class $p$-$wA(s, t)$, normaloid, reguloid, Weyl's theorem, Putnam's inequality

Received March 28, 2017, and in revised form November 24, 2017. (Registered under 20/2017.)

Abstract. Let $G$ denote the group ${\msbm R}$ or ${\msbm T}$, let $\iota $ denote the identity element of $G$, and let $s\in{{\msbm N}}$ be given. Then, a \emph{difference of order} $s$ is a function $f\in L^2(G)$ for which there are $a\in G$ and $g \in L^2({G})$ such that $f= (\delta_{\iota }-\delta_{a})^s\ast g$. Let ${{\cal D}}_s(L^2(G))$ be the vector space of functions that are finite sums of differences of order $s$. It is known that if $f\in L^2({{\msbm R}})$, $f\in{{\cal D}}_s(L^2({{\msbm R}}))$ if and only if $\int_{-\infty }^{\infty }|{\widehat f}(x)|^2|x|^{-2s}dx< \infty $. Also, if $f\in L^2({{\msbm T}})$, $f\in{{\cal D}}_s(L^2({{\msbm T}}))$ if and only if ${\widehat f}(0)=0$. Consequently, ${{\cal D}}_s(L^2(G))$ is a Hilbert space in a (possibly) weighted $L^2$-norm. It is known that every function in ${{\cal D}}_s(L^2(G))$ is a sum of $2s+1$ differences of order $s$. However, there are functions in ${{\cal D}}_s(L^2({{\msbm R}}))$ that are not a sum of $2s$ differences of order $s$, and we call the latter type of fact a \emph{sharpness result}. In ${{\cal D}}_1(L^2({{\msbm T}}))$, it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces ${{\cal D}}_s(L^2({{\msbm T}}))$ that complement the results known for ${{\msbm R}}$, but also to present new results in ${{\cal D}}_s(L^2({{\msbm T}}))$ that do not correspond to known results in ${{\cal D}}_s(L^2({{\msbm R}}))$. Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics.

DOI: 10.14232/actasm-017-522-y

AMS Subject Classification (1991): 42A16, 42A38

Keyword(s): Fourier transform, finite differences, subspaces of $L^2({{\msbm T}})$, combinatorial inequalities, badly approximable vectors in ${{\msbm R}}^n$, sharpness results, Sobolev spaces

Received April 7, 2017 and in final form May 22, 2018. (Registered under 22/2017.)

Abstract. In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.

DOI: 10.14232/actasm-017-773-6

AMS Subject Classification (1991): 47A30, 47A05, 47B15

Keyword(s): closed range operator, Moore-Penrose inverse, group inverse, self-adjoint operator, unitary operator, normal operator, partial isometry operator, isometry operator, operator inequality

Received April 8, 2017, and in final form November 25, 2017. (Registered under 23/2017.)

Abstract. In this paper, we discuss the conditions under which composition operators and weighted composition operators become quasi-$A(n)$ operators, quasi-$*$-$A(n)$ operators, $k$-quasi-$A(n)$ operators and $k$-quasi-$*$-$A(n)$ operators in terms of the Radon--Nikodym derivative $h_n$.

DOI: 10.14232/actasm-017-032-5

AMS Subject Classification (1991): 47B33, 47B20; 46C05

Keyword(s): composition operators, weighted composition operators, quasi-$A(n)$ operators, quasi-$*$-$A(n)$ operators, $k$-quasi-$A(n)$ operators and $k$-quasi-$*$-$A(n)$ operators

Received May 23, 2017, and in revised form November 27, 2017. (Registered under 32/2017.)

Abstract. In this paper we study the problem of the boundedness and compactness of the Toeplitz operator $T_{\varphi }$ on $L_{a}^{2}(\Omega )$, where $\Omega $ is a multiply-connected domain and $\varphi $ is not bounded. We find a necessary and sufficient condition when the symbol is $\mathcal{BMO}.$ For this class we also show that the vanishing at the boundary of the Berezin transform is a necessary and sufficient condition for compactness. The same characterization is shown to hold when we analyze operators which are finite sums of finite products of Toeplitz operators with unbounded symbols.

DOI: 10.14232/actasm-017-283-0

AMS Subject Classification (1991): 47B35; 47B38

Keyword(s): Bergman space, Toeplitz operator, Berezin transform

Received May 29, 2017 and in final form September 2, 2018. (Registered under 33/2017.)

Abstract. In this paper we consider the nonlinear third-order coupled system composed by the differential equations \[\left\{{ -u^{\prime\prime\prime}(t)=f\left(t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime}(t)\right), \atop -v^{\prime\prime\prime}(t) =h\left( t,u(t),u^{\prime }(t),u^{\prime\prime }(t),v(t),v^{\prime }(t),v^{\prime\prime }(t)\right ),}\right. \] with $f,h\colon[0,1] \times\mathbb {R}^{6}\rightarrow\mathbb {R}$ continuous functions, and the boundary conditions \[ \left\{{ u(0)=u^{\prime }(0) =u^{\prime}(1) =0, \atop v(0)=v^{\prime}(0) =v^{\prime}(1) =0. }\right.\] We remark that the nonlinearities can depend on all derivatives of both unknown functions, which is new in the literature, as far as we know. This is due to an adequate auxiliary integral problem with a truncature, applying lower and upper solutions method with bounded perturbations. The main theorem is an existence and localization result, which provides some qualitative data on the system solution, such as, sign, variation, bounds, etc., as it can be seen in the example.

DOI: 10.14232/actasm-017-785-0

AMS Subject Classification (1991): 34B15, 34B27, 34L30

Keyword(s): coupled systems, Green functions, Nagumo-type condition, coupled lower and upper solutions

Received May 29, 2017 and in final form June 10, 2018. (Registered under 35/2017.)

Abstract. Let $A$ be a complex commutative semisimple Banach algebra and let $T$ be a power bounded multiplier of $A$. This paper is concerned with finding necessary and sufficient conditions for the convergence of the sequence $\left\{ T^{n}a\right\} $ $( a\in A) $ in $A.$ Some related problems are also discussed.

DOI: 10.14232/actasm-017-291-5

AMS Subject Classification (1991): 46HXX, 43A20, 43A22

Keyword(s): commutative Banach algebra, multiplier, set of synthesis, convergence

Received June 23, 2017, and in revised form December 6, 2017. (Registered under 41/2017.)

Abstract. We consider the direct sum of a Lie algebroid structure with its dual space and equip this bigger space with a contact form called the generalized almost contact structure and characterize these in the sense of contact morphisms. Attaching an almost generalized complex structure, we recover properties as normality conditions in a direct way and study some aspects of metrical kind of this spaces.

DOI: 10.14232/actasm-017-777-8

AMS Subject Classification (1991): 53D10

Keyword(s): generalized almost contact structure, Lie algebroid, metric structure, normality conditions

Received April 24, 2017, and in revised form September 21, 2017. (Registered under 27/2017.)

Abstract. We obtain a characterization for probability measures on a locally compact Abelian group $X$ based on linear forms of $Q$-independent random elements taking values in $X$ generalizing the earlier work of the author in [12].

DOI: 10.14232/actasm-017-530-3

AMS Subject Classification (1991): 60B15, 62E10

Keyword(s): $Q$-independence, characterization, locally compact Abelian group

Received April 27, 2017 and in final form April 18, 2018. (Registered under 30/2017.)