ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The notion of an island defined on a rectangular board is an elementary combinatorial concept that occurred first in [czedli]. Results of [czedli] were starting points for investigations exploring several variations and various aspects of this notion. In this paper we introduce a general framework for islands that subsumes all earlier studied concepts of islands on finite boards, moreover we show that the prime implicants of a Boolean function, the formal concepts of a formal context, convex subgraphs of a simple graph, and some particular subsets of a projective plane also fit into this framework. We axiomatize those cases where islands have the property of being pairwise comparable or disjoint, or they are distant, introducing the notion of a connective island domain and of a proximity domain, respectively. In the general case the maximal systems of islands are characterised by using the concept of an admissible system. We also characterise all possible island systems in the case of connective island domains and proximity domains.

DOI: 10.14232/actasm-013-279-7

AMS Subject Classification (1991): 06A06

Keyword(s): island system, height function, CD-independent and CDW-independent sets, admissible system, distant system, island domain, proximity domain, point-to-set proximity relation, prime implicant, formal concept, convex subgraph, connected subgraph, projective plane

Received May 8, 2013, and in revised form January 9, 2014. (Registered under 29/2013.)

Abstract. A planar semimodular lattice is slim if it does not contain $SM 3$ as a sublattice. An SPS lattice is a slim, planar, semimodular lattice. Congruence lattices of SPS lattices satisfy a number of properties. It was conjectured that these properties characterize them. A recent result of Gábor Czédli proves that there is an eight element (planar) distributive lattice having all these properties that cannot be represented as the congruence lattice of an SPS lattice. We provide a new proof.

DOI: 10.14232/actasm-014-024-1

AMS Subject Classification (1991): 06C10, 06B10

Keyword(s): fork extension, join-irreducible congruence

Received March 28, 2014, and in revised from May 6, 2014. (Registered under 24/2014.)

Abstract. Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi (k)+1$ an integer and $\Phi_k(X)\in Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,{\cal N})$ is a canonical number system, with ${\cal N}=\{0,1,\dots,|\Phi_k(m)|-1\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$, $n$ and $k$ fixed.

DOI: 10.14232/actasm-013-825-5

AMS Subject Classification (1991): 11A63, 11D61, 11D41

Keyword(s): canonical number systems, radix representations, diophantine equations, Nagell--Ljunggren equation

Received November 5, 2013, and in revised form August 1, 2014. (Registered under 75/2013.)

Abstract. In this article, several matrix norm inequalities are proved by making use of Hiroshima's result on majorization relations.

DOI: 10.14232/actasm-013-821-3

AMS Subject Classification (1991): 15A60, 47A30

Keyword(s): Hiroshima's theorem, matrix inequalities, commuting type inequalities, unitarily invariant norm

Received October 23, 2013, and in revised form November 20, 2013. (Registered under 71/2013.)

Abstract. We introduce and study the partial singular braid monoid $PSB _n$, a monoid that contains both the inverse braid monoid $IB _n$ and the singular braid monoid $SB _n$. Our main results include a characterization of Green's relations, a presentation in terms of generators and relations, and a proof that $PSB _n$ embeds in the semigroup algebra $CIB_n $.

DOI: 10.14232/actasm-014-012-7

AMS Subject Classification (1991): 20F36; 20M05

Keyword(s): braid groups, singular braid monoids, inverse braid monoids, presentations, Birman's conjecture

Received January 29, 2014, and in revised from May 8, 2014. (Registered under 12/2014.)

Abstract. We survey results concerning special elements of eight types (modular, lower-modular, upper-modular, distributive, codistributive, standard, costandard and neutral elements) in the lattice of all semigroup varieties and three of its sublattices, namely, the lattices of commutative varieties, of permutative varieties and of overcommutative ones. These results are due to Ježek, McKenzie, Shaprynski?, Volkov and the author. Several open questions are formulated.

DOI: 10.14232/actasm-013-072-0

AMS Subject Classification (1991): 20M07; 08B15

Keyword(s): semigroup, variety, lattice of varieties, commutative variety, overcommutative variety, permutative variety, modular element, lower-modular element, upper-modular element, distributive element, codistributive element, standard element, costandard element, neutral element

Received October 27, 2013, and in revised form April 29, 2014. (Registered under 72/2013.)

Abstract. We introduce the following variant of an order and semigroup of quotients in inverse semigroups. A subsemigroup $S$ of an inverse semigroup $Q$ is a new left order in $Q$ if for every $q\in Q$, there exist $a,b\in S$ such that $q=a^{-1}b$. Here $a^{-1}$ is the unique inverse of $a$ in $Q$. A new right order is defined dually, and a new order is the conjunction of the two. This concept produces more (left, right) orders in an inverse semigroup than those studied heretofore. A primitive inverse semigroup is a nontrivial inverse semigroup with zero in which all nonzero idempotents are primitive. It can best be characterized as an orthogonal sum of Brandt semigroups. Our main result consists of necessary and sufficient conditions on a semigroup $S$ to be a new left order in a Brandt semigroup. They are five in number and of relatively concrete form. This result (with a long proof) is used to give two characterizations of new orders in Brandt semigroups, and eventually to perform a similar analysis for the same kinds of orders in primitive inverse semigroups. A uniqueness result concludes the work.

DOI: 10.14232/actasm-013-040-7

AMS Subject Classification (1991): 20M10, 20M18

Keyword(s): Brandt semigroup, primitive inverse semigroup, new (left) order, semigroup of (left) quotients, categorical at zero, $0$-cancellative

Received July 2, 2013, and in final form December 12, 2014. (Registered under 40/2013.)

Abstract. We consider a finite mass points perturbation of a measure supported by a system of curves and arcs in the complex plane. We study the corresponding perturbation of the ratio asymptotics of the associated monic orthogonal polynomials. Our main result gives a description of the perturbations for which the ratio asymptotics is not changed.

DOI: 10.14232/actasm-013-335-z

AMS Subject Classification (1991): 26C05

Keyword(s): polynomials

Received December 27, 2013, and in final form May 17, 2014. (Registered under 85/2013.)

Abstract. Suppose that $G\subset C $ is a Jordan domain and let the functions $f_n$ be holomorphic on $\overline{G}$. Assume that $U$ is a neighborhood of a point $z_0 \in\partial G$. In this paper the ``size'' of $f_n (U)$ is estimated. As application of the main result, informations about the boundary behavior of the partial sums, Ces?ro and Riesz means of power series are obtained.

DOI: 10.14232/actasm-013-084-7

AMS Subject Classification (1991): 30B10, 30C15

Keyword(s): boundary behavior, distribution of values, Jentzsch-type theorems

Received December 13, 2013, and in final form August 12, 2014. (Registered under 84/2013.)

Abstract. On a compact subset of the complex plane the supremum norm of a polynomial of degree $n$ with leading coefficient $1$ must be at least the $n$-th power of the logarithmic capacity of the set. In general, nothing more can be said, but if the polynomial also has zeros on the outer boundary, then those zeros may raise the minimal norm. The paper quantifies how much zeros on the boundary raise the norm on sets bounded by finitely many smooth Jordan curves. For example, $k_n$ zeros results in a factor $(1+ck_n/n)$, while $k_n$ excessive zeros on a subarc of the boundary compared to the expected value based on the equilibrium measure introduces an exponential factor $\exp(ck_n^2/n)$. The results are sharp, and they are related to Turán's power-sum method in number theory. It is also shown by an example that the smoothness condition cannot be entirely dropped.

DOI: 10.14232/actasm-014-323-9

AMS Subject Classification (1991): 42C05, 31A15

Keyword(s): monic polynomials, minimal norm, zeros on the boundary, equilibrium measure

Received November 8, 2014. (Registered under 73/2014.)

Abstract. In this paper we study composition and weighted composition operators on double sequence spaces defined by a modulus. We prove the boundedness and closed range properties of composition and weighted composition operators on double sequence spaces.

DOI: 10.14232/actasm-014-506-2

AMS Subject Classification (1991): 46A45, 47B33

Keyword(s): double sequence spaces, composition operators, weighted composition operators

Received January 8, 2014, and in revised form October 7, 2014. (Registered under 6/2014.)

Abstract. Given a Banach space $X$, we define the number $\lambda_0(X) = \inf d(X_2, \ell ^1(2))$, where the infimum is taken over all two-dimensional subspaces $X_2$ of $X$. Here, $d(M,N)$ means the Banach--Mazur distance between Banach spaces $M,N$ defined by $d(M,N) = \inf\{\|T\|\|T^{-1}\|: T\colon M\to N$ is an isomorphism$\}$. We establish some facts about $\lambda_0$ and then consider applications to Banach--Stone type theorems for isomorphisms on continuous, vector-valued function spaces. If $Q,K$ are locally compact Hausdorff spaces, and $X,Y$ are Banach spaces for which both $\lambda_0(X^*)$ and $\lambda_0(Y^*)$ are greater than one, it has been shown that if $T$ is an isomorphism from $C_0(Q,E)$ onto $C_0(K,Y)$ with $\|T\|\|T^{-1}\|$ sufficiently small, then $Q$ and $K$ are homeomorphic, a generalization of the Banach--Stone Theorem for isometries. We examine such results for subspaces of these spaces. A closed subspace $M$ of $C_0(Q,X)$ is said to be a $C_0(Q)$-module if it is closed under multiplication by functions in $C_0(Q)$. If $M$ and $N$ are $C_0(Q), C_0(K)$-modules, respectively, then with assumptions similar to those mentioned above, we are able to obtain results in which the homeomorphism is between the strong boundaries of $N$ and $M$. In this case, the strong boundaries are the subsets of $K$ and $Q$, respectively, upon which the functions in $N$ and $M$ have nonzero values. We also obtain a new theorem concerning isometries.

DOI: 10.14232/actasm-014-255-x

AMS Subject Classification (1991): 46B03, 46E40

Keyword(s): isomorphism, isometry, strong boundary, homeomorphism

Received January 7, 2014, and in revised form April 1, 2014. (Registered under 5/2014.)

Abstract. The sub-direct sum, a generalization of normal sum operation for matrices was introduced by Fallat and Johnson [FaJo99]. Here, the definition of sub-direct sum is extended to operators between Hilbert spaces. Conditions for the sub-direct sum to have a nonnegative Moore--Penrose inverse are obtained when the summands themselves have nonnegative Moore--Penrose inverses. The converse problem is also considered.

DOI: 10.14232/actasm-013-307-8

AMS Subject Classification (1991): 47A05, 47H05, 15A09, 15A24

Keyword(s): sub-direct sum, Moore--Penrose inverse, group inverse, nonnegativity

Received September 2, 2013. (Registered under 57/2013.)

Abstract. Let $T$ and $R$ be absolutely continuous polynomially bounded operators, that is, $H^\infty $-calculus is well-defined for them, and let $X$ and $Y$ be quasiaffinities which intertwine $T$ and $R$: $XT=RX$, $YR=TY$. If there exists a function $g\in H^\infty $ such that $XY=g(R)$, then $\sigma(T)=\sigma(R)$ and $\sigma_{\text{e}}(T)=\sigma_{\text{e}}(R)$. Also, a generalization of the result for contractions of K. Takahashi [14] is given: if a polynomially bounded operator $T$ is a quasiaffine transform of a unilateral shift $S$ of finite multiplicity, then $\sigma_{\text{e}}(T)=\sigma_{\text{e}}(S)$ and $\mathop{\rm ind}T=\mathop{\rm ind}S$, where $\text{ind}$ is the Fredholm index.

DOI: 10.14232/actasm-013-064-8

AMS Subject Classification (1991): 47A10, 47A60, 47A99

Keyword(s): polynomially bounded operator, quasisimilarity, quasiaffine transform, spectrum, essential spectrum, unilateral shift

Received September 16, 2013, and in final form January 24, 2014. (Registered under 64/2013.)

Abstract. Let $T$ be a bounded linear operator on a Banach space ${\cal X}$. In this paper we study uniform Ces?ro ergodicity when $T$ is not necessarily power-bounded, and relate it to the uniform convergence of the Abel averages. When ${\cal X}$ is over the complex field, we show that uniform Abel ergodicity is equivalent to the uniform convergence of the powers of all (one of) the Abel averages $A_\alpha $, $\alpha\in (0,1)$. This is equivalent to uniform Ces?ro ergodicity of $T$ when $\|T^n\|/n \to0$. For positive operators on real or complex Banach lattices, uniform Abel ergodicity is equivalent to uniform Ces?ro ergodicity. An example shows that this is not true in general. For a $C_0$-semi-group $\{T_t\}_{t\ge0}$ on ${\cal X}$ complex satisfying $\lim_{t\to\infty } \|T_t\|/t=0$, we show that uniform ergodicity is equivalent to uniform convergence of $(\lambda R_\lambda )^n$ for every (one) $\lambda >0$, where $R_\lambda $ is the resolvent family of the generator of the semi-group.

DOI: 10.14232/actasm-012-307-4

AMS Subject Classification (1991): 47A35, 47B65; 47B20

Keyword(s): uniform ergodic theorem, Ces?ro bounded operators, Abel convergence, one-point peripheral spectrum

Received August 2, 2012, and in revised form August 13, 2014. (Registered under 57/2012.)

Abstract. We establish the subnormality and reflexivity of certain operator tuples referred to as spherically quasinormal tuples (of which quasinormal tuples are a special case) and show that the duals of such tuples are also spherically quasinormal (and hence reflexive). Our arguments yield in particular new proofs of the subnormality and reflexivity of quasinormal tuples.

DOI: 10.14232/actasm-014-510-5

AMS Subject Classification (1991): 47B20

Keyword(s): quasinormal, spherically quasinormal, reflexive

Received January 22, 2014. (Registered under 10/2014.)

Abstract. We identify the semigroups consisting of bounded composition operators on the Hardy spaces $H^p(U^+)$ of the upper half-plane. We show that any such semigroup is strongly continuous on $H^p(U^+)$ but not uniformly continuous and we identify the infinitesimal generator.

DOI: 10.14232/actasm-013-526-x

AMS Subject Classification (1991): 47D03, 47B33, 30H10

Keyword(s): semigroups, composition operators, Hardy spaces

Received April 17, 2013, and in final form December 12, 2014. (Registered under 26/2013.)

Abstract. For a composition operator $C_\phi $ on the Hardy space $H^2(\mathbb{D})$ with $\phi(0)=0$, the subspaces $z^{k}H^{2}$ are invariant. In this paper, we demonstrate that certain linear fractional maps induce composition operators for which the restrictions to different subspaces $z^{k}H^{2}$ are not unitarily equivalent, even though they have the same norm and spectrum. On the other hand, we show that these restrictions are unitarily equivalent to compact perturbations of each other.

DOI: 10.14232/actasm-013-048-y

AMS Subject Classification (1991): 47B33; 47D06

Keyword(s): operator theory, composition operator, semigroup

Received August 4, 2013, and in final form November 3, 2013. (Registered under 48/2013.)

Abstract. The goal of the paper is to estimate the first four moments of the offspring and innovation distributions of subcritical, time-homogeneous multitype Galton--Watson processes. We apply the CLS (Conditional Least Squares) and the WCLS (Weighted Conditional Least Squares) methods for this purpose. It is also shown that under the proper moment conditions the estimators are strongly consistent and the ones of the first two moments are asymptotically normal.

DOI: 10.14232/actasm-014-056-7

AMS Subject Classification (1991): 60J80, 62F10, 62F12

Keyword(s): branching processes, Galton--Watson, moments, parameter estimation, conditional least squares

Received July 19, 2014, and in revised form February 5, 2015. (Registered under 56/2014.)