ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In the present paper we give an algorithm to compute generators of power integral bases having ``small" coordinates in an integral basis in sextic fields containing a cubic subfield. As an application of the method, we give a sufficient condition for infinite parametric families of number fields of this type to have power integral basis. To illustrate the statement we construct parametric families of fields and describe generators of power integral bases in them.

AMS Subject Classification (1991): 11Y50; 11D57

Received June 6, 2001, and in revised form October 17, 2001. (Registered under 2880/2009.)

Abstract. Let $R$ be a left slender ring and let $\kappa $ and $ \alpha $ be infinite cardinal numbers. Suppose $\phi\colon R^{\kappa }\rightarrow R^{< \alpha }$ is a left $R$-module homomorphism, where $R^{< \alpha }$ consists of all elements $x$ in $R^{\alpha }$ such that $\left |\mathop{\rm supp} (x)\right | < \alpha $. Using new set-theoretic techniques we will show that, if $\left | R\right | < \mu $, the least measurable cardinal number, then the image of $\phi $ is contained in a copy of $R^{\gamma }$, where $\gamma < \alpha $ if $\alpha\leq \mu $ as well as when $\kappa < \mu $.

AMS Subject Classification (1991): 16D80, 20K25; 13C13, 20K30

Received April 2, 2001, and in revised form July 5, 2002. (Registered under 2881/2009.)

Abstract. We study the following Minkowski-type inequality \[D_{a_0,b_0}(x_1+y_1, x_2+y_2)\le D_{a_1,b_1}(x_1, x_2)+D_{a_2,b_2}(y_1,y_2)\qquad(x_1,x_2, y_1,y_2\in{\msbm R}_+)\] and also its reverse for the two variable Stolarsky (or difference) means that are defined (in the case $ab(a-b)(x-y)\ne0$) by \[D_{a,b}(x,y)=\left(\frac{x^a-y^a}{a}\frac{b}{x^b-y^b}\right)^{\frac{1}{a-b}},\] for $a,b\in{\msbm R}$, $x,y\in{\msbm R}_+$. The results obtained extend that of the paper [14] by Losonczi and the second author concerning the case $a_0=a_1=a_2$, $b_0=b_1=b_2$.

AMS Subject Classification (1991): 26D15, 26D07

Keyword(s): Minkowski inequality, two variable homogeneous means, Stolarsky means, Minkowski separator

Received December 27, 2001. (Registered under 2882/2009.)

Abstract. Under some regularity assumptions, a problem of Z. Daróczy on mixed quasi-arithmetic means is solved.

AMS Subject Classification (1991): 26E60, 30B12

Keyword(s): mean, quasi-arithmetic mean, mixing-arithmetic mean, Aczél's theorem, functional equation

Received June 11, 2001. (Registered under 2883/2009.)

Abstract. We characterize those positive measures which can be represented as the variation of a measure with values in a Banach space $X$. We also show that if there exists such a representation, then there is one with $X=c_0$.

AMS Subject Classification (1991): 28B05, 28A12, 46B15

Received November 28, 2001, and in revised form May 10, 2002. (Registered under 2884/2009.)

Abstract. We first find linear fractional solutions to Abel--Schröder functional equations in Hilbert spaces and then use our results and spiral-like mappings to study the Koenigs embedding property for certain linear fractional transformations.

AMS Subject Classification (1991): 30C45, 46G20, 47A60, 47H06, 47H20

Received October 17, 2001, and in final form August 14, 2002. (Registered under 2885/2009.)

Abstract. For $p>1$, the class $N^p$, introduced by I. I. Privalov with the notation $A_q$ in [11], is defined as the space of analytic functions $f$ on the open unit disk $D$ in ${\bf C}$ for which $(\log ^ +| f(z)| )^p$ has a harmonic majorant on $D$. However, the space $N^p$ can be expressed as a union of certain weighted $H^2$ spaces and it is given a locally convex topology ${\cal H}_p$ as the inductive limit of these spaces. We note that this topology coincides with the Mackey topology (strongest locally convex topology yielding the same dual) on $N^p$. We then consider the individual spaces $H^2(w)$, where $w$ is a positive function on the unit circle such that $w$ and $| \log w| ^p$ are summable. Our results are in fact generalizations of those obtained by J. E. McCarthy in [8] for the case $p=1$. In particular, we give asymptotic versions of Szegő's theorem and the Helson--Szegő theorem on these spaces, and characterize the universal multipliers of their duals.

AMS Subject Classification (1991): 30D55, 46J15

Received June 25, 2001. (Registered under 2886/2009.)

Abstract. The main result in this paper is the following: Let $f=u+iv\in H(U)$ with $v(0)=0,$ $p\in[1,\infty )$, $n\in{\msbm N}$ and $\omega $ is an admissible weight with distortion function $\psi.$ Then there are constants $C=C(p,q,n) $ (resp. $C(p,q,n,\omega )$) such that $${\rm(a)} ||u||^q_{\omega,p,q} \leq C\left( |u(0)|^q+\int_0^1M_p^q(f',s)\psi ^q(s)\omega(s)ds\right ) \mbox{ for } q\in[1,\infty );$$ $${\rm(b)} |u(0)|^q+\int_0^1M_p^q(f^{(n)},s)\psi ^{nq}(s)\omega(s)ds \leq C ||u||^q_{\omega,p,q} \mbox{ for } q\in(0,\infty ).$$

AMS Subject Classification (1991): 31A05, 30E99

Keyword(s): Distortion function, admissible weight, integral means, unit disk, conjugate function, harmonic function

Received September 20, 2001, and in revised form April 10, 2002. (Registered under 2887/2009.)

Abstract. In order to solve the identifiability problem for the partial functional differential equation: $$ u_t(t,x) = u_{xx} (t,x)+b u(t,x) +c u_{xx} (t-h,x) + \int_{-h}^0 a(s) u_{xx} (t+s,x) ds $$ we consider the abstract functional differential equation $u'(t)=A u(t)+B u(t)+L(u_t)$ in the Hilbert space $X$. We study eigenspaces of the infinitesimal generators of the solution and the adjoint semigroups. By using structural operators, eigenspaces of the generator of adjoint semigroup are characterized, and the identifiability of parameters for a given partial functional differential equation is obtained.

AMS Subject Classification (1991): 34G10, 34K30, 47D03

Received June 27, 2001, and in revised form March 28, 2002. (Registered under 2888/2009.)

Abstract. We apply recently obtained results on the hyperbolicity of semigroups to delay equations. The main tool is the theory of operator valued Fourier multipliers.

AMS Subject Classification (1991): 34K05, 34K20, 47D06

Received September 3, 2001, and in revised form April 2, 2002. (Registered under 2889/2009.)

Abstract. Most previous rate of convergence comparisons are based on the ordinary limit of the quotient of two null sequences: $x$ converges faster than $z$ provided that $\lim x_{n}/z_{n}=0$. In the present work the quotient limit is weakened to a statistical limit: $x$ converges (stat) faster than $z$ if there is a subset $P\subseteq{\msbm N}$ of natural density one such that $\lim_{n\in P}x_{n}/z_{n}=0$. This study extends results of Bajraktarević and Miller to give conditions on a collection ${\cal A}$ of nonvanishing null sequences that characterize the existence of a nonvanishing null sequence that converges (stat) faster -- or slower -- than each sequence in ${\cal A}$. Other results show when ${\cal A}$ admits a sequence that is statistically completely incomparable to each sequence in ${\cal A}$.

AMS Subject Classification (1991): 40A05, 40C05

Received August 21, 2001. (Registered under 2890/2009.)

Abstract. The authors estimate the error of best polynomial approximation in $L^p$-weighted spaces with weights having zeros inside $[-1,1],$ using a suitable modulus of smoothness. The Jackson and Stechkin inequalities are given. Moreover some estimates of the derivatives of the polynomials of best approximation are proved.

AMS Subject Classification (1991): 41A10, 41A17, 41A27

Keyword(s): Polynomial Approximation, doubling weight, modulus of smoothness, K-functional

Received October 1, 2001, and in revised form July 22, 2002. (Registered under 2891/2009.)

Abstract. In this paper, we get a necessary and sufficient condition for sine series with monotonically decreasing cofficients to belong to some Orlicz space. Moreover, we have analogue results for cosine series.

AMS Subject Classification (1991): 42A32, 46E30

Keyword(s): Trigonometric series, Orlicz space

Received July 27, 2001, and in revised form November 22, 2001. (Registered under 2892/2009.)

Abstract. By the observation that every almost monotone non-increasing sequence $\{a_n\} $ has the same convergence and divergence property as the monotone non-increasing sequence $\{\alpha_n\}, \alpha_n:=\sup_{k\ge n}a_k,$ we can slightly weaken the assumptions on the coefficients of Tandori's divergence theorems from monotonicity to almost monotonicity.

AMS Subject Classification (1991): 42C15, 42C05

Received January 20, 2003. (Registered under 2893/2009.)

Abstract. We formalize and discuss a concept of mean value on a locally compact abelian group. We show that most of the classical notions of a mean value arising in mathematical analysis (e.g., integration on a compact group, the mean value for periodic functions, the mean value for almost periodic functions, etc.) are only various particular expressions of a great unity.

AMS Subject Classification (1991): 43A07; 28A25, 28C10

Keyword(s): Mean value, locally compact abelian groups

Received October 17, 2001, and in revised form October 7, 2002. (Registered under 2894/2009.)

Abstract. In this paper we analyse the compressed shift operators belonging to various subspaces arising in the theory of stochastic realizations without reconstructing the whole picture used in the analysis of stationary processes. Let $H$ be a separable Hilbert space with a bilateral shift operator $U$ of finite multiplicity. Assume that $H=H_1\oplus H_2$, where $H_1$ and $H_2$ are double-invariant subspaces. Let $S\subset H$ be a subspace for which $U^{-1}S\subset S$. We show that under some weak regularity conditions the compressed shift operators defined on the subspaces $\Pr_{H_1}S\ominus\left (S\cap H_1\right )$ and $\Pr_{H_2}S\ominus\left (S\cap H_2\right )$ are quasi-similar. Using Hardy space approach and working directly with inner functions and spectral factors under some more restricted circumstances this result was proven by P. Fuhrmann and A. Gombani [2].

AMS Subject Classification (1991): 45C07, 47A15, 47A40, 60G10, 93E08

Keyword(s): compressed shift operators, quasi-similarity, stochastic realization theory

Received June 8, 2001, and in final form June 5, 2002. (Registered under 2895/2009.)

Abstract. We continue the investigations of the indefinite generalization of L. de Branges theory of Hilbert spaces of entire functions. In this paper we are concerned with the detailed study of degenerated dB-subspaces of a dB-Pontryagin space, and of singularities of maximal chains of matrix functions. These phenomena are typical for the indefinite situation; there are no definite analogues. The main theorem is a continuity result for so-called intermediate Weyl coefficients. As a basic tool we introduce and investigate a certain transformation of maximal chains of matrix functions.

AMS Subject Classification (1991): 46C20, 30H05, 47B50, 47A45

Keyword(s): 46C20, 30H05, 47B50, 47A45

Received August 6, 2001. (Registered under 2896/2009.)

Abstract. We are interested in computing the $\rho $-numerical radius of the truncated shift $S_n$ on $l_2^n$. This value, first computed for $\rho =2$ by Haagerup and de la Harpe, appears as a reference in constrained von Neumann inequalities. In this paper, we give an effective way of computing this value in the general case, and then some exact formulas in particular cases.

AMS Subject Classification (1991): 47A12, 15A60, 15-04; 47A63

Received November 12, 2001, and in revised form September 13, 2002. (Registered under 2897/2009.)

Abstract. We give domain and range characterization of the smallest (Krein--von Neumann) positive self-adjoint extension of Hilbert space operators. There are given new proofs of these facts for the largest (Friedrichs) extension as well. Boundedly or compactly invertible extremal extensions are also characterized as a result of an application of our treatment.

AMS Subject Classification (1991): 47A20, 47B25

Keyword(s): Positive operator, Friedrichs extension, Krein--von Neumann extension, invertible extension

Received October 3, 2001. (Registered under 2898/2009.)

Abstract. The aim of this paper is to settle some inaccuracies occurring in the monograph [NF] in connection with factorizations of operator-valued functions.

AMS Subject Classification (1991): 47A45, 47A15

Received September 17, 2002. (Registered under 2899/2009.)

Abstract. In this paper results are proved concerning the form of generalized inverses or Moore-Penrose inverses of operators in certain algebras of bounded operators. A number of applications to operator theory are given.

AMS Subject Classification (1991): 47A53, 47C05, 47D30

Keyword(s): integral operator, Fredholm inverse, g-inverse, Moore-Penrose inverse

Received July 5, 2001. (Registered under 2900/2009.)

Abstract. For a bounded linear operator $T$ acting on a Banach space let $\sigma_{SBF_{+}^-}(T)$ be the set of all $\lambda\in {\msbm C}$ such that $T-\lambda I$ is upper semi-$B$-Fredholm and ${\rm ind} ({T- \lambda I}) \leq0$, and let $ E^a(T)$ be the set of all isolated eigenvalues of $T$ in the approximate point spectrum $ \sigma_{a}(T)$ of $T$. We say that $T$ satisfies generalized $a$-Weyl's theorem if $\sigma_{SBF_{+}^-}(T)= \sigma_{a}(T) \setminus E^a(T)$. Among other things, we show in this paper that if $T$ satisfies generalized $a$-Weyl's theorem, then it also satisfies generalized Weyl's theorem $\sigma_{BW}(T) = \sigma(T) \setminus E(T)$, where $ \sigma_{BW}(T)$ is the $B$-Weyl spectrum of $T$ and $E(T)$ is the set of all eigenvalues of $T$ which are isolated in the spectrum of $T$.

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

Keyword(s): semi-Fredholm operator, quasi-Fredholm operator, B, semi--Fredholm operator, Weyl's theorem, a, -Weyl's theorem, a, generalized-Weyl's theorem

Received July 27, 2001, and in revised form December 12, 2001. (Registered under 2901/2009.)

Abstract. Let $A$ and $B$ be positive operators on a Hilbert space, and $p$, $q$ and $\lambda $ be positive numbers. Ito and Yamazaki showed relations between the two operator inequalities $(B^{r\over2}A^pB^{r\over2})^{r\over p+r} \ge B^r$ and $A^p \ge(A^{p\over2}B^rA^{p\over2})^{p\over p+r}$. We shall show parallel relations between the two weaker inequalities $$ {rB^{r\over2}A^pB^{r\over2}+p\lambda ^{p+r}I\over(p+r)\lambda ^p}\ge B^r \hbox{ and } A^p \ge{(p+r)\lambda ^pA^{p\over2}B^rA^{p\over2}\over rA^{p\over2}B^rA^{p\over2}+p\lambda ^{p+r}I}, $$ which can be obtained by applying the arithmetic--geometric--harmonic mean inequality to the inequalities studied by Ito and Yamazaki. As an application of these relations, we shall show {\it ``an operator $T$ is normal if $T$ and $T^*$ are paranormal,''} which is an extension of Ando's result, that is, he showed the same result with the additional kernel condition $N(T)=N(T^*)$. We shall also show an extension of a result on normality of $w$-hyponormal operators via Aluthge transformation by Chō, Huruya and Kim.

AMS Subject Classification (1991): 47A63, 47B20

Received July 20, 2001. (Registered under 2902/2009.)

Abstract. Let $B$ denote the open unit ball in ${\msbm R}^n$. We study Hankel operators $H_f$ on harmonic Bergman spaces $L^p_h(B)$ for $1< p< \infty $. We obtain a necessary and sufficient condition for $H_f$ to be bounded or compact on both $L^p_h(B)$ and its dual space. In particular, a necessary and sufficient condition for $H_f$ to be bounded or compact on $L^2_h(B)$ is $f\in\mathop{\rm BMO} ^2$ or $f\in\mathop{\rm VMO} ^2$, respectively. The results of this paper extend those in [9] and [11] and are real-variable analogue of those in [4] and [8].

AMS Subject Classification (1991): 47B35; 47B32, 47B47

Received January 16, 2002, and in revised form August 30, 2002. (Registered under 2903/2009.)

Abstract. The Hausdorff operators defined by suitable signed measures on $R=(-\infty,\infty )$ are shown to be bounded on $L^p(R^n)$, on the real Hardy space $H^1(R^n)$, and on the space of bounded mean oscillation $BMO(R^n)$. An example is given that negatively resolves a related conjecture of Móricz.

AMS Subject Classification (1991): 47B38, 46A30

Keyword(s): Fourier transform, Riesz transforms, Hausdorff operator, Ces?ro operator, Lebesgue spaces, Hardy Spaces, Bounded Mean Oscillation

Received December 14, 2001, and in revised form June 6, 2002. (Registered under 2904/2009.)

Abstract. A representation of the Baer subgeometries of $PG(n,q^2)$ in $PG(2n+1,q)$ is given. Using this representation the possible intersection configurations of two Baer subgeometries are determined.

AMS Subject Classification (1991): 51E20, 05B25, 51E23

Received September 17, 2002, and in revised form April 22, 2003. (Registered under 2905/2009.)

Abstract. Let $P$ be a convex $d$-polytope in $E^d$ such that any point $x \in P$ belongs to finitely many affine diameters of $P$. Then the set of points from $P$, each belonging to an even number of affine diameters of $P$, has $d$-dimensional measure zero.

AMS Subject Classification (1991): 52A20, 52B11

Keyword(s): Antipodal faces, affine diameter, convex polytope

Received September 11, 2002, and in revised form March 6, 2003. (Registered under 2906/2009.)

Abstract. For any given sequence $k_n\to\infty $ of integers such that $k_n/n\to0$, we derive a rate at which the distribution functions of suitably centered and normed sums of the $k_n$ largest winnings in $n$ generalized St.Petersburg games merge together uniformly with that sequence of semistable infinitely divisible distribution functions that approximate well the corresponding distribution functions of the total gains in $n$ games. The rate depends on $n$, $k_n$ and the tail parameter of the underlying game.

AMS Subject Classification (1991): 60F05; 60E07, 60G50

Received July 23, 2002. (Registered under 2907/2009.)