ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. This is an attempt to create a partial elementary logic system which would provide a basis for the construction of cylindric algebras describing properties of partial relations. The intended link between such partial logic and the corresponding cylindric algebra theory should be analogous to that which relates classic elementary logic to "classic" cylindric algebras. After introducing basic notions and elementary properties of the system, we prove a completeness theorem. The proof is based on a standard idea, even if its realization is non-standard due to the partiality of relations in models. The approach presented in this paper is, in some model-theoretic sense, a generalization of the presentation of classic relational systems and algebras (partial or total) in the first order language. Some arguments concerning the last statement are contained in the series of informal remarks of model-theoretic and categorical nature, carried on paralelly to the main stream of the work.

AMS Subject Classification (1991): 03B60, 03C07, 03C35, 03C52, 18A15

Received April 3, 1995 and in revised form February 11, 1997. (Registered under 6101/2009.)

Abstract. An algebra is characteristically simple if it has no nontrivial congruence relations preserved by all automorphisms. We investigate and classify the finite characteristically simple algebras. From the main results, among others, we derive an explicit description of the finite characteristically simple algebras with transitive automorphism groups.

AMS Subject Classification (1991): 08A40

Received September 13, 1995, in final form January 6, 1996. (Registered under 6102/2009.)

Abstract. Let $G$ be a $p$-group and denote by $E(G)$ the set of all elements $g\in\Delta (G)$ such that $g$ has infinite height in the centre of the centralizer $C_G(g)$ of $g$. Under the assumption that $E(G)$ is a subgroup, we describe the first Ulm-subgroup and the maximal divisible subgroup of the centre of the unit group of the group algebra of a $p$-group over a field of characteristic $p$. Every nilpotent and $FC$-group satisfies these properties.

AMS Subject Classification (1991): 16S34, 16U60, 20C07

Keyword(s): group algebra, unit group, divisible group

Received February 14, 1996 and in revised form October 18, 1996. (Registered under 6103/2009.)

Abstract. Let ${p_{mn}}$ be a nonnegative double sequence with $p_{00}>0$ such that $(1,1)$ lies on the boundary of the convergence set of the power series (2.1), and (2.2) holds. The double sequence ${s_{mn}}$ is said to be $bJ_{p}$-limitable to $\sigma $ (in short $bJ_{p^-}\lim s_{mn} = \sigma$) if the power series (2.3) converges for $0< x,y< 1$ and $p_{s}(x,y)/p(x,y)$ is bounded and converges to $\sigma $ as $x,y \to1-$. We show (for example Satz 4.1): If $J_{p}$ is $b$-regular and ${p_{m-i, n-j}/p_{mn}}$ is a double moment sequence, with $i,j\in{-1, 0, 1}$ fixed, then $bJ_{p^-}\lim s_{mn}= \sigma $ implies $$bJ_{p^-}\lim s_{m+i, n+j} = \sigma\cdot \lim(p_{m-i, n-j}/p_{mn}).$$ This result is applied to logarithmic and generalized Abel methods.

AMS Subject Classification (1991): 40B05, 40C15

Received December 5, 1995. (Registered under 6104/2009.)

Abstract. The extremal problem $$\min_{-1\le x_1<\cdots< x_n\le1}\max_{1\le i\le n-1} \Sigma_{k=1}^i(x_{i+1}-x_k)^{-p}$$ is investigated. Its exact lower bounds are provided and the equioscillation characterizations of Bernstein and Erdős of its solution are given.

AMS Subject Classification (1991): 41A05

Received August 30, 1996. (Registered under 6105/2009.)

Abstract. Our main result is a generalization of a Paley type inequality. Namely, the estimation $$\big(\sum_{n=0}^\infty m_n^{1-2/p}M_n^{2-2/p} \sum_{j=1}^{m_n-1}|{\hat f}(jM_n)|^2\big )^{1/2} \leq C_p\|f\|_{H^p_{**}} (*)$$ is proved for martingales $f\in H^p_{**}(G_m)$ ($0< p\leq1$), where $\hat f(.)$ denote the Vilenkin--Fourier coefficients of $f$ and the Hardy space $H^p_{**}(G_m)$ is defined by means of a maximal function. <br /> We formulate the dual inequalities of $(*)$ and its variant for other Hardy- and ${\cal BMO}$-spaces, too. The so called ``bounded case" is also investigated, specially as a corollary we get the known generalization of Khinchin's inequality in this case.

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

Received March 8, 1996 and in revised form November 11, 1996. (Registered under 6106/2009.)

Abstract. We consider summability of orthogonal series (OS) $\sum c_nf_n(x)$, $x\in[0,1]$, a.e. by methods of class $\{ (\varphi,\lambda )\}$, which includes many concrete methods of summability. Assuming that $\sum c_n^2l^2(n)< \infty,$ $0< l(n)\uparrow\infty $, we obtain estimates of the rate of convergence to $0$ of the deviation $|f(x)-\sigma(x,u)|$ in terms of the rate of increase of $l(n)$; here $f$ is the sum of the OS in $L^2[0,1]$, and $\sigma(x,u)$ are the $(\varphi,\lambda )$--means of the OS.

AMS Subject Classification (1991): 42C15

Received May 10, 1996 and in revised form September 30, 1996. (Registered under 6107/2009.)

Abstract. Bretagnolle et al. [4] showed that for $1\leq p\leq2$, a normed real vector space $(E,\|\cdot\|)$ can be embedded in an $L^p$-space if and only if $\|\cdot\|^p$ is negative definite. Recently, the author proved that the requirement that $\|\cdot\|$ be a norm can be replaced with the conditions that $\|tx\|=|t| \|x\|$ for $t\in{\msbm R}$ and $x\in E$ and $\|x\|>0$ for $x\in E\setminus\{0\} $, and that the result so modified holds whenever $0< p< 2$. Zoltán Sasvári then showed that for any $p>0$ which is not an even integer, $(E,\|\cdot\|)$ can be embedded in an $L^p$-space if and only if $\|\cdot\|$ is continuous on finite-dimensional linear subspaces of $E$ and $(-1)^k\|\cdot\|^p\in P(1,k)$ (see definitions below) where $k=\lceil p/2\rceil $. Finally, the author found a necessary and sufficient condition for the case $p=2k$, and proved that it is not necessary to assume continuity on finite-dimensional linear subspaces.

AMS Subject Classification (1991): 28-XX, 43A35, 46-XX, 51-XX

Received July 15, 1996. (Registered under 6108/2009.)

Abstract. We study decompositions of a Banach space operator $T$ in the form $T=K+Q$, where $K$ is compact (or meromorphic) and where the spectrum of $Q$ is contained in the set of all accumulation points of the spectrum of $T$. Many known decomposition results of this type are subsumed in our construction. We prove that every Hilbert space operator has the meromorphic decomposition, and obtain an improvement of a result of Laurie and Radjavi on the West decomposition of Riesz operators in a Banach space.

AMS Subject Classification (1991): 47A10, 47A65, 47B06, 47B99

Received August 21, 1996 and in revised form January 15, 1997. (Registered under 6109/2009.)

Abstract. Let $T$ be a bounded linear operator on a complex Banach space $X$. We establish that if $T$ is supercyclic and $(\|T^n\|)$ is a bounded sequence then $(T^nx)$ converges to $0$ for each $x\in X$, which generalizes a result by V.~Matache. As an application, we show that a composition operator on the Hardy space of the open unit disk $U$ is not supercyclic whenever it is induced by a mapping fixing a point in $U$. We then turn our attention to powers of cyclic operators. We prove that if $T^n$ is cyclic for each positive integer $n$, then the set of cyclic vectors for $T$ is dense in $X$. We also discuss simultaneous cyclicity of $T$ and $T^{-1}$.

AMS Subject Classification (1991): 47A15, 47B38

Received September 27, 1996. (Registered under 6110/2009.)

Abstract. It is well known that every Boolean algebra of projections (on a Banach space) which is complete (resp. $\sigma $-complete), in the sense of W. Bade [1], is the range of some spectral measure (eg. on the Borel (resp. Baire) sets of its Stone space). For {\it equicontinuous} Boolean algebras which are complete or $\sigma $-complete the same is true in the setting of locally convex spaces. However, equicontinuity is unduly restrictive in practice. In this note we characterize precisely those Boolean algebras of projections acting in a (general) locally convex space which are the range of some spectral measure, thereby describing completely the intimate and subtle connection between Boolean algebras of projections and spectral measures.

AMS Subject Classification (1991): 47A67, 47B15

Received March 18, 1996. (Registered under 6111/2009.)

Abstract. We study several properties of composition operators acting on the Dirichlet space, and related weighted Dirichlet spaces, in the disk. In particular we consider the question of when $C_{\varphi }$ has closed range on these spaces.

AMS Subject Classification (1991): 47B38

Received June 3, 1996 and in revised form November 12, 1996. (Registered under 6112/2009.)

Abstract. $AC$-operators were introduced by Berkson and Gillespie as a generalization in the context of well-boundedness of normal operators on Hilbert space. In this paper we explore some of the properties of these operators, such as the uniqueness of their splitting into real and imaginary parts, and their interpolation properties. We also examine the interpolation properties of the important subclass consisting of the trigonometrically well-bounded operators.

AMS Subject Classification (1991): 47B40, 46B70

Keyword(s): AC, -operators, trigonometrically well-bounded operators, functional calculus, interpolation properties

Received March 29, 1996. (Registered under 6113/2009.)

Abstract. In [11, Theorem 3.2], P. J. Maher has shown that, if $N$ is normal and $T\in{\rm Ker}(\delta_N|C_p)$, then $\| T-\delta_N(X) \|_p\ge\| T\|_p$ for any $X\in L(H)$. In the first part, we generalize P. J. Maher's results. We also generalize P. J. Maher's results by minimizing the map $X\to\| T-(AX-XB)\|_p^p$, $1\le p< \infty$ and classifying its critical points.

AMS Subject Classification (1991): 47B47, 47B10

Keyword(s): commutator approximation, generalized derivation, Fuglede-Putnam theorem

Received March 30, 1995 and in revised form April 3, 1996. (Registered under 6114/2009.)

Abstract. Following R. Fefferman, the Hardy space ${\cal H}^1({\msbm R}\times{\msbm R})$ of functions $f\in L^1({\msbm R}^2)$ is defined by the requirement that its Hilbert transforms $H_1f, H_2f$, and $H_1H_2f$ also belong to $L^1({\msbm R}^2)$. The proof of the statement claimed in the title relies on the closed graph theorem and on the fact that if a function $f\in L^1({\msbm R}^2)$ is such that its Fourier transform ${\hat f}(u,v)=0$ unless $u\ge0, v\ge0$, then $f\in{\cal H}^1 ({\msbm R}\times{\msbm R})$. The following reversed statement is also proved: If $f\in{\cal H}^1 ({\msbm R}\times{\msbm R})$, then $f$ can be represented in the form $f=f_1+f_2+f_3+f_4$, where each $f_j\in{\cal H}^1({\msbm R}\times{\msbm R})$ and ${\hat f}_1(u,v)=0$ unless $u\ge0, v\ge0$; ${\hat f}_2(u,v)=0$ unless $u\le0, v\ge0$; ${\hat f}_3(u,v)=0$ unless $u\ge0, v\le0$; and ${\hat f}_4(u,v)=0$ unless $u\le0, v\le0$.

AMS Subject Classification (1991): 47D05

Keyword(s): Hilbert transforms, ${\cal H}^1({\msbm R}\times{\msbm R})$, Hardy space, Fourier transform, Cesàro operator, closed graph theorem

Received September 6, 1996. (Registered under 6115/2009.)

Abstract. Let $A$ be a non-empty approximately $p$-compact convex subset and $B$ be a non-empty closed convex subset of a Hausdorff locally convex topological vector space with a continuous semi-norm $p$. Given a multifunction $T\colon A\to2^B$ and a single valued function $g\colon A\to A$, a best proximity pair theorem which provides sufficient conditions ascertaining the existence of an element $x_0\in A$ such that $d_p(Tx_0,gx_0)=d_p(A,B)$ is established. In the setting of normed linear spaces, this theorem reduces to a fixed point theorem for multifunctions if $T$ is a self-multifunction on $A$ and $g$ is the identity function on $A$. Further, a best approximation theorem is proved for continuous Kakutani factorizable multifunctions which are not necessarily convex valued.

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

Keyword(s): Kakutani factorizable multifunction, Best proximity pair, Best approximation, Pseudo affinity, Quasi affinity

Received November 23, 1996. (Registered under 6116/2009.)

Abstract. Consider an i.i.d. sequence $\{\zeta_i\} _{i=-\infty }^\infty $, a sequence of real numbers $\{b_i\} _{i\geq0}$ and the pertaining infinite-order moving average $\epsilon_i=\sum_{j\leq i}b_{i-j}\zeta_j$, $i=1,2,\ldots $. Under conditions on $\{b_i\} $ which entail that $\{\epsilon_i\} $ is either long-range or short-range dependent, we study the partial-sum process $S_n(t)=\sum_{i=1}^{\lfloor nt\rfloor }\epsilon_i$, $t\geq0$. For $0< b< \infty $, $k\in{\msbm N}$, a suitable norming sequence $\{a_n\} $ and sequences of gap-lengths $l_{1,n}, \ldots, l_{k,n}$ such that $l_{1,n}\to\infty $ and $l_{j,n} - l_{j-1,n}\to\infty $, $j = 2, \ldots, k$, we prove in the first case that the vector process $a_n(S_n(t_0), S_n(l_{1,n}+t_1)-S_n(l_{1,n}), \ldots, S_n(l_{k,n}+t_k)-S_n(l_{k,n}))$, $0\leq t_0,\ldots,t_k\leq b$, converges in distribution in ${\cal D}[0,b]^{k+1}$ to the vector of ${k+1}$ independent fractional Brownian motions. The result is then generalized to the case when $\{\epsilon_i\} $ is replaced by values of an $m$-th Appell polynomial $\{P_m(\epsilon_i)\},m\in{\msbm N}.$ In the short-range dependence case $$n^{-1/2}(S_n(t_0), S_n(l_{1,n}+t_1)-S_n(l_{1,n}), \ldots, S_n(l_{k,n}+t_k)-S_n(l_{k,n}))$$ converges in distribution to the vector of $k+1$ independent Wiener processes, provided $l_{j,n}-l_{j-1,n}\geq b+\gamma $ for some $\gamma >0$, $j=1,\ldots,k$. As applications, in both cases we determine the asymptotic behaviour of the finite-dimensional distributions of kernel estimators in the fixed-design regression model with errors $\epsilon_i$. The results parallel those of Csörgő and Mielniczuk [3], [4] when $\{\epsilon_i\} $ is an instantaneous transformation of a Gaussian sequence.

AMS Subject Classification (1991): 60F05, 62G07

Received December 10, 1996 and in revised form December 30, 1996. (Registered under 6117/2009.)