ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $D$ be a finite distributive lattice with $n$ join-irreducible elements. In Part III, we proved that $D$ can be represented as the congruence lattice of a special type of planar semimodular lattices of $O(n^3)$ elements, we called {\it rectangular}. In this paper, we show that this result is best possible. Let $D$ be a finite distributive lattice whose order of join-irreducible elements is a balanced bipartite order on $n$ elements. Then any rectangular lattice $L$ whose congruence lattice is isomorphic to $D$ has at least $k n^3$ elements, for some constant $k > 0$.

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

Keyword(s): semimodular lattice, planar, congruence, rectangular

Received February 4, 2008, and in revised form February 11, 2009. (Registered under 53/2010.)

Abstract. We prove that every finite semimodular lattice has a cover-preserving embedding into a simple semimodular lattice.

AMS Subject Classification (1991): 06C10, 06B05

Keyword(s): semimodular lattice, cover-preserving, simple, finite

Received December 19, 2008, and in revised form January 16, 2009. (Registered under 82/2008.)

Abstract. The aim of the present paper is to carry on the research of Czédli in determining the maximum number of rectangular islands on a rectangular grid. We estimate the maximum of the number of square islands on a rectangular grid.

AMS Subject Classification (1991): 06D99, 05A05

Keyword(s): lattice, distributive lattice, weakly independent subset, weak basis, full segment, square island, rectangular grid

Received November 4, 2008, and in revised form January 11, 2010. (Registered under 54/2010.)

Abstract. A variety ${\cal V}$ has {\it definable factor congruences} if and only if factor congruences can be defined by a first-order formula $\Phi $ having {\it central elements} as parameters. We prove that if $\Phi $ can be chosen to be existential, then factor congruences in every algebra of ${\cal V}$ are compact.

AMS Subject Classification (1991): 08B05, 03C40

Keyword(s): central element, compact congruence, semidegenerate variety, factor congruences

Received October 7, 2008, and in revised form May 26, 2009. (Registered under 55/2010.)

Abstract. We prove several inverse spectrum theorems for real, nonnegative and positive matrices. The results are of a local character with respect to the topology generated by the matching distance of the spectral lists of matrices. We prove e.g. that the set of spectral lists of positive matrices is an open set in this topology, and extend a result of Minc. A constructive method is used everywhere, which can produce the realizing matrices explicitly.

AMS Subject Classification (1991): 15A18, 15A29, 15A48

Keyword(s): spectral list of a real or nonnegative matrix, local inverse eigenvalue problem, pseudo-metric generated by the matching distance

Received December 23, 2008, and in revised form April 7, 2009. (Registered under 83/2008.)

Abstract. A double Jensen type inequality for $2n$-convex functions is obtained and applied to establish upper and lower bounds for the $f$-divergence measure in Information Theory. Some particular inequalities of interest are stated as well.

AMS Subject Classification (1991): 94Axx, 26D15, 26D10

Keyword(s): $f$-divergence measure, $2n$-convexity, convex functions, absolutely monotonic and completely monotonic functions, analytic inequalities

Received September 10, 2008, and in final form October 13, 2009. (Registered under 56/2010.)

Abstract. The spaces $\Delta c_{0}(p),$ $\Delta c(p) $ and $\Delta\ell _{\infty }(p) $ were defined by Ahmad and Mursaleen [1]. In [2], Altay and Polat also defined the sequence spaces $e_{0}^r(\Delta ) $, $e_{c}^r(\Delta )$ and $e_{\infty }^r(\Delta )$, and determined the absolute Köthe--Toeplitz duals of these spaces. The main purpose of this work is to introduce some new paranormed sequence spaces defined by Euler and difference operators and to compute their $\alpha $-, $\beta $-, $\gamma $-duals. Also some of the matrix transformations has been characterized.

AMS Subject Classification (1991): 46A45, 46B45

Keyword(s): paranormed sequence space, matrix mapping, Köthe--Toeplitz duals, Euler and difference sequence spaces

Received October 18, 2008, and in revised form June 9, 2009. (Registered under 57/2010.)

Abstract. The idea of statistical convergence was introduced in [1] and [17] and since then several generalizations and applications of this concept have been investigated by various authors. Recently Karakus [7] and Gürdal and S. Pehlivan [6] studied statistical convergence in probabilistic normed space and 2-Banach space, respectively. In this paper we propose to study statistical convergence in random 2-normed space which seems to be a quite new and interesting idea.

AMS Subject Classification (1991): 46H25, 46A99, 60H10

Keyword(s): distribution function, t-norm, triangle function, 2-norm, probabilistic normed space, random 2-normed space, statistical convergence, statistically Cauchy

Received November 18, 2008. (Registered under 59/2008.)

Abstract. The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasi-free states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasi-free state is described by the representing block matrix. The proof is based on results on the statistical sufficiency in the non-commutative case. The relation to classical Gaussian Markov triplets is also described.

AMS Subject Classification (1991): 46L53, 60J10, 40C05, 81R15

Keyword(s): Weyl unitaries, Fock representation, quasi-free state, von Neumann entropy, CCR algebra, Markov triplet

Received November 25, 2008, and in final form March 4, 2009. (Registered under 64/2008.)

Abstract. The property $(w)$ is a variant of Weyl's theorem, for a bounded operator $T$ acting on a Banach space. In this note we consider the preservation of property $(w)$ under compact, or Riesz, or algebraic perturbations commuting with $T$. We shall also give some perturbation results on the generalized property $(gw)$.

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

Keyword(s): property $(w)$, SVEP, Riesz operators

Received October 27, 2008, and in revised form August 14, 2009. (Registered under 58/2010.)

Abstract. This note is concerned with weakly hypercyclic vectors and operators and weakly orbit-transitive operators (definitions below). We show that, given a sequence $\{x_{n}\} $ of vectors with $\|x_{n}\|\to\infty $ and $0\in\{x_{n}\} ^{^{\underline{w}}}$, there exists another sequence $\{w_{n}\} $ with approximately equal growth rate that is weakly dense. This complements a result of V. Kadets [11]. Then we apply Kadets' theorem, together with others used previously [10], to show that certain classes of operators consist entirely of non-weakly-orbit-transitive operators, thereby generalizing the results of [10]. Along the way we show that K. Ball's complex-plank theorem [2] is equivalent to a (slightly stronger) version of an old theorem of Beauzamy [3].

AMS Subject Classification (1991): 47A15, 47A16, 47B37

Keyword(s): orbit-transitive, hypertransitive

Received November 24, 2008, and in revised form March 11, 2009. (Registered under 61/2008.)

Abstract. The Cauchy transform $C\colon f(z)\to\int _Kf(w)/(z-w) dw$ over $L^1(K)$ (where $K$ is a compact set in ${\msbm C}$) and the multiplication operators $M_h\colon f(z)\to h(z)f(z)$ have compact comutators $[M_h,C]$ for all $h\in\overline {{\cal R}(K)}$, the closure in $C(K)$ of the rational functions with poles off $K$. In the special case of a polynomial $h$, the eigenvalues and eigenfunctions of $[M_h,C]$ are obtained from the eigenvalues and eigenvectors of a related symmetric matrix.

AMS Subject Classification (1991): 47B47, 47A75

Keyword(s): Cauchy transform, commutators, multiplication operator, compact operator, Hardy space, complex moments

Received January 28, 2009. (Registered under 11/2009.)

Abstract. Characterizations of the classes of $p$-hyponormal, $\infty $-hyponormal and weak hyponormal weighted composition operators are shown. It is shown that some classes of weak hyponormal weighted composition operators can not be separated. It is an extension of the result by C. Burnap, etc.

AMS Subject Classification (1991): 47B33, 47B20

Keyword(s): weighted composition operators, $p$-hyponormal operators, class A($p, r$) operators, absolute-$(p, r)$-paranormal operators, $p$-quasihyponormal operators, conditional expectation

Received August 5, 2008, and in final form December 1, 2009. (Registered under 60/2010.)

Abstract. In this paper we consider the notion of $n$-isometry on a Banach space. We determine the existence of non-isometric $2$-isometries for the classes of weighted shifts and composition operators on $\ell_p$ spaces. We also address similar questions for weighted composition operators acting on the space of scalar valued continuous functions, defined on a compact Hausdorff topological space.

AMS Subject Classification (1991): 47B33, 47B37

Keyword(s): $n$-isometries, weighted composition operators on $l_p$ spaces, composition operators on ${\cal C}(X)$

Received January 12, 2009, and in revised form July 7, 2009. (Registered under 5/2009.)

Abstract. In this work, we obtain trace ideal criteria for holomorphic weighted composition operators acting on the weighted Bergman spaces $A^2_{\lambda }(\Omega )$ of a bounded symmetric domain $\Omega $ in ${\msbm C}^n.$

AMS Subject Classification (1991): 47B33, 46E30, 47B07, 46B70

Keyword(s): Bergman spaces, bounded symmetric domains, Schatten $p$-class operator

Received November 18, 2009. (Registered under 58/2008.)

Abstract. Let $X$ and $Y$ be two infinite dimensional real or complex Banach spaces, and $\phi :{\cal L}(X)\to{\cal L}(Y)$ be an additive surjective mapping. We show that if $\phi $ preserves the minimum modulus or the surjectivity modulus, then either there exist two surjective linear or conjugate linear isometries $U$ and $V$ such that $\phi(T)=UTV$ for all $T$, or there exist two surjective linear or conjugate linear isometries $U'$ and $V'$ such that $\phi(T)=U'T^*V'$ for all $T$.

AMS Subject Classification (1991): 47B48, 47A10, 46H05

Keyword(s): additive preservers, minimum modulus, surjectivity modulus

Received March 11, 2009, and in revised form July 2, 2009. (Registered under 44/2009.)

Abstract. We will give an upper bound of the maximum number of points in the 5-dimensional unit cube so that all mutual distances are at least 1.

AMS Subject Classification (1991): 52C17

Keyword(s): packing, point, cube

Received October 10, 2008, and in revised form March 21, 2009. (Registered under 39/2008.)