ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. J. Tůma proved an interesting ``congruence amalgamation'' result. We are generalizing and providing an alternate proof for it. We then provide applications of this result: \item{(i)} A. P. Huhn proved that every distributive algebraic lattice $D$ with at most $\aleph_1$ compact elements can be represented as the congruence lattice of a lattice $L$. We show that $L$ can be constructed as a locally finite relatively complemented lattice with zero. \item{(ii)} We find a large class of lattices, the {\it $\omega $-congruence-finite} lattices, that contains all locally finite countable lattices, in which every lattice has a relatively complemented congruence-preserving extension.

AMS Subject Classification (1991): 06B05, 06B10, 06D05

Keyword(s): Congruence, amalgamation, lattice, distributive, relative complemented

Received March 10, 1999. (Registered under 2717/2009.)

Abstract. A semigroup $S$ is said to have the Berman property if its $p_n$-sequence (counting the $n$-ary term operations which depend on all their variables) is either eventually strictly increasing or else it is bounded. In an earlier paper the authors showed that every surjective semigroup (i.e. a semigroup $S$ satisfying $S^2=S$) has the Berman property. In the present paper, we consider finite semigroups which are nilpotent ideal extensions of certain types of surjective semigroups and prove the Berman property for them. Our scope includes, for example, finite nilpotent extensions of completely regular semigroups (unions of groups) and of commutative semigroups. Also, the Berman property is proved for finite strict nilpotent extensions of arbitrary surjective semigroups.

AMS Subject Classification (1991): 20M10, 08A40

Received March 24, 1999. (Registered under 2718/2009.)

Abstract. A semigroup is called right commutative if it satisfies the identity $axy=ayx$. We say that a semigroup $S$ is a $\Delta $-semigroup if the lattice of all congruences of $S$ is a chain. In this paper we prove that a semigroup is a right commutative $\Delta $-semigroup if and only if it is isomorphic to either $G$ or $G^0$, where $G$ is a subgroup of a quasicyclic $p$-group ($p$ is a prime) or $L$ or $L^0$, where $L$ is a two-element left zero semigroup or $N$, where $N$ is a non-trivial right commutative nil semigroup whose lattice of ideals is a chain or a full $\Delta $-overact of a null semigroup by a commutative nil $\Delta $-semigroup with an identity adjoined. We also give a construction of full $\Delta $-overacts of null semigroups by commutative nil $\Delta $-semigroups with an identity adjoined.

AMS Subject Classification (1991): 20M35

Received March 4, 1998, and in final form September 29, 1999. (Registered under 2719/2009.)

Abstract. Let $(X,{\cal F},\mu )$ be a probality space and let $L^2(X,0)$ be the collection of all $f \in L^2(X)$ with zero integrals. A collection ${\cal A}$ of linear operators on $L^2(X)$ is said to satisfy the Gaussian distribution property (G.D.P.) if $L^2(X,0)$ is invariant under ${\cal A} $ and there exists a constant $C < \infty $ such that the following condition holds: Whenever $T_1,\ldots,T_k$ are finitely many operators in ${\cal A}$, and $f$ is a function in $L^2$ with zero integral, then, for any required degree of approximation, there is another $L^2$ function $g$ with $\|g\|_2 \leq C\|f\|_2$, such that all the inner products $({\rm Re} T_ig, {\rm Re} T_jg)$ are approximately equal to the corresponding inner products $({\rm Re} T_if, {\rm Re} T_jf)$, $i,j=1,\ldots,k$, and such that the joint distribution of the functions ${\rm Re} T_1g,\ldots,{\rm Re} T_kg$ is approximately Gaussian. It has been proved recently ([3]) that if $(S_n)^\infty_1$ is a sequence of uniformly bounded linear operators on $L^2 (X)$ with $S_n {\bf1} = {\bf1}$, $n=1,2,\ldots $, and such that $(S_n)^\infty_1$ satisfies the Bourgain's infinite entropy condition and the G.D.P., then there exists an $h\in L^2(X)$ such that $\lim_{n\to\infty } S_n h$ fails to exist $\mu $-a.e. as a finite limit on $X$. The purpose of this paper is to show that various classes of operators commonly encountered in ergodic theory do actually satisfy the G.D.P.. In particular, we will show that continuous ergodic automorphism of a compact abelian group, translations and endomorphisms of the $n$-tori, multiple Riemann-sum operators, and operators arising from the conjectures of Bellow and Khintchine all satisfy the G.D.P.. Applications to convergence problems are also given.

AMS Subject Classification (1991): 28D99, 60F99

Received May 26, 1997, and in final form November 12, 1999. (Registered under 2720/2009.)

Abstract. The authors consider the equation (E) $(x(t)+\lambda x(t-\tau ))^{(n)} + f(t,x(g(t))) = 0,$ where $ \lambda\not= 0$, $\tau >0$, $f$ and $g$ are continuous, $g(t) \le t$, and $\lim_{t\to +\infty }g(t) = + \infty $. They give sufficient conditions for (E) to have an oscillatory solution, and in addition, give an asymptotic expression for this oscillatory solution involving the sine function and an exponential.

AMS Subject Classification (1991): 34K15, 34K40

Received May 8, 1998, and in revised form April 2, 1999. (Registered under 2721/2009.)

Abstract. In this paper we investigate the exponential stability of the trivial solution of the state-dependent delay differential equation $\dot x(t)=a(t)x(t-\tau(t,x(t)))$. It is shown that, under some conditions, this state-dependent equation is exponentially stable, if the trivial solution of $\dot y(t)=a(t)y(t-\tau(t,0))$ is exponentially stable. Assuming the existence of bounded partial derivatives of the delay function, the reverse statement will also be proved.

AMS Subject Classification (1991): 34K, 34D

Received March 22, 1999, and in revised form July 4, 1999. (Registered under 2722/2009.)

Abstract. Suppose that $f (z) = \sum ^\infty_{\nu = 0} a_\nu z^\nu $ is a power series with radius of convergence 1 and let $A = [\alpha_{n \nu }]$ be a lower triangular matrix. We are interested in the behaviour of the $A$-transforms $$ \sigma_n (z) = \sum ^n_{\nu = 0} \alpha_{n \nu } \sum ^{\nu }_{\mu = 0} a_\mu z^\mu, $$ where we do not assume that the matrix $A$ is regular or satisfies some quasi-regular properties. Among others we deal with the following problems. \item{$\bullet $} Let it be given an $R > 0$. Under which necessary and sufficient conditions on $A$ is $\{\sigma_n (z) \} $ compactly convergent in $|z| < R$ and which limit functions are possible. \item{$\bullet $} The growth-properties of $\{\sigma_n (z)\} $ are investigated. \item{$\bullet $} Some problems concerning the value-distribution of the $A$-transforms are studied. For instance the location of the limit points of $w_0$-values of $\sigma_n (z)$, the number of $w_0$-values in a circle $|z| < S$ and properties of the mapping $w = \sigma_n (z)$ are investigated.

AMS Subject Classification (1991): 40A25, 40G99

Received April 15, 1999. (Registered under 2723/2009.)

Abstract. We consider the circle methods of summability theory (in discrete and continuous form). The basic Tauberian properties ($b$-equivalence, $O$-theorem) are treated in a short and lucid way. Thereby we employ logarithmic derivatives and M untz approximations.

AMS Subject Classification (1991): 40E10, 40G99, 40H05

Received February 15, 1999. (Registered under 2724/2009.)

Abstract. When we approximate a continuous function $f$ which changes its monotonicity finitely many, say $s$ times, in $[-1,1]$, we wish sometimes that the approximating polynomials follow these changes in monotonicity. However, it is well known that this requirement restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of $\omega_2(f,1/n)$ and even this not with a constant (dependent only on $s$), rather with a constant which depends on the location of the interior extrema. Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to $1/n$ about the interior extrema of the function and in intervals of length $1/n^2$ near the endpoints, what we called nearly comonotone approximation, allows the polynomials to achieve a pointwise approximation rate of $\omega_3$ (moreover, with a constant which depends only on $s$). We show here that even when we relax the requirement of monotonicity of the polynomials on sets of measures approaching $0$ (no matter how slowly or how fast), $\omega_4$ is not reachable. We conclude the paper with results on the sizes of the deleted sets which allow the $\omega_3$ degree of approximation in the norm; and when $f$ is differentiable, allow estimates involving the $k$th modulus of smoothness of $f'$.

AMS Subject Classification (1991): 41A10, 41A25, 41A29

Keyword(s): Approximation by polynomials, Shape preserving approximation

Received February 15, 1999. (Registered under 2725/2009.)

Abstract. We study the rate of pointwise convergence for a sequence of positive linear operators $R_n^{[\beta ]}$ approximating functions on the infinite interval $[0,\infty )$ which were investigated by K. Balázs and Szabados. They are a special case of more general operators introduced by K. Balázs. The paper presents the complete asymptotic expansion for operators $R_{n}^{\left[ \beta\right ] }$ as $n$ tends to infinity. All coefficients are calculated explicitly. It turns out that Stirling numbers of first and second kind play an important role. Our work generalizes previous results.

AMS Subject Classification (1991): 41A25, 41A36, 41A60

Received October 20, 1998, and in revised form June 2, 1999. (Registered under 2726/2009.)

Abstract. Let $\mu $ be a positive measure on $[-1,1]$ satisfying the Szegő condition $$ \int_{-1}^1 {\log\mu ^\prime(t) \over\sqrt {1-t^2}} dt > -\infty $$ with $\mu ^\prime $ being the Radon--Nikodym derivative with respect to the Lebesgue measure, and let further $w_{2n} \in{\cal P}_{2n}$, $n\in{\msbm N}$, be a sequence of polynomials of degree at most $2 n$, with zeros $a_{2n,1},\ldots,a_{2n,m} \in{\msbm C}\setminus[-1,1]$, $m = \deg(w_{2n})$, and being positive on $[-1,1]$. We study the asymptotic behavior of the polynomials $p_n = p_n(\mu_n; \cdot )$ orthonormal with respect to the varying measure $d\mu_n := w_{2n}^{-1} d\mu $. Let $\varphi\colon \overline{\msbm C}\setminus[-1,1] \rightarrow{\msbm D}$ be the conformal mapping of $\overline{\msbm C}\setminus[-1,1]$ onto ${\msbm D}$ with $\varphi(\infty ) = 0$ and $\varphi ^\prime(\infty ) > 0$. Under the assumption that $$(\ast)\qquad\qquad\lim _{n \to\infty } \left[ (2n - \deg(w_{2n})) + \sum_{j = 1}^{\deg(w_{2n})} (1 - | \varphi(a_{2n,j})| ) \right ] = \infty $$ we prove strong asymptotic relations in ${\msbm C}\setminus[-1,1]$ for the orthonormal polynomials $p_n (\mu_n; \cdot )$ as $n \to\infty $. An analogue of this result is proved for polynomials that are orthonormal with respect to varying measures on the unit circle ${\msbm T}= \partial{\msbm D}$. Also in this case the original measure of orthogonality has to belong to the Szegő class (on ${\msbm T}$) and an assumption analoguous to (*) has to hold true. The necessity of this assumption will be discussed in some detail for the case of orthogonality on ${\msbm T}$. The strong asymptotic relations for $p_n (\mu_n;\cdot )$ lead to correspondingly precise asymptotic error estimates for multipoint Padé approximants (i.e., rational interpolants) to Markov functions $f(z) = \int(t-z)^{-1} d\mu(t)$. The error estimates for multipoint Padé approximants have been the primary motivation for studying strong asymptotics in the present paper.

AMS Subject Classification (1991): 42C05, 41A25

Keyword(s): Orthonormal polynomials with varying weights, Szegő asymptotics, strong asymptotics, multipoint Padé approximants, Markov functions, Markov's Theorem

Received February 15, 1999. (Registered under 2727/2009.)

Abstract. The purpose of this work is to show a Paley type inequality for two-parameter Vilenkin system. For the one-dimensional analogue see Simon, Weisz [7]. Hence we will verify the estimation $$(*)\ \ \ C_p\|f\|_{H^p} \geq\left(\sum_{n,k=0}^\infty(m_nm_k)^{1 - 2 / p}(M_nM_k)^{2 - 2 / p} \sum_{j=1}^{m_n-1}\sum_{l = 1 \atop \alpha \leq jM_n / lM_k \leq \beta}^{m_k-1} |{\hat f}(jM_n,lM_k)|^2\right)^{1/2}$$ for martingales $f\in H^p(G_m^2)$ $(2/5\leq p\leq1)$. Here $0<\alpha\leq 1\leq\beta$ and $\hat f(u,v)$ ($u, v\in{\msbm N}$) is the $(u,v)$-th (two-parameter) Vilenkin--Fourier coefficient of $f$. The Hardy space $H^p(G_m^2)$ is defined by means of a diagonal maximal function. By usual interpolation argument it follows an $L^p$-variant of (*) for $1< p\leq2$. As the dual inequality of (*) we formulate a $BMO$-result as well as some analogous statements for other Hardy- and ${\cal BMO}$-spaces. The non-improving of the assumption $2/5\leq p$ as well as the diagonal case of (*) will be investigated.

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

Received August 4, 1998. (Registered under 2728/2009.)

Abstract. The orthonormal Franklin spline system on ${\msbm R}$ is treated as a system of functions in the metric space $L^p({\msbm R})$, $0< p\le1$. It is proved that a given Franklin series converges unconditionally in this metric space if and only if the corresponding square function is in $L^p({\msbm R})$. Moreover, the latter takes place if and only if the maximal function for the partial sums is in $L^p({\msbm R})$.

AMS Subject Classification (1991): 42C10

Received February 18, 1999. (Registered under 2729/2009.)

Abstract. In this paper we introduce Sobolev type spaces associated to Bessel operators. Also we study bounded linear operators that commute with Hankel translations on $L_p$-spaces.

AMS Subject Classification (1991): 46F12

Keyword(s): Sobolev spaces, Bessel operator, Hankel convolution, Hankel transformation

Received February 5, 1999. (Registered under 2730/2009.)

Abstract. Some interesting algebras of operators are not complete in any algebra norm. Two examples: The algebra of all compact operators on an incomplete normed linear space; and the algebra of all bounded Carleman integral operators on $L^2[0,1]$. Although not Banach algebras, both of these algebras are functional algebras, i.e. the usual holomorphic functional calculus operates in them. This paper is concerned with functional algebras and related topics. Examples and applications involving algebras of bounded linear operators are given.

AMS Subject Classification (1991): 46H30, 46H35, 47A10

Keyword(s): holomorphic functional calculus, left and right ideals, range inclusion

Received February 15, 1999. (Registered under 2731/2009.)

Abstract. In this paper we study bounded weak approximate identities for Segal algebras on LCA groups. In particular we show that, if a Segal algebra $A$ on a non-compact LCA group $G$ belongs to some familiar classes, we can construct bounded weak approximate identities for $A$ of norm 1. From this result we get at once that Segal algebras in these classes are BSE. Examples of Segal algebras that have no bounded weak approximate identities are also given.

AMS Subject Classification (1991): 46J25, 46J40, 43A25

Received December 7, 1998. (Registered under 2732/2009.)

Abstract. Let $A$ be a unital $C^*$-algebra and let $L\colon A\rightarrow B(H)$ be a linear map. Applying Proposition 2.2 and Theorem 2.4 of the paper, we can define operator radii $w_{\rho }(a)$ and induced completely bounded norms $\|L\| _{w_{\rho }cb}$, where $1\leq\rho \leq2$. When $\rho =2$, we prove that $$\|L\| _{wcb}=\inf\left\{\|\phi\| :\pmatrix{\phi & L \cr L^* & \phi} \hbox{ is completely positive}\right\}.$$ In general, we have the inequalities $|L|\leq\|L\| _{cb}\leq\|L\| _{w_{\rho}cb}\leq\|L\|_{wcb}\leq2|L|$, where $|L|=\inf\{\|\phi\|:\phi\pm\mathop{\rm Re}\alpha L$ is completely positive for all $|\alpha|=1\}$. We also give equivalent conditions for $\|L\| _{wcb}\leq1$ and extend the main Theorem of T. Ando and K. Okubo [1, p. 183] from $n\times n$ complex matrices to $n\times n$ matrices of operators on some Hilbert space.

AMS Subject Classification (1991): 46L05

Received February 4, 1999, and in revised form August 26, 1999. (Registered under 2733/2009.)

Abstract. A classical result of Riesz [8] says that the product of two commuting positive operators is also positive. Wigner's generalization states that the selfadjoint product of atmost three positive operators is positive too. Bernau proved that the selfadjoint product of a positive operator and another operator with positive spectrum is automatically positive. We show that these statements remain true under further essential weakening of the assumptions. The selfadjoint product of a positive operator and another operator with spectrum without negative reals is also positive (Theorem 4). Consequently the selfadjoint product of two positive operators and an operator with closed numerical range without negative reals is automatically positive (Theorem 6).

AMS Subject Classification (1991): 47A05, 47A12

Received April 8, 1999, and in revised form July 5, 1999. (Registered under 2734/2009.)

Abstract. Let $\{D_n\} $ be a sequence of bounded invertible operators on a Hilbert space ${\cal H}$. It is shown that the collection of operators $T$ for which the norm-limit $\lim D_nTD_n^{-1}$ exists is an algebra. Furthermore, some sufficient conditions on this sequence are established for the corresponding algebra to have a nontrivial invariant subspace. By considering specific sequences of operators several invariant subspace results are obtained.

AMS Subject Classification (1991): 47A15

Received December 3, 1998, and in revised form September 13, 1999. (Registered under 2735/2009.)

Abstract. In this paper we consider the problem of the existence of an interwining lifting or extension for ($*$-)regular isometric (unitary) dilations of two bicontractions. It is known that in such a generality, the commutant lifting theorem fails (see [10]). We show that such a lifting exists if the intertwining operator doubly intertwines one of the components.

AMS Subject Classification (1991): 47A20, 47A13

Keyword(s): Bicontractions, distinguished dilations, *, (-)regular dilations, intertwining operators, extensions

Received June 30, 1998, and in revised form September 9, 1999. (Registered under 2736/2009.)

Abstract. In the category of Hilbert modules $M$ over a function algebra $A$ we introduce the notion of Nagy--Foiaş diagram which, in case it exists, connects in a special way a minimal subspectral resolution of $M$ with its corresponding minimal subspectral resolution of the adjoint module $M_{*}$ associated via the minimal spectral dilation of $M$. We show that there is a one-to-one correspondence between Nagy--Foiaş diagrams and a class of $A$-module maps. In case $A$ is the disk algebra, the Nagy--Foiaş diagram expresses the geometry of the space of the minimal unitary dilation of the contraction $T$ which generates the $A$-module structure on $M$, while the class of $A$-module maps is the class of the purely contractive analytic functions. The above correspondence is in this case the Nagy--Foiaş model based on the characteristic function.

AMS Subject Classification (1991): 47A20, 46E20

Keyword(s): Hilbert module, spectral dilation, Silov resolution, Nagy--Foiaş model

Received November 17, 1998, and in final form September 27, 1999. (Registered under 2737/2009.)

Abstract. In this note we obtain some sufficient conditions that a trace-class commutator have trace zero, and we use these results together with two theorems of Voiculescu to establish some structure theorems for hyponormal-like operators and to give short proofs of theorems of G. Weiss and Helton--Howe.

AMS Subject Classification (1991): 47B20

Keyword(s): Commutators, trace zero, almost hyponormal operators

Received June 22, 1999. (Registered under 2738/2009.)

Abstract. In this paper, we describe a class of maps of the unit ball in ${\msbm C}^N$ into itself that generalize the automorphisms and deserve to be called linear fractional maps. They are special cases or generalizations of the linear fractional maps studied by Kre?n and Smul'jan, Harris and others. As in the complex plane, a linear fractional map on ${\msbm C}^N$ is represented by an $(N+1)\times(N+1)$ matrix. Basic connections between the properties of the map and the properties of this matrix viewed as a linear transformation on an associated Kre?n space are established. These maps are shown to induce bounded composition operators on the Hardy spaces $H^p({\msbm B}_N)$ and some weighted Bergman spaces and we compute the adjoints of these composition operators on these spaces. Finally, we solve Schroeder's equation $f\circ\varphi = \varphi '(0)f$ when $\varphi $ is a linear fractional self-map of the ball fixing 0.

AMS Subject Classification (1991): 47B38, 32A30, 30C45

Received September 2, 1998, and in revised form March 22, 1999. (Registered under 2739/2009.)

Abstract. In this note we characterize the weighted composition operators on $CV_0(X,E)$ induced by the mappings $\phi\colon X\to X$ and $\pi\colon X\to B(E)$. A study of the invertible weighted composition operators on $CV_0(X,E)$, the weighted spaces of vector-valued continuous functions, is also given. Some examples are presented to illustrate the theory.

AMS Subject Classification (1991): 47B38, 47A05, 47A56, 46E10

Keyword(s): Weighted spaces, Weighted composition operators, Invertible operators

Received April 8, 1999. (Registered under 2740/2009.)

Abstract. We generalize the complete lift of real functions introduced by Yano--Ishihara [17], and the complete lift of maps between Euclidean spaces defined by Ou [12], to obtain harmonic maps and morphisms on semi-Riemannian manifolds. A new characterization of harmonic morphisms between (semi-)Riemannian manifolds is obtained. For some quadratic forms which generalize the quadratic maps, we give necessary and sufficient conditions under which they are harmonic maps or morphisms.

AMS Subject Classification (1991): 53C20, 58E20

Keyword(s): Harmonic maps and morphisms, tangent bundle, semi-Riemannian metric, complete lift

Received October 15, 1998, and in revised form June 17, 1999. (Registered under 2741/2009.)

Abstract. We give a probabilistic representation of all possible limiting distributions of sums of independent identically distributed random variables along subsequences of ${\msbm N}$ satisfying a geometrical growth condition and describe the distributions partially attracted to them. We investigate the structure of these domains of partial attraction and also discuss the effect of light trimming.

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

Keyword(s): Semistable laws, domains of partial attraction, lightly trimmed sums

Received April 23, 1999, and in revised form August 30, 1999. (Registered under 2742/2009.)