ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Two years ago, I characterized the order $\Princl L$ of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and $\gf $ is a \zo homomorphism of $K$ into $L$, then there is a natural isotone \zo map $\Princl\gf $ from $\Princl K$ into $\Princl L$. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map $\gy $ of $P$ into $Q$, we represent $P$ and $Q$ as $\Princl K$ and $\Princl L$ for bounded lattices $K$ and $L$ with a \zo homomorphism $\gf $ of $K$ into $L$, so that $\gy $ is represented as $\Princl\gf $.

DOI: 10.14232/actasm-015-056-y

AMS Subject Classification (1991): 06B10

Keyword(s): bounded lattice, congruence, principal, order

Received July 20, 2015, and in revised form September 18, 2015. (Registered under 56/2015.)

Abstract. We prove that every finite lattice $L$ can be embedded in a three-generated \emph{finite} lattice $K$. We also prove that every \emph{algebraic} lattice with accessible cardinality is a \emph{complete} sublattice of an appropriate \emph{algebraic} lattice $K$ such that $K$ is completely generated by three elements. Note that ZFC has a model in which all cardinal numbers are accessible. Our results strengthen P. Crawley and R. A. Dean's 1959 results by adding finiteness, algebraicity, and completeness.

DOI: 10.14232/actasm-015-586-2

AMS Subject Classification (1991): 06B99, 06B15

Keyword(s): three-generated lattice, equivalence lattice, partition lattice, complete lattice embedding, inaccessible cardinal

Received December 12, 2015, and in final form September 19, 2016. (Registered under 86/2015.)

Abstract. Orthomodular lattices were introduced to get an algebraic description of the propositional logic of quantum mechanics. In this paper, we set up axiomatization of this logic as a Hilbert style implicational logical system $\LOM $, i.e., we present a set of axioms and derivation rules formulated in the signature $\{\to,0\}$. The other logical operations $\vee, \wedge, \neg $ are expressed in terms of implication (which is the so-called Dishkant implication) and falsum. We further show that the system $\LOM $ is algebraizable in the sense of Blok and Pigozzi, and that orthomodular lattices provide an equivalent algebraic semantics for it.

DOI: 10.14232/actasm-015-813-6

AMS Subject Classification (1991): 06C15, 03G12

Keyword(s): algebraizable logic, axiom system, derivation rule, Dishkant implication, logic of quantum mechanics, orthomodular implication algebra, orthomodular lattice, semi-orthomodular lattice, weak BCK-algebra

Received August 16, 2015, and in final form January 16, 2016. (Registered under 63/2015.)

Abstract. Faithful representations of regular $\ast $-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between classes of spaces and classes of representable structures is analyzed; for a class $\mathcal{S}$ of spaces which is closed under ultraproducts and non-degenerate finite-dimensional subspaces, the class of representable structures is shown to be closed under complemented [regular] subalgebras, homomorphic images, and ultraproducts. Moreover, this class is generated by its members which are isomorphic to subspace lattices with involution [endomorphism $\ast $-rings, respectively] of finite-dimensional spaces from $\mathcal{S}$. Under natural restrictions, this result is refined to a $1$-$1$-correspondence between the two types of classes.

DOI: 10.14232/actasm-015-283-5

AMS Subject Classification (1991): 06C20, 16E50, 16W10, 51D25

Keyword(s): sesquilinear space, endomorphism ring, regular ring with involution, lattice of subspaces, complemented modular lattice with involution, representation, semivariety, variety

Received May 8, 2015, and in revised form May 22, 2016. (Registered under 33/2015.)

Abstract. We define the analogue of $q$-additivity for the canonical number systems in the Gaussian ring of integers. We characterize all those functions $f\colon\zz [i]\to\cc $ which are $\theta =-A+i$-additive and completely multiplicative (Theorem 1). For $\theta =-1+i$ we give all functions which are $\theta $- and $\overline{\theta }$-additive (Theorem 2).

DOI: 10.14232/actasm-015-052-9

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

Keyword(s): completely additive, completely multiplicative, $q$-additive function, Gaussian integers, canonical number system

Received July 4, 2015, and in revised form June 25, 2016. (Registered under 52/2015.)

Abstract. In this paper, we investigate the Zassenhaus conjecture for exceptional groups of Lie type $G_2(q)$ for $q=\{3,4\}$. Consequently, we prove that the Prime graph question is true for these groups.

DOI: 10.14232/actasm-015-048-6

AMS Subject Classification (1991): 16S34, 20C05

Keyword(s): Zassenhaus Conjecture, torsion unit, partial augmentation, integral group ring

Received July 1, 2015, and in revised form November 2, 2015. (Registered under 48/2015.)

Abstract. The goal of this paper is to study the stabilization of chaos in dynamical systems by adding nonlinear feedback. We analyze what happens when two control parameters, the gain and the parameter memory, are considered. It is shown that the introduction of the additional control of the memory parameter does not extend the class of admissible maps. It appears, however, that a stabilizing control may use a variety of time shifts. In this case, one can change the nature of the decay of a chaotic regime, making it smoother, which may be of significance in the management of biological, economical and medical systems.

DOI: 10.14232/actasm-014-522-z

AMS Subject Classification (1991): 42A05, 39A30

Keyword(s): trigonometric polynomials, dynamical systems, optimal control of chaos

Received March 26, 2014, and in revised form August 15, 2016. (Registered under 22/2014.)

Abstract. We extend the definition of Jamison sequences in the context of topological abelian groups. We then study these sequences when the group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux [BadeaGrivaux2] about Jamison sequences of integers. In particular, we prove that the sequence consisting in all the elements of the group is a Jamison sequence. In the opposite, a sequence which generates a subgroup of infinite index in the group is never a Jamison sequence. We also generalize a result of Nikolskii by showing that the growth of the norms of a representation is influenced by the Haar measure of its unimodular point spectrum.

DOI: 10.14232/actasm-015-020-3

AMS Subject Classification (1991): 47A10, 37C85, 43A40, 28C10

Keyword(s): unimodular point spectrum, Jamison sequences, discrete abelian groups, characters and dual group, Haar measures

Received February 28, 2015, and in final form December 20, 2015. (Registered under 20/2015.)

Abstract. We prove that every infinite, discrete abelian group admits a pair of $I_{0}$ sets whose union is not $I_{0}$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

DOI: 10.14232/actasm-016-518-4

AMS Subject Classification (1991): 43A46

Keyword(s): Sidon set, $I_{0}$ set, Kronecker set

Received March 24, 2016, and in revised form August 5, 2016. (Registered under 18/2016.)

Abstract. Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ be the $\cle $-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well-known Beurling--Lax--Halmos theorem states that every shift invariant subspace of $H^2_{\cle }(\D )$ other than $\{0\}$ has the form $\Theta H^2_{\cle_*}(\D )$, where $\Theta $ is an operator-valued inner multiplier in $H^\infty_{B(\cle_*;\mathcal{E})}(\mathbb{D})$ for some Hilbert space $\cle_*$. In this paper we identify $H^2(\mathbb{D}^n)$ with the $H^2(\mathbb{D}^{n-1})$-valued Hardy space $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ and classify all such inner multipliers $\Theta\in H^\infty_{\mathcal{B}(H^2(\mathbb{D}^{n-1}))}(\mathbb{D})$ for which $\Theta H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ is a Rudin type invariant subspace of $H^2(\mathbb{D}^n)$.

DOI: 10.14232/actasm-015-773-y

AMS Subject Classification (1991): 47A13, 47A15, 46E20, 46M05

Keyword(s): Hardy space, inner sequence, operator-valued inner function, invariant subspace, unitary equivalence

Received March 17, 2015. (Registered under 23/2015.)

Abstract. Our study is in the set ${{\cal S}}(H)$ of all semiclosed operators in a Hilbert space $H$. We show that the set ${{\cal S}}_{sa}(H)$ of all selfadjoint operators is relatively open in the set ${{\cal S}}_{sym}(H)$ of all semiclosed symmetric operators. We calculate the value of a radius of minus-Laplacian $-\Delta $. As a topological approach, we show the selfadjointness of the Schrödinger operator with a Kato--Rellich potential.

DOI: 10.14232/actasm-015-044-4

AMS Subject Classification (1991): 47A65, 47A05

Keyword(s): De Branges space, semiclosed symmetric operators, selfadjoint operators, the $q$-metric

Received June 18, 2015, and in revised form January 17, 2016. (Registered under 44/2015.)

Abstract. In [17], M. Uchiyama gave necessary and sufficient conditions for contractions to be quasiaffine transforms, quasisimilar, or similar to unilateral shifts of finite multiplicity in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In this paper, the result for contractions to be quasiaffine transforms of unilateral shifts is generalized to power bounded operators. It is shown that the result for contractions to be quasisimilar or similar to unilateral shifts can't be extended to power bounded operators: a counterexample is given. No curvature of the holomorphic vector bundle generated by eigenvectors of operators is computed.

DOI: 10.14232/actasm-015-060-1

AMS Subject Classification (1991): 47A05, 47B99, 47B32, 30H10

Keyword(s): power bounded operator, unilateral shift, quasiaffine transform, quasisimilarity, contraction, analytic family of eigenvalues, similarity

Received August 11, 2015, and in revised form September 6, 2016. (Registered under 60/2015.)

Abstract. In this note we give conditions for the invertibility of a bounded linear operator $T$ defined on a Banach space $X$ such that $X$ decomposes into a (non direct) sum of two closed $T$-invariant subspaces.

DOI: 10.14232/actasm-015-534-x

AMS Subject Classification (1991): 47A10, 47A15,47A05, 15A29

Keyword(s): invariant subspace, spectrum of an operator

Received May 13, 2015, and in final form April 6, 2016. (Registered under 34/2015.)

Abstract. We consider the weighted shifts of infinite multiplicity with quasi-affine weights. We obtain a necessary and sufficient condition for the Deddens algebra associated to such a shift to have a nontrivial invariant subspace, or to be dense. Our technique is based on the study of compressions of operators in the Deddens algebra to some subspaces, and the relations between such compressions.

DOI: 10.14232/actasm-014-789-3

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

Keyword(s): Deddens algebra, weighted shift, invariant subspace

Received May 6, 2014, and in final form June 23, 2016. (Registered under 39/2014.)

Abstract. The question if a 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 the paper, cyclic polynomially bounded operators which are not similar to contractions and are quasisimilar to $C_0$-contractions or to isometries are constructed. The construction is based on a perturbation of the sequence of finite dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, constructed by Pisier.

DOI: 10.14232/actasm-016-016-4

AMS Subject Classification (1991): 47A65, 47A60, 47A16, 47A20, 47A55

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

Received March 10, 2016, and in revised form August 15, 2016. (Registered under 16/2016.)

Abstract. An order relation for contractions on a Hilbert space can be introduced by stating that $A\Prec B$ if and only if $A$ is unitarily equivalent to the restriction of $B$ to an invariant subspace. We discuss the equivalence classes associated to this relation, and identify cases in which they coincide with classes of unitary equivalence. The results extend those for completely nonunitary partial isometries obtained by Garcia, Martin, and Ross.

DOI: 10.14232/actasm-015-068-5

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

Keyword(s): contractions on Hilbert space, preoder relations, unitary equivalence

Received September 6, 2015, and in final form May 23, 2016. (Registered under 68/2015.)

Abstract. Let $T=U|T|$ be the polar decomposition of a bounded linear operator on a complex Hilbert space. $T $ is called a class $p$-$wA(s,t)$ operator if $(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{tp}{s+t}}\geq |T^{*}|^{2tp}$ and $(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{sp}{s+t}}\leq |T|^{2sp}$ where $0 < s, t $ and $0 < p \leq1$. We investigate spectral properties of a class $p$-$wA(s,t)$ operator $T$. We prove that if $s + t = 1$ and $\lambda\not = 0 $ is an isolated point of the spectrum $\sigma(T)$ then the Riesz idempotent $E$ with respect to $\lambda $ is self-adjoint and $ {\rm ran } E = \ker(T- \lambda ) = \ker((T-\lambda )^{*})$. Also, we prove relating results.

DOI: 10.14232/actasm-015-275-0

AMS Subject Classification (1991): 47B20

Keyword(s): class $A$ operator, class $p$-$wA(s, t)$ operator, Riesz idempotent

Received March 27, 2015, and in revised form May 6, 2015. (Registered under 25/2015.)

Abstract. It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_{\alpha }(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.

DOI: 10.14232/actasm-015-801-z

AMS Subject Classification (1991): 47B33

Keyword(s): composition operator, adjoint, weighted Hardy space

Received July 3, 2015, and in revised form August 31, 2015. (Registered under 51/2015.)

Abstract. An affirmative answer to the question in the title is proved in the plane by showing that any real analytic multicurve can be uniquely determined from its generalized visual angles given at every point of an open ring around the multicurve.

DOI: 10.14232/actasm-015-299-1

AMS Subject Classification (1991): 0052, 0054, 52A10; 44A12

Keyword(s): visual angle, masking function, Steinhaus, Crofton

Received July 1, 2015, and in revised form September 22, 2015. (Registered under 49/2015.)