ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We examine the minimal distance (number of differing entries) between different group tables of the same order $n$. Here group table means a matrix of order $n$ with entries from a fixed set of $n$ symbols, which (with suitable border elements) is the multiplication table of a group. (The border elements are not considered part of the table. A group is defined up to isomorphism by its multiplication table without border elements.) With the exception of some pairs of groups of orders $4$ and $6$, which are listed explicitly, different group tables of order $n$ differ in at least $2n$ places; and with the exception of some pairs of groups of orders $4$, $6$, $8$ and $9$, which are listed explicitly, tables of non-isomorphic groups of order $n$ always differ in strictly more than $2n$ places.

AMS Subject Classification (1991): 05B15, 20N05, 20A99

Received October 14, 1994 and in revised form September 5, 1996. (Registered under 5769/2009.)

Abstract. The purpose of this paper is to prove the following criterion for distributivity: Let $L$ be a lattice which has no infinite chains. Then $L$ is distributive if and only if any ideal $I$ of $L$ is equal to its second meander $I^2$.

AMS Subject Classification (1991): 06B10, 06B99, 06D99

Received January 27, 1997 and in revised form June 2, 1997. (Registered under 5770/2009.)

Abstract. In this paper we establish some equivalent conditions for a $C$-lattice to be an almost discrete valuation lattice. We characterize weak invertible lattices (or {\it WI}-lattices) in terms of Baer lattices and quasiregular lattices. We give some equivalent conditions for a $C$-lattice to be a Dedekind lattice.

AMS Subject Classification (1991): 06F10, 06F05, 06F99, 13A15

Keyword(s): multiplicative lattice, Noether latttice, principal element, Dedekind, invertible

Received February 20, 1997 and in revised form April 28, 1997. (Registered under 5771/2009.)

Abstract. Algebras having only trivial subalgebras and reducts are determined up to equivalence. They are simple, and their clones are simple, too; thus, in a sense, they are the smallest algebras.

AMS Subject Classification (1991): 08A30, 08A40

Received January 27, 1997. (Registered under 5772/2009.)

Abstract. In 1967 Marchenko and Pastur studied the limit of the eigenvalue distribution of the sum of $p(n)$ rank one random projections in the $n$ dimensional space when $p(n)/n \to a$ as $n\to\infty $. More recently this Marchenko--Pastur distribution occured in the free analogue of the Poisson limit theorem. In this paper we derive a recursive as well as an explicite formula for the moments of the Marchenko--Pastur distribution which turn out to be polynomials of $a$. Moreover, an elementary combinatorial proof is given to the known fact that a variant of the Marchenko--Pastur distribution describes the asymptotical eigenvalue density of sample covariance matrices.

AMS Subject Classification (1991): 15A52, 60F05, 60H25

Received November 18, 1996 and in revised form May 5, 1997. (Registered under 5773/2009.)

Abstract. The character groups of the Butler groups are investigated by making use of the Pontrjagin--duality.

AMS Subject Classification (1991): 20K15, 20K45, 22B05

Received August 30, 1996. (Registered under 5774/2009.)

Abstract. We find a multiplication on the lattice of existence varieties of locally inverse semigroups which extends the classical multiplication of group varieties and the recently found multiplication of inverse semigroup varieties. As a special case we get an associative multiplication on the lattice of varieties of completely simple semigroups.

AMS Subject Classification (1991): 20M07, 08B20

Received April 11, 1996. (Registered under 5775/2009.)

Abstract. We describe semigroup varieties, on whose relatively free members either all fully invariant congruences or all fully invariant congruences contained in the least semilattice congruence permute.

AMS Subject Classification (1991): 20M07

Received January 21, 1997. (Registered under 5776/2009.)

Abstract. Let $S$ be a finite commutative groupfree semigroup. Then $H^2(S,A) = 0$ for all functors $A$ if and only if $S$ is a semilattice.

AMS Subject Classification (1991): 20M50, 20M14

Received March 30, 1995. (Registered under 5777/2009.)

Abstract. Hardy [2] characterized summability $(C, 1)$ of a single series of complex numbers by the ordinary convergence of another series. We extend this result to double series, while convergence in Pringsheim's sense should be replaced by regular convergence, introduced also by Hardy [1] and rediscovered by the first named author [3] (see also [5]). We also present an example showing that the assumption of regularity is indispensable for the validity of our extension.

AMS Subject Classification (1991): 40B05, 40G05

Keyword(s): Double series and sequences, convergence in Pringsheim's sense, regular convergence, (C, summability, 1, 1), (C, 1, 0), (C, and, 0, 1), regular summability, Cauchy convergence principle, double summation by parts

Received January 31, 1997. (Registered under 5778/2009.)

Abstract. This paper shows that optimal Lebesgue function type sum $\Lambda_n(\chi ^*,x)$ must equioscillate and gives the optimal set of nodes and the exact value of $\lambda_n(s)$.

AMS Subject Classification (1991): 41A05

Received October 25, 1996 and in revised form May 7, 1997. (Registered under 5779/2009.)

Abstract. The aim of this paper is to offer representation of bilinear forms defined either on ${\rm C}(T)$ or on ${\rm L}^\infty(T)$ and satisfying an a priori estimate. These representation theorems turn out to be useful in describing the generalized Hessians of $C^{1,1}$ functionals that are defined on the above spaces.

AMS Subject Classification (1991): 28A99, 46A22, 49J52

Keyword(s): Bilinear forms, representation theory, generalized Hessians

Received November 28, 1996. (Registered under 5780/2009.)

Abstract. On the unit ball of an Orlicz function space the (weak star) denting points and (weak star) quasi-denting points coincide. But in Orlicz sequence space the (weak star) quasi-denting points are different from (weak star) denting points. We also show that in Orlitz space the weak star drop property and the Kadec-Klee property are equivalent. Hence the weak drop property and the weak star drop property are independent in Banach spaces.

AMS Subject Classification (1991): 46B20, 46E30

Received August 6, 1996 and in revised form March 7, 1997. (Registered under 5781/2009.)

Abstract. In this paper, some criteria of weakly exposed points and weakly* exposed points on the unit sphere of Orlicz sequence spaces are obtained.

AMS Subject Classification (1991): 46B20, 46E30

Keyword(s): Orlicz sequence space, weakly exposed point, weakly* exposed point

Received September 3, 1996. (Registered under 5782/2009.)

Abstract. It is known from the Sz.-Nagy--Foias theory of operators that if $T$ is a Hilbert space contraction of class $C_{1\bullet }$ and if the unitary spectrum $\sigma(T)\cap{\partial }{\bf D}$ is of Lebesgue measure zero, then $T$ is a singular unitary operator. We extend this statement to polynomially bounded operators acting on arbitrary Banach spaces, presenting also its local version. It is shown how the method applied provides Katznelson--Tzafriri type theorems. One-parameter semigroups of Hilbert space contractions are also considered.

AMS Subject Classification (1991): 47A11, 47A10, 47D03

Received November 22, 1996 and in revised form May 5, 1997. (Registered under 5783/2009.)

Abstract. In this paper we study the situation in which the numerical range of the generalized derivation coincides with the convex hull of its spectrum. We also turn our attention to the range of the inner derivation.

AMS Subject Classification (1991): 47B47

Received November 4, 1996. (Registered under 5784/2009.)

Abstract. Applying appropriate normalizing gauge functions and using the concept of almost convergence, results connected with asymptotic behaviour of power bounded operators are extended to a large class of operators.

AMS Subject Classification (1991): 47A15, 47A10, 47A20

Received December 10, 1996 and in revised form February 12, 1997. (Registered under 5785/2009.)

Abstract. We show that the set $\cup_{j\in J}F^n_j $ is arcwise connected, where $ J$ is a subset of $ \overline{\msbm N} $ and $ F^n_j (n\leq j)$ is the set of semi-Fredholm operators of index $n $ and kernel dimension $ j$. The distance of an arbitray operator to $\cup_{j\in J}F^n_j$ is also determined. We show that dist($T,\cup_{j\in J}F^n_j )= $ dist$ (T, F^n_{j_0})$, where $ j_0 =$ inf$\{j: j \in J\} $. Thus $ F^n_{j_0} $ is dense in $\cup_{j\in J}F^n_j$. We also find the boundary of $\cup_{j\in J}F^n_j$ and $ \overline{\cup_{j\in J}F^n_j } $.

AMS Subject Classification (1991): 47A53

Received March 10, 1997. (Registered under 5786/2009.)

Abstract. Let ${\cal H}(p)$ (resp. ${\cal H}U(p)$), $0< p< 1/2$, denote the class of $p$-hyponormal operators (resp. $p$-hyponormal operators with equal defect and nullity) on a Hilbert space $H$. Given $A\in{\cal H}(p)$, define $\hat A$ and $\tilde A$ by $\hat A=| A|^{1/2}U| A|^{1/2}$ and $\tilde A=|\hat A|^{1/2}V|\hat A|^{1/2}$, where $U$ and $V$ are as in the polar decompositions $A=U| A|$ and $\hat A=V|\hat A|$. $\tilde A$ is then hyponormal with $\sigma(|\tilde A|)=\sigma(| A|)$ and, if $A\in{\cal H} U(p)$, $\sigma_e(|\tilde A|)=\sigma_e(| A|)$. We use this to prove that ${\cal H}(p)$ operators share a number of spectral properties with hyponormal operators. It is shown that $f(A)$, $A\in{\cal H}U(p)$ and $f$ analytic on $\sigma(A)$, satisfies Weyl's theorem, and $$\| |A|^{2p}-|A^*|^{2p}\|\le\min\big\{\frac p\pi\int_{\sigma(A)}r^{2p-1}dr d\theta, \frac1{\pi^p}\big(\int_{\sigma(A)}rdrd\theta\big)^p\big\}$$ for $A\in{\cal H}(p)$. Also, if an $A\in{\cal H}(p)$ is the sum of a normal and a Hilbert-Schmidt operator, then $A$ is normal.

AMS Subject Classification (1991): 47B20, 47A30

Keyword(s): p, -hyponormal operator, Weyl and essential spectra, area inequality

Received October 11, 1996. (Registered under 5787/2009.)

Abstract. We extend a recent result of Korenblum and Zhu on Toeplitz operators to spaces of harmonic functions in higher dimensions.

AMS Subject Classification (1991): 47B35, 44A15

Received October 28, 1996 and in revised form May 13, 1997. (Registered under 5788/2009.)

Abstract. Given a vector space of Hilbert space valued integrable functions $H$, it is of interest to know whether every scalar integrable function can be written as a pointwise scalar product of functions in $H$. We give a result of this type in which the assumptions on $H$ are rather weak. In particular, we weaken the requirement on the existence of weakly null `special' sequences in $H$. We show how the abstract factorization result can be applied to the study of certain algebras related with subnormality.

AMS Subject Classification (1991): 47D25, 47B20

Received August 12, 1996 and in revised form February 25, 1997. (Registered under 5789/2009.)

Abstract. The amplitude-phase method has been proposed recently for computing the rapidly oscillating solutions of second-order ordinary differential equations on a semi-infinite interval in [1]. In the paper error estimates are obtained for this. The method and the error estimations introduced below can be applied to a wide class of problems. The numerical solutions are evaluated by solving differential problems posed for auxiliary functions on a finite interval, only. The error estimation problem of approximations at the right end point is reduced to the estimation of certain improper integrals. By proving the Leibniz property of sequences closely related to these integrals, an a priori estimate will be given. As a direct consequence of the result, a new definition of the shifting function will be introduced which allows the simultaneous stabilization of the auxiliary functions.

AMS Subject Classification (1991): 65D20, 65L70

Received October 9, 1996. (Registered under 5790/2009.)

Abstract. Let $\{X,X_n;n\ge0\} $ be a sequence of $B$-valued independent random variables. We consider the random variables $\xi(\beta )=\sum_{n=0}^\infty\beta ^nX_n$ for $0<\beta < 1$, and establish the LIL for $\xi(\beta )$ as $\beta\nearrow 1$. In particular, in the case when $\{X,X_n;n\ge0\} $ are independent and identically distributed, we prove that the LIL for $\xi(\beta )$ holds if and only if the same result is true for $S_n$, where $S_n=X_1+\cdots +X_n$, $n\ge1$. We also prove a central limit theorem for $\xi(\beta )$.

AMS Subject Classification (1991): 60F05, 60F15

Received June 12, 1996 and in revised form May 15, 1997. (Registered under 5791/2009.)

Abstract. This note corrects the proof of Theorem 1(ii) given in [2].

AMS Subject Classification (1991): 47A67

Received June 25, 1997. (Registered under 5792/2009.)