ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We obtain the general form of a bijective order-preserving 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.)

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 $p-1$. 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.)

Abstract. For two given orthogonal projectors $P_A$ and $P_B$ of the same order, we investigate how to express Moore--Penrose 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, Moore--Penrose inverse, weighted Moore--Penrose inverses, orthogonal projector

Received May 28, 2002, and in revised form February 6, 2003. (Registered under 2915/2009.)

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 $$ f_\lambda\colon {\msbm R}^2 \to{\msbm R}^2 (x,y) \mapstochar\rightarrow (y,2\lambda y-x), $$ 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.)

Abstract. By the classical Cauchy--Lipschitz 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, blow-up, 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 non-negative 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 Lorentz--Zygmund space. Moreover, we have analogous results for cosine series.

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

Keyword(s): Trigonometric series, Lorentz--Zygmund 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 Abel--Poisson mean for a periodic, Lebesgue integrable function in two variables. We also prove an estimate of the mixed partial derivative of the Abel--Poisson 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, Abel--Poisson 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.)

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_k-c_{\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.)

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., complex-valued 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 two-variable 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 multi-shifts.

AMS Subject Classification (1991): 47B20, 44A60, 47A20, 47B37

Received February 11, 2002, and in revised form May 10, 2002. (Registered under 2930/2009.)

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.)

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.)

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.)

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.)

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.)

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 face-lattice. 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 4-polytopes. 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.)