ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Let $\vec{H}$ and $\vec{K}$ be finite composition series of a group $G$. The intersections $H_i\cap K_j$ of their members form a lattice ${\rm CSL}(\vec{H},\vec{K})$ under set inclusion. Improving the Jordan--Hölder theorem, G. Grätzer, J. B. Nation and the present authors have recently shown that $\vec{H}$ and $\vec{K}$ determine a unique permutation $\pi $ such that, for all $i$, the $i$-th factor of $\vec{H}$ is ``down-and-up projective'' to the $\pi(i)$-th factor of $\vec{K}$. Equivalent definitions of $\pi $ were earlier given by R. P. Stanley and H. Abels. We prove that $\pi $ determines the lattice ${\rm CSL}(\vec{H},\vec{K})$. More generally, we describe slim semimodular lattices, up to isomorphism, by permutations, up to an equivalence relation called ``sectionally inverted or equal''. As a consequence, we prove that the abstract class of all ${\rm CSL}(\vec{H},\vec{K})$ coincides with the class of duals of all slim semimodular lattices.

AMS Subject Classification (1991): 06C10, 20E15

Keyword(s): composition series, Jordan--Hölder Theorem, group, slim lattice, semimodularity, planar lattice, permutation

Received May 8, 2013, and in revised form May 16, 2013. (Registered under 30/2013.)

Abstract. Let $FG$ be the group algebra of a finite group $G$ over a field $F$ of characteristic $p$. We give the maximal number of the non-isomorphic unitary subgroups with respect to the involutions of $FG$ which arise from $G$. Furthermore, we characterize the group algebras with Hamiltonian unitary subgroup under the canonical involution, where $G$ is a finite $p$-group and $F$ is a finite field of characteristic $p$. Let $FG$ denote the group algebra of a non-abelian group of order $8$ over a finite field of characteristic two. We also describe the structure of the non-isomorphic unitary subgroups of $FG$ linked to all the involutions which arise from $G$.

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

Keyword(s): group ring, involution

Received July 22, 2012, and in revised form August 7, 2013. (Registered under 54/2012.)

Abstract. An asymptotically sharp Bernstein-type inequality is proven for trigonometric polynomials in integral metric. This extends Zygmund's classical inequality on the $L^p$ norm of the derivatives of trigonometric polynomials to the case when the set consists of several intervals. The result also contains a recent theorem of Nagy and Toókos, who proved a similar statement for algebraic polynomials.

AMS Subject Classification (1991): 31A15, 41A17

Keyword(s): Bernstein inequality, integral norm, sharp constants

Received April 4, 2013, and in revised form July 3, 2013. (Registered under 23/2013.)

Abstract. In this paper we develop a new method to determine the essential spectrum of coupled systems of singular differential equations. Applications to problems from magnetohydrodynamics and astrophysics are given.

AMS Subject Classification (1991): 47A10, 34L05, 47A55, 76E99

Keyword(s): essential spectrum, singular differential systems, matrix differential operators

Received December 1, 2012, and in revised form July 22, 2013. (Registered under 100/2012.)

Abstract. It is proved a necessary and sufficient type theorem concerning the absolute summability of the generalized de la Vallée Poussin means of orthogonal series.

AMS Subject Classification (1991): 42C15, 40G99

Keyword(s): de la Vallée Poussion summability

Received March 13, 2012, and in revised form September 3, 2013. (Registered under 15/2012.)

Abstract. We show the existence of ``large" weak orbits of $C_0$-semigroups with generators satisfying natural spectral assumptions. We give also certain applications of our results to harmonic analysis and discuss related results.

AMS Subject Classification (1991): 47D03, 47A10, 42A38

Keyword(s): weak orbit, $C_0$-semigroup, domination, decay, Fourier transform

Received December 6, 2012. (Registered under 108/2012.)

Abstract. We consider the families of finite Abelian groups ${\msbm Z}/p{\msbm Z}\times{\msbm Z}/p{\msbm Z}$, ${\msbm Z}/p^2{\msbm Z}$ and ${\msbm Z}/p{\msbm Z}\times{\msbm Z}/q{\msbm Z}$ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality $k$ while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of $k$ in the third case. Such equality cases were previously known when $k$ divides the cardinality of the group, or for groups ${\msbm Z}/p{\msbm Z}$.

AMS Subject Classification (1991): 42A99

Keyword(s): uncertainty principle, finite Abelian groups, Fourier matrices

Received January 26, 2010, and in final form September 9, 2013. (Registered under 5/2010.)

Abstract. The notion of framings, recently emerging in [caz] as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\it théor?me principal} of B. Sz-Nagy [app] go much beyond it.

AMS Subject Classification (1991): 42C15, 47A20

Keyword(s): framings, dilations

Received January 17, 2013, and in revised form June 14, 2013. (Registered under 6/2013.)

Abstract. New growth conditions on the Ces?ro means of higher order are investigated for Banach space operators with peripheral spectrum reduced to $\{1\} $. Certain consequences concerning the powers of such operators are derived. The uniform and strong convergence of the differences of consecutive Ces?ro means are studied, and several examples are presented. These topics are related to the boundedness and convergence of Ces?ro means of higher order, and also to Gelfand--Hille and Esterle--Katznelson--Tzafriri type theorems. In particular, if $V$ denotes the classical Volterra operator, then our results provide a simultaneous conceptual proof showing that the operator $I-V$ is Ces?ro ergodic on $L^p(0,1)$ for $1\le p< \infty $, completing the known cases $p=1$ and $p=2$. Even every power of the latter operator is Ces?ro ergodic, though the operator itself is not power-bounded if $p\not=2$. Analogous examples, with respect to uniform ergodicity, are given as well. We also obtain improvements on the general 1939 Lorch theorem, within the above spectral picture.

AMS Subject Classification (1991): 47A10, 47A35

Keyword(s): Abel mean, Ces?ro mean, Kreiss resolvent condition, spectrum, power-dominated operator, nilpotent operator, ascent, descent

Received March 7, 2013, and in revised form July 29, 2013. (Registered under 20/2013.)

Abstract. We prove the following properties of the numerical range of a KMS matrix $J_n(a)$: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\not=0$, (2) its boundary $\partial W(J_n(a))$ contains a line segment if and only if $n\ge3$ and $|a|=1$, and (3) the intersection of the boundaries $\partial W(J_n(a))$ and $\partial W(J_n(a)[j])$ is either the singleton $\{\min\sigma (\mathop{\rm Re}J_n(a))\} $ if $n$ is odd, $j=(n+1)/2$ and $|a|>1$, or the empty set $\emptyset $ if otherwise, where, for any $n$-by-$n$ matrix $A$, $A[j]$ denotes its $j$th principal submatrix obtained by deleting its $j$th row and $j$th column ($1\le j\le n$), $\mathop{\rm Re}A$ its real part $(A+A^*)/2$, and $\sigma(A)$ its spectrum.

AMS Subject Classification (1991): 15A60

Keyword(s): Numerical range, KMS matrix, $S_n$-matrix, $S_n^{-1}$-matrix

Received October 4, 2012, and in revised form May 11, 2013. (Registered under 79/2012.)

Abstract. In a recent paper of Tarkhanov and Wallenta [TW] a definition of Lefschetz numbers for morphisms $a = (a^\bullet )$ of Fredholm quasicomplexes $E^\bullet = (E^\bullet, d^\bullet )$ with trace class curvature is proposed. In the present note we show that there always exist trace class perturbations of $a$ and $E^\bullet $ to a cochain mapping $A = (A^\bullet )$ of a Fredholm complex $(E^\bullet,D^\bullet )$, and we clarify the relation between the Lefschetz number of $A$ relative to the perturbed complex $(E^\bullet,D^\bullet )$ and the Lefschetz number of $a$ relative to the original quasicomplex $(E^\bullet,d^\bullet )$. Furthermore, we prove that the Lefschetz numbers relative to $E^\bullet $ satisfy a natural commutativity property.

AMS Subject Classification (1991): 47A53; 46M20, 47A13

Keyword(s): Fredholm complexes, quasicomplexes of Banach spaces, Lefschetz numbers

Received September 22, 2012, and in revised form June 15, 2013. (Registered under 78/2012.)

Abstract. The Sz.-Nagy--Foias model theory for $C_{\cdot0}$ contraction operators combined with the Beurling--Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot0}$ Hilbert-space contraction operators. We discuss an analogue of all these ideas in the context of weighted Hardy spaces over the unit disk and an associated class of hypercontraction operators.

AMS Subject Classification (1991): 47A57

Keyword(s): Operator-valued functions, weighted Hardy space, Bergman inner functions, Beurling--Lax theorem, hypercontraction operators, dilation theory, characteristic function

Received December 7, 2012, and in final form May 19, 2013. (Registered under 109/2012.)

Abstract. We consider the optimal quantization problem with Rényi-$\alpha $-entropy constraints for centered Gaussian measures on a separable Banach space. For $\alpha = \infty $ we can compute the optimal quantization error by a moment on a ball. For $\alpha\in {} ]1,\infty ]$ and large entropy bound we derive sharp asymptotics for the optimal quantization error in terms of the small ball probability of the Gaussian measure. We apply our results to several classes of Gaussian measures. The asymptotical order of the optimal quantization error for $\alpha > 1$ is different from the well-known cases $\alpha = 0$ and $\alpha = 1$.

AMS Subject Classification (1991): 60G15, 62E17, 94A17

Keyword(s): Gaussian measures, Rényi-$\alpha $-entropy, functional quantization, high-resolution quantization

Received February 18, 2010, and in revised form December 19, 2012. (Registered under 11/2010.)

Abstract. In matching theory of contracts the substitutes condition plays an essential role to ensure the existence of stable matchings. We study many-to-many matchings where groups of individuals, of size possibly greater than two, are matched to a set of institutions. Real-world examples include orphan brothers accepting an adoptive family conditional on all of them being included; hiring contracts that may only be chosen together; or a situation where a firm accepts to hire several workers only if they accept to work on different days (part-time jobs). We demonstrate by several examples that such extra conditions may alter the natural choice maps so that stable matchings cannot be obtained by applying the standard theorems. We overcome this difficulty by introducing a new construction of choice maps. We prove that they yield stable matchings if the construction respects an ``anti-trust'' rule on the supply side of the market.

AMS Subject Classification (1991): 91B68, 90C27

Keyword(s): games, matchings, choice maps, blocs, substitutes condition

Received July 19, 2012, and in revised form May 31, 2013. (Registered under 55/2012.)