
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Florence Micol,
Géza Takách

314

Abstract. We show that the earlier characterizations of pasting are strong enough to describe a construction, not only a decomposition. This construction is also compared with gluing and $S$glued sum.
AMS Subject Classification
(1991): 06B05, 06C05
Keyword(s):
pasting,
gluing,
Sglued sum
Received June 27, 2001, and in final form August 16, 2005. (Registered under 5903/2009.)
Abstract. We study the basic Galois connection induced by the ``satisfaction" relation between external operations $A^n\rightarrow B$ defined on a set $A$ and valued in a possibly different set $B$ on the one hand, and ordered pairs $(R,S)$ of relations $R\subseteq A^m$ and $S\subseteq B^m$, called relational constraints, on the other hand. We decompose the closure maps associated with this Galois connection, in terms of closure operators corresponding to simple closure conditions describing the corresponding Galois closed sets of functions and constraints. We consider further Galois correspondences by restricting the sets of primal and dual objects to fixed arities. We describe the restricted Galois closure systems by means of parametrized analogues of the simple closure conditions, and present factorizations of the corresponding Galois closure maps into simpler closure operators.
AMS Subject Classification
(1991): 08A02
Keyword(s):
Galois connections,
external operations,
homomorphisms,
function classes,
class composition,
relations,
constraints,
preservation,
constraint satisfaction,
minors,
factorizations,
operator decompositions
Received February 25, 2005, and in revised form December 31, 2005. (Registered under 5904/2009.)
Abstract. We determine the number of unary polynomial functions on all Frobenius complements and on all finite solvable groups all of whose abelian subgroups are cyclic.
AMS Subject Classification
(1991): 08A40
Received October 14, 2004, and in revised form November 12, 2005. (Registered under 5905/2009.)
Przemysław Koprowski

5158

Abstract. We discuss here the problem of the existence of polynomials with prescribed cycle lengths. From Šarkovskii's theorem we know that for every real polynomial $f$, the set of lengths of all cycles generated by $f$ is of the form $\mathop{\rm Cycl} (f)= \{m\in{\msbm N}\colon m\succeq n\} $ for some $n\in{\msbm N}$, where $\succeq $ denotes Šarkovskii's ordering. We show that for every odd integer $n\geq3$ there exists a polynomial $f$ such that $\mathop{\rm Cycl} (f)=\{m\in{\msbm N}\colon m\succeq n\} $. Moreover, our proof works not only over ${\msbm R}$ but over any real closed field.
AMS Subject Classification
(1991): 12D15, 11C08, 26A18, 39B12
Keyword(s):
polynomial cycles,
iterations,
cycle lengths,
Šarkovskii's theorem,
real closed fields
Received October 22, 2004, and in final form February 1, 2006. (Registered under 5906/2009.)
Ralph McKenzie,
David Stanovský

5964

Abstract. Every quasigroup (loop, Bol loop, group, respectively) is isomorphic to the factor of a subdirectly irreducible quasigroup (loop, Bol loop, group, respectively) over its monolithic congruence.
AMS Subject Classification
(1991): 20N05, 08B26
Keyword(s):
subdirectly irreducible,
quasigroup,
loop,
wreath product
Received February 15, 2005, and in final form October 17, 2005. (Registered under 5907/2009.)
Josip Pečarić,
Ivan Perić

6572

Abstract. The multidimensional generalization of the LupasOstrowski inequality, bounding the Chebyshev functional for the functions whose derivatives are square integrable, is given.
AMS Subject Classification
(1991): 26D15
Keyword(s):
The Chebyshev functional,
the Fourier expansion,
square integrable functions,
locally absolutely continuous functions
Received January 26, 2004. (Registered under 5908/2009.)
M. Pavlović,
J. A. Peláez

7393

Abstract. In this paper we work with the class of differentiable weights $\omega $ in the unit disc ${\msbm D}$ such that $$\sup_{0< r<1}{\omega '(r)\over\omega (r)^2}\int_r^1\omega(x) dx < \infty $$ and $${\omega '(r)\over\omega (r)^2}\int_r^1\omega(x) dx \ge 1, 0< r< 1. $$ We prove that if $\omega $ is one of these weights, $N$ is a positive integer, and $0< p< \infty,$ $0< q\le\infty $, then the equivalence $$ \int_0^1 M_q^p(r,f)\omega(r) dr \asymp\sup _{z< 1/2}f(z)^p+ \int_0^1 M_q^p(r,f^{(N)})(\psi_\omega(r))^{Np} \omega(r) dr, $$ holds for all analytic functions $f$ in ${\msbm D}$. The above result generalizes a classical equivalence due to Flett and extends a previous result of the authors to derivatives of higher order. We also extend a result of the first author and prove some results on Hadamard products.
AMS Subject Classification
(1991): 30D55, 32A36, 46E15
Keyword(s):
Weighted integrals,
sucessive derivatives,
Hadamard products
Received August 11, 2005. (Registered under 5909/2009.)
Abstract. We consider a wellknown almost periodic function, and prove exact upper and lower bounds on its local maxima and minima. We also show that the local maximum and local minimum values form dense sets in the proven intervals.
AMS Subject Classification
(1991): 42A05, 42A75
Received November 28, 2005. (Registered under 5910/2009.)
Abstract. There are many classical sufficient conditions for the absolute convergence. Convergence of the series $\sum_{j=0}^{\infty }2^jE_2(f;2^j)$, where $E_2(f;2^j)$ is the best approximation in $L^2$ norm of the function $f(x)$ by WalshFourier polynomials of degree not higher than $2^j$, implies the absolute convergence of the WalshFourier series of this function, which is the result of Bernstein and Steckin. We establish a similar result and also give several corollaries of it for the double WalshFourier series. Our results are expressed in terms of the best approximations and Besov spaces.
AMS Subject Classification
(1991): 42A20, 42C10
Received September 13, 2005, and in revised form January 24, 2006. (Registered under 5911/2009.)
Abstract. We study the interrelation between the strong class $S_p(\lambda )$ and the Nikol'skii class $W^rH_\beta ^\omega $.
AMS Subject Classification
(1991): 42A24, 42A25
Keyword(s):
Strong approximation,
Fourier series,
Embedding theorems
Received June 16, 2005, and in revised form September 9, 2005. (Registered under 5912/2009.)
Linda J. Patton,
Mihai Putinar

129134

Abstract. A nonnegative pluriharmonic polynomial ${\msbm R}e p(z)$ on the unit ball of ${\msbm C}^n$ is used as a weight against the rotationally invariant measure on the unit sphere. The resulting Hardy space carries the canonical ntuple $S$ of multiplication by the coordinate functions. By means of compressions of $S$ to coanalytically invariant subspaces, and known estimates of the numerical radius of a nilpotent matrix we obtain bounds for the coefficients of $p$, in terms of the arithmetic mean and degree of $p$, and dimension $n$.
AMS Subject Classification
(1991): 42B05, 47A12, 31C10
Received June 4, 2005, and in revised form September 21, 2005. (Registered under 5913/2009.)
Árpád Jenei,
Ferenc Móricz

135145

Abstract. We give sufficient conditions for the convergence of the symmetric as well as unsymmetric rectangular partial sums of the double Fourier series of a complexvalued function $f\in L^1 ({\msbm T}^2)$ at a given point $(x_0, y_0) \in{\msbm T}^2$. It turns out that this convergence essentially depends on the convergence behavior of the single Fourier series of the socalled marginal functions $f(x,y_0)$, $x\in{\msbm T}$, and $f(x_0, y)$, $y\in{\msbm T}$, at $x:= x_0$ and $y:= y_0$, respectively. Our theorems apply to functions in the multiplicative Lipschitz classes as well as Zygmund classes.
AMS Subject Classification
(1991): 42B05
Keyword(s):
Dini test,
double Fourier series,
symmetric and unsymmetric rectangular partial sums,
pointwise convergence,
RiemannLebesgue lemma,
multiplicative Lipschitz classes and Zygmund classes
Received August 4, 2005, and in final form January 6, 2006. (Registered under 5914/2009.)
Abstract. We prove that under mild growth conditions, uniqueness holds for a multiple Walsh series whose square dyadic partial sums converge almost everywhere to an integrable function. We apply this result to obtain a new uniqueness result for Cesàro summable multiple Walsh series.
AMS Subject Classification
(1991): 42C10, 43A75
Received February 28, 2005. (Registered under 5915/2009.)
U. Goginava,
G. Tkebuchava

159177

Abstract. In this paper we discuss some convergence and divergence properties of subsequences of partial sums and logarithmic means of WalshFourier series of functions in the uniform, and in the $L$ Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function. It is also proved that there exists a bounded function for which the logarithmic means converge and the partial sums diverge.
AMS Subject Classification
(1991): 42C10
Keyword(s):
WalshFourier series,
Norm convergence,
Logarithmic means
Received January 17, 2005, and in revised form April 6, 2005. (Registered under 5916/2009.)
Daniel Jupiter,
David Redett

179203

Abstract. In this article we examine Dirichlet type spaces in the unit polydisc, and multipliers between these spaces. These results extend the corresponding work of G. D. Taylor in the unit disc. In addition, we consider functions on the polydisc whose restrictions to lower dimensional polydiscs lie in the corresponding Dirichet type spaces. We see that such functions need not be in the Dirichlet type space of the whole polydisc. Similar observations are made regarding multipliers.
AMS Subject Classification
(1991): 46E22, 46E20, 47B32
Keyword(s):
Dirichlet type spaces,
multipliers
Received January 21, 2005, and in revised form October 28, 2005. (Registered under 5917/2009.)
C. Martin Edwards

205235

Abstract. Presymmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GLprojection on a presymmetric space represents an operation on the corresponding system, and has as its range a further presymmetric space which represents the state space of the resulting system. Every Lprojection is a neutral GLprojection, and such a projection represents a classical operation. Two neutral GLprojections $R$ and $S$ on the presymmetric space $A_*$ represent decoherent operations when their ranges are rigidly collinear. It is shown that if $R$ and $S$ each satisfy a condition, a possible physical interpretation of which is that the information lost in their measurement is partially recoverable, then $R$ and $S$ have as supremum $R + S$ and the operations corresponding to $R$, $S$ and $R+S$ are simultaneously performable. Furthermore, it is shown that the smallest Lprojections majorizing $R$, $S$ and $R + S$ coincide, and the greatest Lprojection majorized by $R+S$ is identified.
AMS Subject Classification
(1991): 46L70, 17C65, 81P15
Keyword(s):
^*,
JBWtriple,
presymmetric space,
contractive projection,
inner ideal,
decoherence
Received January 13, 2006. (Registered under 5918/2009.)
Abstract. We consider the isometric equivalence problem for various classical matrix operators on $l^p$. We extend some of these results to invertible operator weighted shifts on $l^p({\cal H})$, $1\leq p< \infty$, where $\cal H$ is a complex Hilbert space. Furthermore, we consider the isometric equivalence problem for the Cesàro operator on rearrangementinvariant function spaces.
AMS Subject Classification
(1991): 47A05, 47B37, 47B49
Received July 12, 2005, and in final form February 24, 2006. (Registered under 5919/2009.)
Kei Ji Izuchi,
Kou Hei Izuchi

251270

Abstract. Nakazi, Seto, and the first author gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which the cross commutators $[S_z,S^*_w]$ and $[S_{z^2},S^*_w]$ vanish, respectively. A characterization is given of backward shift invariant subspaces on which $[S_{z^n},S^*_w] = 0$ for a positive integer $n\ge2$.
AMS Subject Classification
(1991): 47A15, 32A35
Keyword(s):
Hardy space,
backward shift invariant subspace,
cross commutator
Received December 13, 2005, and in revised form March 22, 2006. (Registered under 5920/2009.)
Animikh Biswas,
Ciprian Foias

271298

Abstract. We consider the general intertwining lifting problem as formulated in [F1] and which is connected to interpolation problems in reproducing kernel Hilbert spaces. We reduce this general problem to the case where the operators involved are $n \times n$ block uppertriangular. As a consequence, we show that the causal commutant lifting (see [FT]) and the general intertwining lifting (or extension) problems are equivalent. We also obtain a seemingly new commutant lifting result for the case where one of the operators involved is nilpotent and the other canonical block Jordan. Finally, as an application, we obtain a completely new proof for the CeausescuCarswellSchubert result (see [Ce], [CaS]).
AMS Subject Classification
(1991): 47A20, 47A57, 47A45
Keyword(s):
Cummutant lifting,
interpolation,
intertwining lifting
Received May 10, 2005, and in revised form November 8, 2005. (Registered under 5921/2009.)
A. E. Frazho,
S. ter Horst,
M. A. Kaashoek

299318

Abstract. A new description is given of all solutions to the relaxed commutant lifting problem. The method of proof is also different from earlier ones, and uses only an operatorvalued version of a classical lemma on harmonic majorants.
AMS Subject Classification
(1991): 47A20, 47A57, 31A05, 47A56
Keyword(s):
commutant lifting,
positive real functions,
harmonic majorants,
parameterization
Received September 13, 2005, and in revised form March 14, 2006. (Registered under 5922/2009.)
Laurian Suciu,
Nicolae Suciu

319343

Abstract. In this paper we study the relation of Harnack domination for two Hilbert space contractions, defined relative to an operator acting between the two spaces. We construct the sequence of $n$step intertwining extensions for the underlying operator and we show that this sequence gives, to limit, the intertwining extension of the involved operator. This fact leads to a norm characterization of the relation of Harnack domination, which is used to derive some results in the case when one of the contractions is uniformly stable. As applications, we obtain versions of the commutant dilation theorem of Sz.NagyFoiaş for commuting pairs of contractions, which are related to other similar results.
AMS Subject Classification
(1991): 47A20, 47A45
Keyword(s):
Harnack domination,
dilation,
intertwining extension
Received October 25, 2005, and in revised form February 1, 2006. (Registered under 5923/2009.)
Ioana Serban,
Flavius Turcu

345351

Abstract. In this paper it is shown that if $A$ and $B$ are closed range operators in a Hilbert space for which the equation $B=XA$ has at least one solution, then the compactness of $AB$ is equivalent to the existence of a solution $X$ such that $XI$ is compact. This result has several consequences on the description of the compact perturbations of particular classes of operators.
AMS Subject Classification
(1991): 47A55, 47A05
Received March 23, 2005, and in revised form August 2, 2005. (Registered under 5924/2009.)
Abstract. We generalize the wellknown fact that the spectrum of the unilateral shift is the closed unit disk centered at the origin in the complex plane to more general operators.
AMS Subject Classification
(1991): 47B20, 47A10
Received December 13, 2005, and in final form February 14, 2006. (Registered under 5925/2009.)
Abstract. Let $H$ denote a complex Hilbert space and $B(H)$ denote the algebra of all bounded linear operators on $H$. In this paper, we study the class of pairs of operators $A,B\in B(H)$ that have the following property: $AT=TB$ implies $B^*T=TA^*$ for all $T\in C_{1}(H)$ (trace class operators). The main result is the equivalence between this character and the fact that the ultraweak closure of the range of a generalized derivation is closed under taking adjoints which is also equivalent to the generalized Dsymmetric operators.
AMS Subject Classification
(1991): 47B47, 47A30, 47B20, 47B10
Keyword(s):
Generalized derivation Elementary operators,
Trace class operators
Received April 14, 2005, and in final form March 7, 2006. (Registered under 5926/2009.)
Abstract. A family of sets ${\cal F} = \{C_\alpha\} $ in Euclidean space ${\msbm E}^d$ has the {\it strong binary intersection} (SBI) property provided for any selection $\{v_\alpha\} $ of vectors in ${\msbm E}^d$ the family of translates ${\cal F}' = \{v_\alpha + C_\alpha\mid C_\alpha\in {\cal F}\} $ has nonempty intersection if and only if any two members of ${\cal F}'$ have nonempty intersection. We show that a finite family of at least five convex bodies in ${\msbm E}^d,$ $d=2,3$, has the SBI property if and only if the members of ${\cal F}$ are isothetic parallelotopes, i.e., the edges of these parallelotopes are parallel to some $d$ linearly independent vectors in ${\msbm E}^d$.
AMS Subject Classification
(1991): 52A35
Received April 15, 2005, and in revised form November 17, 2005. (Registered under 5/2005.)
Leonardo Biliotti

387405

Abstract. In this paper we study the local and global properties of a complete Hilbert manifold, proving results of finite dimensional Riemannian geometry in the context of Hilbert manifolds.
AMS Subject Classification
(1991): 58B20, 53C21
Keyword(s):
Riemannian geometry,
Hilbert manifold,
exponential map,
sectional curvature
Received January 31, 2005, and in revised form September 23, 2005. (Registered under 5927/2009.)

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