ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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, S-glued 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.)

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.)

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.)

Abstract. The multidimensional generalization of the Lupas--Ostrowski 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.)

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 well-known 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 Walsh--Fourier polynomials of degree not higher than $2^j$, implies the absolute convergence of the Walsh--Fourier 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 Walsh--Fourier 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.)

Abstract. A non-negative 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 n-tuple $S$ of multiplication by the coordinate functions. By means of compressions of $S$ to co-analytically 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.)

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 complex-valued 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 so-called 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, Riemann--Lebesgue 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.)

Abstract. In this paper we discuss some convergence and divergence properties of subsequences of partial sums and logarithmic means of Walsh--Fourier 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): Walsh--Fourier series, Norm convergence, Logarithmic means

Received January 17, 2005, and in revised form April 6, 2005. (Registered under 5916/2009.)

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.)

Abstract. Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GL-projection on a pre-symmetric space represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system. Every L-projection is a neutral GL-projection, and such a projection represents a classical operation. Two neutral GL-projections $R$ and $S$ on the pre-symmetric 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 L-projections majorizing $R$, $S$ and $R + S$ coincide, and the greatest L-projection majorized by $R+S$ is identified.

AMS Subject Classification (1991): 46L70, 17C65, 81P15

Keyword(s): ^*, JBW-triple, pre-symmetric 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 rearrangement-invariant function spaces.

AMS Subject Classification (1991): 47A05, 47B37, 47B49

Received July 12, 2005, and in final form February 24, 2006. (Registered under 5919/2009.)

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.)

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 upper-triangular. 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 Ceausescu--Carswell--Schubert 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.)

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 operator-valued 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.)

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.-Nagy--Foiaş 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.)

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 $A-B$ is equivalent to the existence of a solution $X$ such that $X-I$ 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 well-known 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 D-symmetric 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.)

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.)