
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

481481
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Abstract. We obtain the general form of a bijective orderpreserving map defined on the poset of all $n\times n$ idempotent matrices over a field with at least 3 elements.
AMS Subject Classification
(1991): 06A11; 15A30
Received June 2, 2003, and in revised form July 14, 2003. (Registered under 2910/2009.)
Abstract. We characterize lattices $L$ with 1 where for each element $p$ the interval $[p,1]$ is a pseudocomplemented lattice. Moreover, if for $x,y\in L$ the relative pseudocomplement $x\ast y$ exists then it is equal to the pseudocomplement of $x\vee y$ in $[y,1]$. However, the latter exists for each $x,y$ also e.g. in $N_5$ contrary to the case of relatively pseudocomplemented lattices which are distributive, see e.g. [1], [2], [3].
AMS Subject Classification
(1991): 06D15, 06D20
Keyword(s):
Relative pseudocomplement,
pseudocomplement,
semidistributive lattice
Received March 11, 2002, and in revised form May 9, 2002. (Registered under 2911/2009.)
B. N. Waphare,
Vinayak Joshi

497504

Abstract. In this paper an equivalent criterion for distributive elements in lattices is established. Standard elements and distributive elements are characterized in $SSC$ lattices. Moreover, central elements in atomistic $SSC^*$ lattices are characterized in terms of dually distributive elements and also in terms of $p$elements.
AMS Subject Classification
(1991): 06D22, 06D50, 06D99
Keyword(s):
Standard element,
distributive element,
neutral element,
p,
element
Received November 20, 2001, and in revised form March 18, 2003. (Registered under 2912/2009.)
Abstract. We show that a minimal clone has a nontrivial weakly abelian representation iff it has a nontrivial abelian representation, and that in this case all representations are weakly abelian.
AMS Subject Classification
(1991): 08A40, 20N02
Keyword(s):
clone,
minimal clone,
(weakly) abelian algebra,
groupoid
Received April 2, 2002. (Registered under 2913/2009.)
Abstract. Let $p$ be a prime, and $F(x_1,x_2,\ldots, x_n)\in{\msbm F} _p[x_1,\ldots,x_n]$ be a nonconstant polynomial such that the degree of $F$ in each variable $x_i$ is at most $p1$. The {\it rank} of $F$ is the least integer $r$ for which there exists an invertible homogeneous linear change of variables which carries $F$ into a polynomial with precisely $r$ variables. In [6] Rédei proposed the following conjecture: if the rank of $F$ is at least $\deg F$, then the equation (congruence) $F(x_1,\ldots,x_n)=0$ has a solution in ${\msbm F} _p^n$. We disprove the conjecture by giving counterexamples. On the other hand, we show that it holds for some important special cases, including generalized diagonal equations.
AMS Subject Classification
(1991): 11T06, 11D79
Keyword(s):
Finite fields,
equations,
solvability
Received July 29, 2002, and in revised form April 30, 2003. (Registered under 2914/2009.)
Shizhen Cheng,
Yongge Tian

533542

Abstract. For two given orthogonal projectors $P_A$ and $P_B$ of the same order, we investigate how to express MoorePenrose inverses of the differences $P_A  P_B$ and $P_AP_B  P_BP_A$, and then consider some related topics.
AMS Subject Classification
(1991): 15A09, 15A24
Keyword(s):
difference,
MoorePenrose inverse,
weighted MoorePenrose inverses,
orthogonal projector
Received May 28, 2002, and in revised form February 6, 2003. (Registered under 2915/2009.)
Pierre Antoine Grillet

543567

Abstract. This article defines cancellative coextensions and generalized coset extensions for commutative semigroups and studies their first properties, including their relationship to Ponizovsky families and completions.
AMS Subject Classification
(1991): 20M14
Keyword(s):
cancellative coextension,
generalized coset,
congruence,
commutative semigroup,
subelementary semigroup,
subcomplete semigroup,
Ponizovsky family,
completion
Received August 14, 2002. (Registered under 2916/2009.)
Abstract. We introduce the notion of a factorisable and an almost factorisable straight locally inverse semigroup, and prove that every straight locally inverse semigroup can be obtained as a subsemigroup of an idempotent separating homomorphic image of a Pastijn product of a semilattice by a completely simple semigroup. Moreover, we give an alternative proof of the fact that each straight locally inverse semigroup has a weakly $E$unitary cover, implicitely due to Pastijn.
AMS Subject Classification
(1991): 20M17, 20M10
Received March 19, 2002, and in revised form November 10, 2002. (Registered under 2917/2009.)
Abstract. In this paper we deal with regularity properties of functions $f$ and $g$ satisfying a functional inequality of the following type $$ f(a(x,y))f(a(x,z))\le g(b(x,y))g(b(x,z)) ((x,y),(x,z) \in D), $$ where the real valued functions $a$ and $b$ defined on an open set $D\subset{\msbm R}^2$ enjoy certain sufficiently strong regularity properties. One of the main results states that if $g$ is pointwise Lipschitz on a dense subset of $b(D)$ (for instance if $g$ is differentiable on a dense subset) then $f$ is locally Lipschitz on $a(D)$. Another result says that if $f$ admits an inverse pointwise Lipschitz condition on a dense subset of $a(D)$ (for instance, if $f$ is differentiable on a dense subset with nonzero derivative), then $g$ is locally invertible with a locally Lipschitz inverse. The results so obtained have applications in the regularity theory of composite functional equations.
AMS Subject Classification
(1991): 26D15, 26D07
Keyword(s):
composite functional equation,
regularity theory,
local Lipschitz property,
inverse local Lipschitz property
Received February 28, 2002. (Registered under 2918/2009.)
Abstract. Carleson type measures with additional logarithmic terms are characterized by using functions in $BMOA$ and the Bloch space. The results are applied to a kind of integral operators and pointwise multipliers on $BMOA$ and the Bloch space, as well as Toeplitz operators on the weighted Bergman $1$space.
AMS Subject Classification
(1991): 30D50; 30D45, 47B35
Keyword(s):
logarithmic Carleson measure,
BMOA,
pointwise multipliers,
Bloch space
Received February 20, 2002, and in revised form July 19, 2002. (Registered under 2919/2009.)
Abstract. We investigate the dynamics and spectral properties of the unitary operators $U_\lambda :=e^{i\lambda x^2}F$, where $\lambda\in {\msbm R}$ and $F$ is the Fourier transform. We show that $U_\lambda $ is a quantization of the classical map $$ f_\lambda\colon {\msbm R}^2 \to{\msbm R}^2 (x,y) \mapstochar\rightarrow (y,2\lambda yx), $$ and that the phase transition at $\lambda =1$ for $f_\lambda $ corresponds to a similar phase transition for $U_\lambda $, which changes at those values from having a pure point to a continuous spectrum.
AMS Subject Classification
(1991): 32H50, 37N20
Keyword(s):
Unitary dynamics
Received June 6, 2002, and in revised form November 13, 2002. (Registered under 2920/2009.)
Abstract. H. L. Smith and H. R. Thieme showed that if a scalar retarded functional differential equation generates a strongly order preserving semiflow with respect to the exponential ordering, then there are certain similarities between the behavior of the solutions of the functional differential equation and the ordinary differential equation obtained by ignoring the delays. In this paper we present further results of this type concerning the boundedness of the solutions and the local and global stability of equilibria.
AMS Subject Classification
(1991): 34K12, 34K20, 34K25
Received June 26, 2003. (Registered under 2921/2009.)
Vilmos Komornik,
Patrick Martinez,
Michel Pierre,
Judith Vancostenoble

651657

Abstract. By the classical CauchyLipschitz theory of ordinary differential equations, no maximal solution of $x'=f(t,x)$ can belong to some compact subset of the domain of definition $D$ of $f$. In the finite dimensional case it follows that the maximal solutions are defined up to the boundary of $D$. Dieudonné and later Deimling gave counterexamples in some infinite dimensional spaces: the maximal solution can remain bounded while it blows up in finite time. We give a complete, elementary and natural proof of this result for {\it all} infinite dimensional Banach spaces.
AMS Subject Classification
(1991): 34K30, 34K35
Keyword(s):
Ordinary differential equation,
blowup,
bounded solutions
Received July 5, 2002, and in the final form April 18, 2003. (Registered under 2922/2009.)
Abstract. In this paper, we shall give sufficient conditions for the convergence of the lacunary Fourier series having the following form $$\sum_{k= \infty }^{\infty }k^\delta\varphi \left(\hat{f}(n_k) \right )< \infty, $$ where $\delta $ is a nonnegative number and $\varphi $ is an increasing and concave function.
AMS Subject Classification
(1991): 42A28, 42A55
Keyword(s):
Absolute convergence,
Lacunary series
Received September 11, 2002, and in revised form December 30, 2002. (Registered under 2923/2009.)
Abstract. In this paper, we get a necessary and sufficient condition for sine series with monotonically decreasing coefficients to belong to some generalized LorentzZygmund space. Moreover, we have analogous results for cosine series.
AMS Subject Classification
(1991): 42A32, 46E30
Keyword(s):
Trigonometric series,
LorentzZygmund space
Received May 28, 2002, and in revised form October 14, 2002. (Registered under 2924/2009.)
Abstract. A theorem of Ferenc Lukács determines the jumps of a periodic, Lebesgue integrable function $f$ in terms of the partial sum of the conjugate series to the Fourier series of $f$. The aim of this paper is to prove an analogous theorem in terms of the AbelPoisson mean for a periodic, Lebesgue integrable function in two variables. We also prove an estimate of the mixed partial derivative of the AbelPoisson mean of the conjugate series to the Fourier series of an integrable function $F(x,y)$ at such a point, where $F$ is smooth. The two results are closely related.
AMS Subject Classification
(1991): 42B05, 42A16
Keyword(s):
Fourier series,
conjugate series,
rectangular partial sum,
AbelPoisson mean,
generalized jump,
smoothness of a function in two variables,
\lambda_* ({\msbm T}^2),
\Lambda_*({\msbm T}^2),
Zygmund classesand
Received April 15, 2002, and in revised form August 14, 2002. (Registered under 2925/2009.)
N. TanovićMiller

687732

Abstract. The integrability classes for even trigonometric and Walsh systems, such as Fomin's classes ${\cal F}_p$, $p>1$, and their enlargements $dv^2$, $\overline{cv}^2$ and $cv^2$, are not necessarily integrability classes for general Vilenkin systems. Recently Aubertin and J.F. Fourier have resolved the problem for the class $dv^2$ proving that for $(c_k) \in dv^2$, (*) $\sum c_kc_{\tilde k}/k< \infty $ is a necessary and sufficient condition in order that the sum of the Vilenkin series $\sum c_k \chi_k$ be an integrable function. Here $\tilde k$ is that index for which $\chi_{\tilde k} = \overline{\chi }_k$, and hence depends on the characteristic sequence of primes $p=(p_{j+1})$. We improve and extend this result from $dv^2$ to new larger classes $lv^2(p)$, $\overline{mv}^2(p)$ and $mv^2(p)$, where $dv^2 \subset lv^2(p) \subset\overline {mv}^2(p)\cap bv$ and $\overline{cv}^2 \subset\overline {mv}^2(p) \subset mv^2(p)$. We prove that for $(c_k)\in mv^2(p)\cap bv$, (*) is also a necessary and sufficient condition for the integrability of $\sum c_k \chi_k $ and derive new equivalent forms of (*). Applications of these results yield several known theorems on integrability of Vilenkin series.
AMS Subject Classification
(1991): 42C10, 43A55
Received April 17, 2002, and in revised form June 21, 2002. (Registered under 2926/2009.)
Torben Maack Bisgaard

733738

Abstract. It is shown that if $S$ is a commutative involution semigroup then the set ${\cal H}(S)$ of all moment functions on $S$ (i.e., complexvalued functions defined on $SS:=\{ st\mid s,t\in S \} $ and admitting a disintegration as an integral of hermitian multiplicative functions) is closed under pointwise convergence in ${\bf C}^{SS}$ if and only if for each $s$ in $S$ there is a positive integer $n$ such that $(s^*s)^n$ is the product of $2n+1$ elements of $S$. In fact, if the condition is satisfied then an `indeterminate method of moments' holds, asserting that if $(\varphi_i)$ is a net of moment functions, converging pointwise to some function $\varphi $, and if for each $i$ in the index set $\mu_i$ is a disintegrating measure of $\varphi_i$ then some subnet of $(\mu_i)$ converges to a disintegrating measure of $\varphi $ (implying in particular that $\varphi $ is a moment function).
AMS Subject Classification
(1991): 43A35, 44A60
Received May 28, 2002, and in revised form December 12, 2002. (Registered under 2927/2009.)
Abstract. For every inner function $\psi\in H^2(D)$, the Jordan block $S(\psi )$ is the compression of the unilateral shift to the quotient space $H^2(D)\ominus\psi H^2(D)$. On the Hardy space over the bidisk $H^2(D^2)$, the Toeplitz operators $T_{z}$ and $T_{w}$ are unilateral shifts of infinite multiplicity. For every subspace $M\subset H^2(D^2)$ invariant under $T_{z}$ and $T_{w}$, the associated {\it twovariable Jordan block} $S(M):=(S_{z}, S_{w})$ is the compression of the pair $(T_{z}, T_{w})$ to the quotient space $H^2(D^2)\ominus M$. This paper proves that $S(M)$ has no reducing subspace for any $M$, and gives a detailed study of $S_{w}$ when $S_{z}$ is a strict contraction. The one variable Jordan block $S(\psi )$ and the Toeplitz algebra are special cases of the work in this paper.
AMS Subject Classification
(1991): 46E20, 47A20, 47A13
Received May 14, 2002, and in revised form October 25, 2002. (Registered under 2928/2009.)
Abstract. In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *isomorphism and a *antiisomorphism between the underlying von Neumann algebras.
AMS Subject Classification
(1991): 46L60, 47B49
Received August 18, 2003. (Registered under 2929/2009.)
Abstract. We give necessary and sufficient conditions for a tuple of unbounded operators to be (jointly) subnormal. This criterion is then applied to characterize the subnormality of some special tuples of operators, in particular commuting bilateral weighted multishifts.
AMS Subject Classification
(1991): 47B20, 44A60, 47A20, 47B37
Received February 11, 2002, and in revised form May 10, 2002. (Registered under 2930/2009.)
Muneo Cho,
Young Min Han,
Tadasi Huruya

789800

Abstract. In this paper, we show that the following spectral mapping theorem holds: Let $T = H + iK$ be hyponormal and $\varphi $ be a strictly monotone increasing continuous function on $\sigma(H)$. We define $\tilde{\varphi }(x+iy)=\varphi(x)+iy$ for $x \in\sigma (H), y \in{\msbm R}$ and $\tilde{\varphi }(T)=\varphi(H)+iK$. Then $$ \sigma_{na}(\tilde{\varphi }(T)) = \tilde{\varphi }(\sigma_{na}(T)), \sigma_{a}(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma_{a}(T)) \mbox{ and } \sigma(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma(T)). $$ We also show that Weyl's theorem holds for $\tilde{\varphi }(T)$ and study the G$_1$ property of the operator $\tilde{\varphi }(T)$.
AMS Subject Classification
(1991): 47B20
Received March 11, 2002. (Registered under 2931/2009.)
Abstract. We consider the infinite dimensional linear dynamical systems of composition operators defined on the Hardy space $H^2 (D)$. We investigate the scalar multiplied composition operators which are chaotic in Devaney's sense.
AMS Subject Classification
(1991): 47B33, 58F13
Received January 22, 2002, and in revised form April 15, 2002. (Registered under 2932/2009.)
Christopher Hammond

813829

Abstract. For any analytic map $\varphi\colon {\msbm D}\rightarrow{\msbm D}$, the composition operator $C_{\varphi }$ is bounded on the Hardy space $H^2$, but there is no known procedure for precisely computing its norm. This paper considers the situation where $\varphi $ is a linear fractional map. We determine the conditions under which $\C_{\varphi }\$ is given by the action of either $C_{\varphi }$ or $C_{\varphi }^{\ast }$ on the normalized reproducing kernel functions of $H^2$. We also introduce a new set of conditions on $\varphi $ under which we can calculate $\C_{\varphi }\$; moreover, we identify the elements of $H^2$ on which such an operator $C_{\varphi }$ attains its norm. Several specific examples are provided.
AMS Subject Classification
(1991): 47B33
Received February 20, 2002, and in revised form June 18, 2002. (Registered under 2933/2009.)
Abstract. As is known, the corona theorem is in general not true for a function $F \in H^{\infty }(L(H))$, where $L(H)$ is the space of bounded operators on an infinite dimensional separable Hilbert space $H$. Combined with a relatively compact range $F({\msbm D})$, the approximation property (AP), either in $H^{\infty }$ or in $L(H)$, provides functions satisfying the corona theorem, see [Vit]. Here we prove by counterexamples that these two methods are independent. We also give some new examples of subspaces of $L(H)$ and quotient spaces $H^{\infty }/ BH^{\infty }$ satisfying (AP). To finish, we give a version of the corona theorem for functions in the operator Nevanlinna class having a relatively compact range.
AMS Subject Classification
(1991): 47A56, 47A20, 46B28, 30D55
Received April 16, 2002, and in revised form March 6, 2003. (Registered under 2934/2009.)
Masatoshi Ito,
Takeaki Yamazaki,
Masahiro Yanagida

853862

Abstract. Let $A$ and $B$ be positive operators. We remark that $A$ and $B$ are not necessarily invertible. Recently, Ito and Yamazaki showed relations between the two inequalities $$ (B^{r\over2}A^pB^{r\over2})^{r\over p+r} \ge B^r \mbox{ and } A^p \ge(A^{p\over2}B^rA^{p\over2})^{p\over p+r}, $$ for fixed positive numbers $p \ge0$ and $r \ge0$. In this paper, as extensions of these results, we shall show relations between the two inequalities $$ (B^{r\over2}A^pB^{r\over2})^{r\delta\over p+r} \ge B^{r\delta } \mbox{ and } A^{p\over2}B^{\delta }A^{p\over2} \ge(A^{p\over2}B^rA^{p\over2})^{\delta +p\over p+r}, $$ for fixed positive numbers $r \ge\delta \ge0$ and $p \ge0$. We shall also show a relation between the two inequalities $$ A^{p\gamma } \ge(A^{p\over2}B^rA^{p\over2})^{p\gamma\over p+r} \mbox{ and } (B^{r\over2}A^pB^{r\over2})^{\gamma +r\over p+r} \ge B^{r\over2}A^{\gamma }B^{r\over2}, $$ for fixed positive numbers $p \ge\gamma \ge0$ and $r \ge0$. Furthermore, we shall show a slight extension of a result on transitive properties of the first two inequalities by Yanagida as an application of these results.
AMS Subject Classification
(1991): 47A63
Keyword(s):
Positive operators,
Furuta inequality
Received April 24, 2002, and in revised form October 24, 2002. (Registered under 2935/2009.)
Shizuo Miyajima,
Isao Saito

863869

Abstract. The authors [5] called a bounded linear operator $T$ $\infty $hyponormal if $T$ is $p$hyponormal for every $p>0$. They investigated the spectral properties of a pure $\infty $hyponormal operator $T$ under the condition that $T$ has dense range and has no nontrivial reducing subspace. In this paper it is shown that these properties of a pure $\infty $hyponormal operator $T$ still hold without this condition.
AMS Subject Classification
(1991): 47B20, 47A10
Received April 24, 2002, and in revised form January 21, 2003. (Registered under 2936/2009.)
Brent J. Carswell,
Barbara D. MacCluer,
Alex Schuster

871887

Abstract. We determine the holomorphic mappings of ${\msbm C}^n$ that induce bounded composition operators on the Fock space in ${\msbm C}^n$. Furthermore, we determine which of these composition operators are compact, and we compute the operator norm of all bounded composition operators in this setting. We also consider extensions of these results to various generalizations of the Fock space.
AMS Subject Classification
(1991): 47B33; 32A37
Received May 28, 2002, and in revised form November 15, 2002. (Registered under 2937/2009.)
A. Böttcher,
S. Grudsky

889900

Abstract. We prove that the spectral norm of a finite Toeplitz matrix can be estimated from below through the Fejér mean of the generating function. This result has applications to the problem of finding the most probable values of $\A_n x\$ in case $A_n$ is a large finite Toeplitz matrix and $x$ is uniformly distributed on the unit sphere of ${\bf C}^n$.
AMS Subject Classification
(1991): 47B35; 15A12, 42A05, 60H25
Received July 29, 2002, and in revised form December 12, 2002. (Registered under 2938/2009.)
Abstract. A polytope is perfect if its shape cannot be changed without changing the action of its symmetry group on its facelattice. Perfect polytopes are completely known only in dimensions 2 and 3, while exploring their various possible classes in dimension $4$ is still in progress. Here a new class is constructed which is closely related to regular 4polytopes. In addition, there are some interesting coincidences between the $f$vectors of some of them, which are briefly discussed at the end of the paper.
AMS Subject Classification
(1991): 20F55, 52B05, 52B15
Received May 28, 2002, and in revised form March 24, 2003. (Registered under 2939/2009.)

911943
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