ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. A sectionally pseudocomplemented poset $P$ is one which has the top element and in which every principal order filter is a pseudocomplemented poset. The sectional pseudocomplements give rise to an implication-like operation on $P$ which coincides with the relative pseudocomplementation if $P$ is distributive. We characterise this operation and study some elementary properties of upper semilattices, lower semilattices and lattices equipped with this as well as two weaker kinds of implication. We also clarify connections of these algebras with Hilbert algebras and with relatively pseudocomplemented posets and semilattices. Sectionally pseudocomplemented lattices have already been studied in the literature.

AMS Subject Classification (1991): 03G25, 06A12, 06D15, 06F35

Received August 14, 2007, and in final form November 24, 2007. (Registered under 6027/2009.)

Abstract. Let $n\geq3$. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an $n$-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an $n$-dimensional vector space over an arbitrary finite field is generated by five elements.

AMS Subject Classification (1991): 06C05, 50D30, 14N20, 51D25

Keyword(s): Arguesian lattice, subspace lattice of a vector space, generating set of a subspace lattice

Received February 4, 2008, and in revised form February 8, 2008. (Registered under 6028/2009.)

Abstract. Let $R$ be a commutative Noetherian ring, $\Phi $ a system of ideals of $R$, and $M$ a finitely generated $R$-module. Suppose that ${\eufm a}\in\Phi $ and $t$ is a non-negative integer. It is shown that if $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^j(M))$ is finitely generated for all $i$ and all $j< t$, then $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^t(M))$ is finitely generated for $i=0,1$. In particular, if $R$ is a local ring of dimension at most $2$, then $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^j(M))$ is finitely generated for all $i,j$.

AMS Subject Classification (1991): 13D45, 13E99

Keyword(s): General local cohomology modules, Cofinite modules

Received December 13, 2007, and in revised form March 30, 2008. (Registered under 6029/2009.)

Abstract. It is shown that the semigroup variety generated by the monoid $$ A_{0}^1 = \langle a, b : a^2 = a, b^2 = b, ba = 0 \rangle\cup \{1 \} $$ of order five is hereditarily finitely based and contains countably infinitely many subvarieties. In contrast, the monoid variety generated by $A_{0}^1$ is shown to contain only eight subvarieties.

AMS Subject Classification (1991): 20M07, 08B15, 03C05

Keyword(s): semigroups, monoids, varieties, finitely based, hereditarily finitely based

Received July 4, 2007, and in revised form October 31, 2007. (Registered under 6030/2009.)

Abstract. A semigroup variety is called {\it modular} [{\it upper-modular, lower-modular, neutral}] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. We classify all lower-modular varieties in the class of varieties of semigroups with a completely regular power, in the class of varieties of index $\le2$, and in the class of varieties satisfying an identity of the form $x_1x_2 \cdots x_n=x_{1\pi }x_{2\pi }\cdots x_{n\pi }$, where $\pi $ is a permutation on the set $\{1,2,\ldots,n\} $ with $1\pi\not=1$ and $n\pi\not=n$. It turns out that every lower-modular variety is modular in all these three classes. Moreover, for varieties of index $\le2$, the properties of being lower-modular, modular and neutral are equivalent. We completely determine also all semigroup varieties that are both upper-modular and lower-modular. It turns out that all such varieties are neutral.

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

Keyword(s): Semigroup, variety, lattice of varieties, [lower-, upper-]modular element, neutral element

Received October 16, 2007, and in revised form February 14, 2008. (Registered under 6031/2009.)

Abstract. In this paper, we show how to extend the holomorphic connection induced by a scalar Fuchsian equation on an open subset of the projective line as a logarithmic connection on the projective line in such a way that the eigenvalues of the residue of the integrable connection in a logarithmic point agree with the exponents of the equation.

AMS Subject Classification (1991): 14H60, 34M50

Received October 15, 2007. (Registered under 34/2007.)

Abstract. The aim of the paper is to generalize two fundamental theorems of Boas. One of them deals with conditional integrability, another proves $p$th power integrability. Both theorems consider Fourier series with nonnegative coefficients and classical weight $x^\gamma $. The weight functions in our theorems are more general, they merely have $\beta $-power-monotone properties.

AMS Subject Classification (1991): 26D15, 40A05, 40A10, 42A32

Received January 15, 2008, and in revised form March 21, 2008. (Registered under 6032/2009.)

Abstract. In this paper we deal with the linear functional equation $$ \int_0^1 f(x+t(y-x))d\mu(t)=0 ((x,y)\in\Omega ), $$ which can be considered as a generalization of the Fréchet functional equation that characterizes polynomials. Here $f\colon I\to{\msbm R}$ is a continuous function, $\mu $ is a given signed Borel measure on $[0,1]$ and $\Omega\subseteq I\times I$ is a given open set containing the diagonal of $I\times I$. Our main result shows that $f$ is a solution of the above equation if and only if $f$ is a polynomial of degree at most $n-1$, where $n$ is the smallest nonnegative integer such that the $n$th moment of the measure $\mu $ does not vanish. The main result is also used to solve certain composite functional equations.

AMS Subject Classification (1991): 39B22, 39B12

Keyword(s): Linear functional equation, composite functional equation, convolution smoothing

Received July 11, 2008, and in revised form August 27, 2008. (Registered under 6033/2009.)

Abstract. In 1992, Móricz and Siddiqi [4] studied the approximation properties of Walsh--Nörlund means in $L^p$ spaces. This study embraced earlier results in this area by Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesáro means. The main objective of this paper is to improve, and to extend the results of Móricz and Siddiqi [MS] to dyadic homogeneous Banach spaces, and to dyadic Hardy spaces $H^p$, $p<1$.

AMS Subject Classification (1991): 42C10, 41A25, 41A65, 60G42

Keyword(s): Walsh system, Nörlund means, Homogeneous Banach spaces, Hardy spaces

Received July 10, 2008, and in revised form August 27, 2008. (Registered under 6034/2009.)

Abstract. We obtain a Tauberian theorem for a class of regular lower triangular matrices operating on Walsh--Fourier series for functions $f \in L^1(0, 1)$. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

AMS Subject Classification (1991): 42C10; 42A24, 40G05

Keyword(s): generalized bounded variation, Hausdorff matrices, Nörlund matrices, orthonormal systems Walsh--Fourier series

Received July 11, 2008, and in final form July 31, 2008. (Registered under 6035/2009.)

Abstract. The main aim of this paper is to prove that the maximal operator $\widetilde{\sigma }^*f:=\sup_{n\in P}{| \sigma_{n}f| \over\log^2(n+1)}$ is bounded from the Hardy space $H_{1/2}$ to the space $L_{1/2}$.

AMS Subject Classification (1991): 42C10

Keyword(s): Walsh function, Hardy space, Fejér means

Received August 27, 2007. (Registered under 6036/2009.)

Abstract. We define a new type of spectrum, called the $\epsilon$-condition spectrum, of an element $a$ in a complex unital Banach algebra $A$ as $$\sigma_\epsilon(a):=\big\{\lambda\in{\msbm C} : \lambda-a \text{ is not invertible or } \|{\lambda-a}\|\|{(\lambda-a)^{-1}}\| \geq\frac{1}{\epsilon}\big\}. $$ This is expected to be useful in solving operator equations. We show that this is a particular case of the generalized spectrum defined by Ransford [10]. This $\epsilon$-condition spectrum shares some properties of the usual spectrum such as nonemptiness and compactness. But at the same time it has many properties that are different from the properties of the usual spectrum. For example, the $\epsilon$-condition spectrum always has only a finite number of components. Also if $a$ is not a scalar multiple of 1 then $\sigma_\epsilon(a)$ has no isolated points. Several examples are given to illustrate the main ideas.

AMS Subject Classification (1991): 46H05; 46J05

Keyword(s): Condition spectrum, Ransford spectrum, condition number

Received June 12, 2007, and in revised form September 21, 2007. (Registered under 6037/2009.)

Abstract. We obtain some intrinsic algebraic characterizations of Hilbert space operators which are similar to elements of some classes of power partial isometries. A power partial isometry is a partial isometry, all whose powers are also partial isometries. We study operators which are similar to direct sums whose summands are finite sums of truncated shifts, isometries or coisometries, as well as operators similar to subnormal or normal partial isometries.

AMS Subject Classification (1991): 47A05, 47A62

Received February 2, 2007, and in revised form July 21, 2008. (Registered under 6038/2009.)

Abstract. In this article we study left and right poles of the resolvent of a bounded operator defined on an infinite-dimensional complex Banach space. In particular, some spectral decompositions associated with left and right poles of the resolvent are established, and the components of the upper and lower semi B-Fredholm regions are studied by means of the localized single valued extension property.

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

Keyword(s): localized SVEP, B, semi-Browder operators, semi B-Weyl operators, left and right Drazin invertibility

Received October 2, 2007, and in revised form December 17, 2007. (Registered under 6039/2009.)

Abstract. We present a self-contained method initiated by A. M. Davie to prove the existence of nontrivial hyperinvariant subspace for Bishop-type operator $T_\alpha $ on $L^2(0,1)$ associated with an irrational $\alpha\in (0,1)$. Using all the strength of the Denjoy--Carleman theorem, we prove that our method works except on a set of Hausdorff measure equal to zero. We also show how to construct Liouville numbers $\alpha $ for which $T_\alpha $ has nontrivial hyperinvariant subspaces.

AMS Subject Classification (1991): 47A15; 47A10, 47A60

Received May 9, 2007, and in final form November 12, 2007. (Registered under 6040/2009.)

Abstract. The functional calculus approach of Davie is extended to show the existence of hyperinvariant subspaces for a collection of weighted composition operators that includes certain products of Bishop operators. Moreover it is shown that such operators have trivial point spectrum.

AMS Subject Classification (1991): 47A15; 47A10, 47A60, 47A15

Received October 15, 2007, and in revised form March 21, 2008. (Registered under 6041/2009.)

Abstract. Some results concerning reducibility and triangularizability of some sets of algebraic and of compact operators on locally convex spaces are given.

AMS Subject Classification (1991): 47A15, 47B99, 46A13, 46A08

Keyword(s): Locally convex space, compact operator, algebraic operator, invariant subspace, triangularization

Received August 14, 2007, and in revised form October 10, 2007. (Registered under 6042/2009.)

Abstract. In this note we construct an operator $T$ on a (separable, complex) Hilbert space such that for every nonzero vector $x$, the sequence $\{\|T^nx\|\} _{n\in{\msbm N}}$ is dense in ${\msbm R}_{+}$, but despite this, $T$ is not hypercyclic (i.e., no vector in ${\cal H}$ has a dense orbit). In addition, this operator has the property that there are subsequences $\{r_{n}\} $ and $\{q_{n}\} $ of ${\msbm N}$ such that $T^{r_{n}}\rightarrow0$ and $T^{q_{n}}\rightarrow +\infty $ (properly defined) in the strong operator topology. Finally, neither $T$ nor $T^*$ has point spectrum. This partially answers a question in [5] and provides a counterexample to some reasonable conjectures.

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

Received December 12, 2007, and in revised form January 17, 2008. (Registered under 6043/2009.)

Abstract. It is known that if $T$ is a contraction of class $C_{10}$ and $I-T^\ast T$ is of trace class, then $T$ is a quasiaffine transform of a unilateral shift. Also it is known that if the multiplicity of a unilateral shift is infinite, the converse is not true. In this paper the converse for a finite multiplicity is proved: if $T$ is a contraction and $T$ is a quasiaffine transform of a unilateral shift of finite multiplicity, then $I-T^\ast T$ is of trace class. As a consequence we obtain that if a contraction $T$ has finite multiplicity and its characteristic function has an outer left scalar multiple, then $I-T^\ast T$ is of trace class.

AMS Subject Classification (1991): 47A45

Received August 27, 2007, and in revised form September 3, 2008. (Registered under 6044/2009.)

Abstract. In [AMB] we introduced and studied property $(gw),$ which is an extension to the context of B-Fredholm theory, of the property $(w)$ introduced by Rakočević in [RA1]. In this paper we continue the study of property $(gw)$ and we consider its preservation under perturbations by finite rank and nilpotent operators.

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

Keyword(s): B, -Fredholm operator, Weyl's theorem, generalized Weyl's theorem, a, -Weyl's theorem, a, generalized-Weyl's theorem, (gw), property

Received June 21, 2007, and in final form January 31, 2008. (Registered under 6045/2009.)

Abstract. The first part is devoted to establish a useful mapping formula for functional calculus associated with $\rho $-contractions. In the second part, we give a general Julia's lemma for operators whose spectrum is contained in the closed unit disc. Finally, some applications, which are concerned with the hyperbolic metric on the Harnack parts of $\rho $-contractions, are given.

AMS Subject Classification (1991): 47A63, 47A15; 32D15, 47A11

Keyword(s): Integral formula for functional calculus, Julia's lemma for operators, \rho, -kernel, invariant subspaces

Received July 20, 2007, and in revised form July 10, 2008. (Registered under 6046/2009.)

Abstract. Composition operators on Fock type spaces have simple forms, and they are just operators composed through linear functions. The purpose of this paper is to study basic problems and to give answers for composition operators on Fock type spaces concerning with spectra, cyclicity, hypercyclicity and invariant subspaces.

AMS Subject Classification (1991): 47B33, 32A37

Keyword(s): Fock type space, Composition operator, translation operator, invariant subspace

Received May 10, 2007, and in revised form February 26, 2008. (Registered under 6047/2009.)

Abstract. In this paper, we estimate the essential norm of a weighted composition operator between different weighted Bergman spaces on the unit ball in the complex $N$-dimensional Euclidean space. In our results, we use Carleson type measures and certain integral operators to estimate the essential norm.

AMS Subject Classification (1991): 47B38, 32A36

Received September 8, 2007, and in revised form February 18, 2008. (Registered under 6048/2009.)

Abstract. For a wide family of multivariate Hausdorff operators, a new condition for the boundedness of an operator from this family on the real Hardy space $H^1({\msbm R}^n)$ is proved.

AMS Subject Classification (1991): 47B38, 42B10; 42B35

Keyword(s): Hausdorff operator, real Hardy space, atomic decomposition, eigenvalues

Received April 24, 2008. (Registered under 6049/2009.)

Abstract. Let $ {\cal B}({\cmss H})$ be the algebra of all bounded linear operators on a complex separable Hilbert space $ {\cmss H}$, and denote by $\gamma(T)$ the reduced minimum modulus of $T\in{\cal B}({\cmss H})$. Mbekhta [Mbekhta2007] conjectured that a surjective linear map $\phi\colon {\cal B}({\cmss H}) \rightarrow{\cal B}({\cmss H}) $ verifying $\gamma(T) = \gamma(\phi(T) ) $ for every $T\in{\cal B}({\cmss H})$ if and only if $\phi $ takes one of the following forms: $\phi(T) = UTV $ for every $T\in{\cal B}({\cmss H})$, or $\phi(T) = UT^{tr}V $ for every $T\in{\cal B}({\cmss H})$, where $U\in{\cal B}({\cmss H})$ and $ V\in{\cal B}({\cmss H}) $ are unitary operators. We answer in the affirmative a problem raised by the conjecture.

AMS Subject Classification (1991): 47B48, 47A30

Keyword(s): reduced minimum modulus, generalized spectrum, unitary operator, linear preservers

Received October 16, 2007, and in revised form December 18, 2007. (Registered under 6050/2009.)

Abstract. We prove that a $C_0$-semigroup $(T(s))_{s \geq0}$ with $\|T(s)\|\geq1$ $(s \geq0)$ has a regular norm-function if and only if its regularity constant is positive. This extends the characterization of single operators with regular norm-sequences.

AMS Subject Classification (1991): 47D06

Received July 30, 2008, and in final form September 4, 2008. (Registered under 6051/2009.)

Abstract. In this article, we consider a map $A\colon M\rightarrow X$ and a multivalued map $B\colon M\rightarrow CB(X)$ where $M$ is a closed convex subset of a Banach space $X$. We give sufficient conditions for the existence of a fixed point $x_0\in M$ of the multivalued operator $A+B$ satisfying $Ax_0+Bx_0=\{x_0\} $. This result includes the well-known Krasnoselskii's fixed point theorem for the sum of two nonlinear single valued operators.

AMS Subject Classification (1991): 54H25, 47H10

Keyword(s): fixed point, Hausdorff metric, multivalued contraction, measure of noncompactness

Received October 18, 2007, and in final form March 12, 2008. (Registered under 6052/2009.)

Abstract. For a point $P$ distinct from the vertices of an affinely regular $d$-gon in the affine plane $AG(2,q)$ with $q$ odd, let $n_P$ denote the number of chords through $P$. The trivial upper bound is $n_P\leq d/2$. In this paper it is shown that this can be improved to $n_P\leq d/3+2$, apart from some exceptional cases described explicitly.

AMS Subject Classification (1991): 51E15, 51E21

Keyword(s): affinely regular polygon, conic, Stöhr-Voloch bound

Received July 20, 2007, and in revised form May 6, 2008, (Registered under 19/2007.)

Abstract. Let $\{X_n\} \subset L_p({\bf P})$, $1< p\le2$, $q=p/(p-1)$, be a sequence of martingale differences. We prove that the Komlós--Révész type weighted averages ${\sum_{k=1}^n (X_k/\|X_k\|_p^q)\over\sum _{k=1}^n (1/\|X_k\|_p^q)}$ converge a.s. and in the $L_p$-norm, and the limit is $0$ if and only if $\sum_{n=1}^\infty(1/\|X_n\|_p^q)=\infty $. We show also that convergence need not hold when we deal with a centered uncorrelated sequence (whether the series $\sum_{n=1}^\infty(1/\|X_n\|_2^2)$ converges or not). Furthermore, for $1< p< 2$ all the results of Komlós--Révész are extended to symmetric independent $p$-stable random variables.

AMS Subject Classification (1991): 60F15, 60F25; 60G42, 60G52, 62F12

Keyword(s): independent random variables, martingale differences, p, -stable random variables, weighted averages, a.s. convergence, norm convergence, consistent estimation of a common mean

Received November 8, 2007, and in revised form March 6, 2008. (Registered under 6053/2009.)

Abstract. The purpose of this paper is to extend a recent result on the Lyapunov-exponent of a stationary, ergodic sequence of block-triangular random matrices to the problem of $L_q$-stability for i.i.d. sequences of block-triangular random matrices. A known sufficient condition for $L_q$-stability of an i.i.d. sequence of random matrices $A_n$, with $q$ even, is that $\rho[ {\rm E}(A^{\otimes q}) ] < 1$, where $\rho $ is the spectral radius. It is shown that the validity of this condition for the diagonal blocks of $A$ implies its validity for the full matrix, see Theorem 1.1. A brief survey of results on $L_q$-stability, and a simple proof of the above sufficient condition will be given. Two major areas of applications, modelling and estimation of bilinear time series and stochastic volatility processes will be also briefly described.

AMS Subject Classification (1991): 93E15, 34D08

Keyword(s): random matrix products, Lyapunov exponents, higher order moments, bilinear models, GARCH processes

Received July 24, 2007, and in final form June 19, 2008. (Registered under 6054/2009.)