ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. For a bounded lattice $L$, the principal congruences of $L$ form a bounded ordered set $\princ L$. G. Grätzer proved in 2013 that every bounded ordered set can be represented in this way. Also, G.Birkhoff proved in 1946 that every group is isomorphic to the group of automorphisms of an appropriate lattice. Here, for an arbitrary bounded ordered set $P$ with at least two elements and an arbitrary group $G$, we construct a selfdual lattice $L$ of length sixteen such that $\princ L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$.

DOI: 10.14232/actasm-015-817-8

AMS Subject Classification (1991): 06B10

Keyword(s): principal congruence, lattice congruence, lattice automorphism, ordered set, bounded poset, quasi-colored lattice, preordering, quasiordering, monotone map, simultaneous representation, independence, automorphism group

Received September 6, 2015, and in revised form September 28, 2015. (Registered under 67/2015.)

Abstract. Slim rectangular lattices were introduced by G. Grätzer and E. Knapp in Acta Sci. Math. 75, 29--48, 2009. They are finite semimodular lattices $L$ such that the poset $\jir L$ of join-irreducible elements of $L$ is the cardinal sum of two nontrivial chains. Using deep tools and involved considerations, a 2013 paper by G. Czédli and the present authors proved that a slim semimodular lattice is rectangular iff so is the Jordan--Hölder permutation associated with it. Here, we give an easier and more elementary proof.

DOI: 10.14232/actasm-015-271-y

AMS Subject Classification (1991): 06C10

Keyword(s): rectangular lattice, semimodularity, slim lattice, planar lattice, combinatorics of permutations

Received June 26, 2014, and in revised form March 2, 2015. (Registered under 21/2015.)

Abstract. Let $\mathbb{F}_{p^{k}}A_{5}$ be the group algebra of $A_{5}$, the alternating group of degree $5$, over $\mathbb{F}_{p^{k}}=GF(p^{k})$, where $p$ is a prime. Using the theory developed by Ferraz in [Ferraz08], we give an explicit description for the Wedderburn decomposition of $\mathbb{F}_{p^{k}}A_{5}$ modulo its Jacobson radical.

DOI: 10.14232/actasm-014-311-2

AMS Subject Classification (1991): 16S34; 16U60

Keyword(s): group algebra, Wedderburn decomposition, unit group

Received August 19, 2014. (Registered under 61/2014.)

Abstract. Let $L=M(G,2)$ be a $RA2$ loop and $F[L]$ be its loop algebra over a field $F$. In this article, we obtain the unit loop of $F[L]/J(F[L]),$ where $L=M(D_{2p},2)$ is obtained from the dihedral group of order $2p$ ($p$ odd prime), $J(F[L])$ is the Jacobson radical of $F[L]$ and $F$ is a finite field of characteristic $2$. The structure of $1+J(F[L])$ is also determined.

DOI: 10.14232/actasm-015-506-6

AMS Subject Classification (1991): 20N05, 17D05

Keyword(s): loop algebra, Moufang loop, Zorn's algebra, general linear loop, loops $M(G, 2)$

Received February 4, 2014, and in revised form August 27, 2015. (Registered under 6/2015.)

Abstract. In the present paper, we consider minimal generating sets of infinite full transformation semigroups with restricted range modulo specific subsets. In particular, we determine relative ranks.

DOI: 10.14232/actasm-015-502-4

AMS Subject Classification (1991): 20M20, 54H15

Keyword(s): generating sets, transformation semigroups, restricted range, relative ranks

Received January 7, 2015, and in revised form September 15, 2015. (Registered under 2/2015.)

Abstract. We first characterize those composition operators that are essentially normal on the weighted Bergman space $A^2_s(D)$ for any real $s>-1$, where induced symbols are automorphisms of the unit disk $D$. Using the same technique, we investigate automorphic composition operators on the Hardy space $H^2(B_N)$ and the weighted Bergman spaces $A^2_s(B_N)$ ($s>-1$). Furthermore, we give some composition operators induced by linear fractional self-maps of the unit ball $B_N$ that are not essentially normal.

DOI: 10.14232/actasm-014-060-x

AMS Subject Classification (1991): 47B33; 32A35, 32A36

Keyword(s): composition operator, essentially normal, automorphism, linear fractional maps

Received August 19, 2014, and in revised form October 19, 2014. (Registered under 60/2014.)

Abstract. The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \Eq{*}{ d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad(x,y\in{\msbm R}, k\in\{0,\ldots,n\}) } is studied, where $\Delta_n:=\big\{(i,j)\in\Z \times\Z \mid0\leq i,j\mbox{ and }i+j\leq n\big\}$ and $\Gamma\colon \Delta_n\to{\msbm R} $ is a symmetric function such that $\Gamma(i,j)=1$ whenever $i\cdot j=0$. On the other hand, the linear dependence and independence of the additive solutions $d_{0},d_{1},\dots,d_{n}\colon{\msbm R} \to{\msbm R} $ of the above system of equations is characterized. As a consequence of the main result, for any nonzero real derivation $d\colon{\msbm R} \to{\msbm R} $, the iterates $d^0,d^1,\dots,d^n$ of $d$ are shown to be linearly independent, and the graph of the mapping $x\mapsto(x,d^1(x),\dots,d^n(x))$ to be dense in ${\msbm R} ^{n+1}$.

DOI: 10.14232/actasm-014-534-6

AMS Subject Classification (1991): 16W25, 39B50

Keyword(s): derivation, higher order derivation, iterates, linear dependence

Received April 23, 2014, and in revised form August 19, 2014. (Registered under 34/2014.)

Abstract. It is well known that for a non-negative sequence $\{a_n\}_{n=1}^\infty $ the continuity of the sum $\sum ^\infty_{n=1}a_n\cos nx$ is equivalent to the convergence of the series $\sum ^\infty_{n=1}a_n$. We prove that for generalized monotone $\{a_n\}_{n=1}^\infty $ the last condition implies the so-called $p$-absolute continuity in the sense of L. C. Young and E. R. Love, where $1< p< \infty $. In this case we give estimates for the $p$-variation moduli of continuity and best approximations in terms of Fourier coefficients of a function. As a corollary of the above results some Konyushkov-type theorems on the equivalence of $O$- and $\asymp $-relations are established.

DOI: 10.14232/actasm-014-574-4

AMS Subject Classification (1991): 42A32, 42A10, 42A16, 41A25

Keyword(s): $p$-variation, $L^p$, best approximation, fractional moduli of continuity, Fourier coefficients, equivalence of $O$- and $\asymp $-relations

Received November 12, 2014, and in revised form August 15, 2015. (Registered under 74/2014.)

Abstract. The inverse wavelet transform is studied with the help of the summability means of Fourier transforms. Norm and almost everywhere convergence of the inversion formula is obtained for $L_p$ functions. The points of the set of the almost everywhere convergence are characterized as the Lebesgue points.

DOI: 10.14232/actasm-014-295-8

AMS Subject Classification (1991): 42C40; 42C15, 42B08, 42A38, 46B15

Keyword(s): continuous wavelet transform, $\theta $-summability, inversion formula

Received June 10, 2014, and in revised form January 6, 2015. (Registered under 45/2014.)

Abstract. We shall show in this paper a quite useful formula connecting the essential minimum modulus and minimum modulus of any linear bounded operator defined on a separable Hilbert space. Since this formula does not depend on the structure of Hilbert space, this result enables us to define the essential minimum modulus of linear operators in the more general context of Banach spaces. The connection between our definition and that given by Zemánek [Geometric interpretation of the essential minimum modulus, Operator Theory: Adv. Appl., 6, (1982), 225-227] is discussed. Moreover, the notion of the essential surjectivity modulus and left (resp. right) essential minimum modulus on Banach spaces are also defined and will be studied in this paper. The asymptotic formula for the essential spectrum of a semi-Fredholm operator with index zero in terms of the left and right essential minimum moduli is proved.

DOI: 10.14232/actasm-014-538-8

AMS Subject Classification (1991): 47A53, 47A55, 47A60, 46B04

Keyword(s): Banach space, minimum modulus, surjectivity modulus, essential minimum modulus, Calkin algebra, essential spectrum, semi-Fredholm, index

Received May 14, 2014, and in final form January 17, 2016. (Registered under 38/2014.)

Abstract. We introduce a new class of operators that we call b-limited operators. Properties of b-limited operators, the relationship between the b-limited operators and various classes of operators are studied.

DOI: 10.14232/actasm-014-570-2

AMS Subject Classification (1991): 46B42, 47B60, 47B65

Keyword(s): order limited operator, b-AM-compact operator, limited operator, (b)-property, discrete Banach lattice

Received October 13, 2014, and in final form January 10, 2015. (Registered under 70/2014.)

Abstract. We provide sufficient and necessary conditions guaranteeing equations $(A+B)^*=A^*+B^*$ and $(AB)^*=B^*A^*$ concerning densely defined unbounded operators $A,B$ between Hilbert spaces. We also improve the perturbation theory of selfadjoint and essentially selfadjoint operators due to Nelson, Kato, Rellich, and Wüst. Our method involves the range of two-by-two matrices of the form $M_{S,T}=\soperator{-T}{S}$ that makes it possible to treat real and complex Hilbert spaces jointly.

DOI: 10.14232/actasm-015-809-3

AMS Subject Classification (1991): 47A05, 47A55, 47B25

Keyword(s): closed operator, adjoint, selfadjoint operator, operator product, operator sum, perturbation

Received June 10, 2014, and in revised form January 6, 2015. (Registered under 59/2015.)

Abstract. In this paper we solve the isometric equivalence problem for composition operators on Hardy spaces of the bi-disk, for generalized composition operators on the Bloch space and for elementary operators on the symmetric subspace of $\mathcal{B}(\mathcal{H})$.

DOI: 10.14232/actasm-014-044-0

AMS Subject Classification (1991): 47A06, 47B25, 47B33

Keyword(s): isometric equivalence, Hardy space, Bloch space, spaces of symmetric operators, composition operators, elementary operators

Received May 31, 2014, and in revised form October 20, 2014. (Registered under 44/2014.)

Abstract. Some classical perturbation results on Fredholm theory are proved and extended by using the stability of the localized single-valued extension property under Riesz commuting perturbations. In the last part, we give some results concerning the stability of property $(gR)$ and property $(gb)$.

DOI: 10.14232/actasm-014-785-1

AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55

Keyword(s): localized SVEP, operators with topological uniform descent, Riesz operators, property $(gR)$ and property $(gb)$

Received April 17, 2014, and in revised form December 5, 2014. (Registered under 35/2014.)

Abstract. We completely characterize the spectrum of a weighted composition operator \W on \HtD when \ph has Denjoy--Wolff point $a$ with $0<|\ph '(a)|< 1$, the iterates, $\ph_n$, converge uniformly to $a$, and $\psi $ is in \Hi(the space of bounded analytic functions on \D ) and continuous at $a$. We also give bounds and some computations when $|a|=1$ and $\ph '(a)=1$ and, in addition, show that these symbols include all linear fractional \ph that are hyperbolic and parabolic non-automorphisms. Finally, we use these results to eliminate possible weights $\psi $ so that \W is seminormal.

DOI: 10.14232/actasm-014-542-y

AMS Subject Classification (1991): 47B33, 47B35, 47A10, 47B20, 47B38

Keyword(s): weighted composition operator, spectrum of an operator, hyponormal operator

Received May 21, 2014, and in revised form September 18, 2014. (Registered under 42/2014.)

Abstract. We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class $C_{0}$, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.

DOI: 10.14232/actasm-015-538-z

AMS Subject Classification (1991): 47A15

Keyword(s): invariant subspaces, Horn inequalities

Received May 19, 2015. (Registered under 38/2015.)

Abstract. The paper deals with Schatten--von Neumann operators in a Hilbert space. A sharp perturbation bound for invariant subspaces is established. It can be considered as a particular generalization of the Davis--Kahan theorem.

DOI: 10.14232/actasm-015-514-y

AMS Subject Classification (1991): 47A15, 47A55, 47B10

Keyword(s): Hilbert space, compact operators, invariant subspaces, perturbations

Received February 22, 2015, and in revised form April 27, 2015. (Registered under 14/2015.)

Abstract. A result of Eckstein asserts that for any $\rho $-contraction $T$ on a Hilbert space $\mathcal{H}$ the sequence $(||T^{n}h||)_{n}$ is convergent for any $h\in\mathcal {H}$. We show that this remains true for a natural generalization of the class of $\rho $-contractions, which we call the class of $(\rho,N)$-contractions (notation: $\mathcal{C}_{\rho,N}(\mathcal{H})$). Our argument follows the lines of Mlak's proof of Eckstein's result, but is somewhat simplified by a study of coisometric $(\rho,N)$-dilations of these operators, which seems to be of independent interest. Along the way we also point out that Gavruta's example extends to the class of $(\rho,N)$-contractions. Namely, let $\mathcal{C}_{\infty,\infty }(\mathcal{H}) :=\cup_{\rho,N}\mathcal{C}_{\rho,N}(\mathcal{H})$; then, for any integer $p>1$, there exists an operator $T$ such that $T^{p}=I$ and $T\notin\mathcal {C}_{\infty,\infty }(\mathcal{H})$.

DOI: 10.14232/actasm-014-004-2

AMS Subject Classification (1991): 47A20

Keyword(s): contractions, coisometric dilations, similarity

Received July 18, 2014, and in final form January 5, 2015. (Registered under 4/2014.)

Abstract. Brodskii and Milman proved that there exists a point in $C(A)$, the set of all Chebyshev centers of $A$, which is fixed by every surjective isometry from $A$ into $A$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. Motivated by this result, Lim et al. proved that every isometry from $A$ into $A$ has a fixed point in $C(A)$ whenever $A$ is a nonempty weakly compact convex set having normal structure in a Banach space. In this paper, we prove that every relatively isometry map $T\colon A\cup B \rightarrow A\cup B$, satisfying $T(A) \subseteq B$ and $T(B) \subseteq A$, has a best proximity point in $C_{A}(B)$, the set of all Chebyshev centers of $B$ relative to $A$, whenever the nonempty weakly compact convex proximal pair $(A, B)$ has proximal normal structure and rectangle property. Also, we prove that, under suitable assumptions, an analogous result of Brodskii and Milman for relatively isometry mappings holds. In case of $A = B$, we obtain the results of Brodskii and Milman, and Lim et al. as a particular case of our results.

DOI: 10.14232/actasm-014-833-1

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

Keyword(s): asymptotic center, Chebyshev center, best proximity points, proximal pairs, relatively nonexpansive maps, rectangle property

Received December 30, 2014. (Registered under 83/2014.)

Abstract. We examine the convexity of the hitting distribution of the real axis for symmetric random walks on $\duz ^2$. We prove that for a random walk starting at $(0,h)$, the hitting distribution is convex on $[h-2,\infty )\cap\duz $ if $h\ge2$. We also show an analogous fact for higher-dimensional discrete random walks. This paper extends the results of a recent paper [NT].

DOI: 10.14232/actasm-014-526-1

AMS Subject Classification (1991): 60G50; 05A20

Keyword(s): discrete random walk, integer lattice, convexity

Received April 2, 2014. (Registered under 26/2014.)

Abstract. We study asymptotic properties of some (essentially conditional least squares) parameter estimators for the subcritical Heston model based on discrete time observations derived from conditional least squares estimators of some modified parameters.

DOI: 10.14232/actasm-015-016-0

AMS Subject Classification (1991): 91G70, 60H10, 62F12, 60F05

Keyword(s): Heston model, conditional least squares estimation

Received February 23, 2015, and in revised form October 24, 2015. (Registered under 16/2015.)