ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We survey eleven papers by György Pollák published from 1973 to 1989 and devoted to various aspects of the theory of semigroup varieties: hereditarily finitely based varieties, permutation identities, covers in varietal lattices.

AMS Subject Classification (1991): 20M07, 08B05, 08B15

Received February 18, 2002. (Registered under 2824/2009.)

Abstract. We introduce a triangular scheme for congruences which is satisfied in any congruence distributive algebra ${\cal A}$. A condition called Weak Triangular Principle is studied, which is equivalent to the distributivity of ${\mathop{\rm Con} {\cal A}}$ for an arbitrary algebra ${\cal A}$. It follows that if ${\cal A}$ is congruence permutable then the Triangular Scheme is equivalent to the distributivity of $ {\mathop{\rm Con} {\cal A}}$. We define the Triangular Principle as well, which is shown to hold in congruence distributive varieties.

AMS Subject Classification (1991): 08A30, 08B10

Keyword(s): congruence distributivity, congruence permutability, Shifting Lemma, Triangular Scheme, Triangular Principle

Received October 30, 2000, and in final form July 7, 2001. (Registered under 2826/2009.)

Abstract. Let $\underline k$ denote a $k$-element chain, $k \geq3$. Let $M$ denote the clone generated by all unary isotone operations on $\underline k$ and let $\mathop{\rm Pol} \leq $ denote the clone of all isotone operations on $\underline k$. We investigate the interval of clones $[M, \mathop{\rm Pol} \leq ]$. Among other results, we describe completely those clones which contain only join (or meet) homomorphisms, and describe the interval completely for $k \leq4$.

AMS Subject Classification (1991): 08A40, 03B50

Keyword(s): clone, isotone operations, monoidal interval

Received October 11, 2000, and in revised form February 27, 2001. (Registered under 2827/2009.)

Abstract. We compute the unary polynomial functions on groups that arise as semidirect products of two groups $A, B$ with the property that every element of $B$ operates on $A$ by conjugation either trivially or as a fixed-point-free automorphism; in many cases, we obtain the number of polynomial functions on these groups.

AMS Subject Classification (1991): 08A40, 16Y30

Received November 17, 2000, and in revised form May 9, 2001. (Registered under 2828/2009.)

Abstract. In this paper we continue the study, initiated in [4], of finite semigroups having a small number of term operations, that is, whose $p_n$-sequences are bounded above by a polynomial function of $n$. We characterize finite semigroups whose $p_n$-sequences are bounded above by a polynomial of a given degree. Further, we show that, given a finite semigroup $S$ with a polynomially bounded $p_n$-sequence, the least natural $k$ for which the inequality $p_n(S)\leq cn^k$ holds is effectively computable in polynomial time. Also, we elaborate the structural features of the considered class of finite semigroups.

AMS Subject Classification (1991): 08A40, 20M07, 20M10

Keyword(s): semigroup, term operation, p_n, -sequence

Received September 24, 2001, and in final form January 24, 2002. (Registered under 2829/2009.)

Abstract. We prove that if $\{x,y\},\{u,v\} $ are two sets of generating pairs for a free group $F$ satisfying the equation $ [x,y^n] = [u,v^m]$ then $n = m$. Further if $n = m \ge2$ then $y$ is conjugate in $F$ to $v^{\pm1}$. This theorem rose out of a question concerning Schottky groups. The method of proof is used to consider certain related equations in free groups and generalizations to genus one Fuchsian groups.

AMS Subject Classification (1991): 20E05

Keyword(s): Free Groups, Equations, Test Elements, Scottky Groups

Received January 19, 2001, and in revised form July 12, 2001. (Registered under 2830/2009.)

Abstract. We consider several natural quasi-orderings on free semigroups and describe minimal elements with respect to these quasi-orderings in some important sets of words.

AMS Subject Classification (1991): 20M05

(Registered under 2831/2009.)

Abstract. Existence, uniqueness and data dependence results for the solution of a Fredholm integral equation with deviating argument are given.

AMS Subject Classification (1991): 34K05, 34K10

Keyword(s): fixed points, nonlinear integral equations

Received January 10, 2001, and in revised form November 5, 2001. (Registered under 2832/2009.)

Abstract. Suppose that $\sum ^\infty_{k = 0} a_k$ is a series with $a_k \ge0$. In a previous paper [Luh] some asymptotic properties were obtained for series of the type $\sum ^\infty_{k = 0} a_k \varphi( \sum ^k_{\nu = 0} a_\nu )$ and $\sum ^\infty_{k = 0} a_k \varphi( \sum ^\infty_{\nu = k} a_\nu )$ where the function $\varphi $ satisfies some natural conditions. It has been first shown by L. Leindler [2] that these asymptotics are best possible if especially the functions $\varphi(t) = t^{- \alpha }$ are considered. In a recent paper the authors have proved the sharpness in the general case too. It is the object of this note to show that the asymptotics obtained in [3] are also best possible with respect to other properties.

AMS Subject Classification (1991): 40A05

Received December 5, 2000, and in revised form February 15, 2001. (Registered under 2833/2009.)

Abstract. A periodic-basis function (PBF) is a function of the form $$ \phi(u)=\sum_{k\in{\msbm Z}}\widehat{\phi }(k) e^{iku}, u\in{\msbm R}, $$ where the sequence of Fourier coefficients $\{\widehat{\phi }(k) : k\in{\msbm Z}\} $ satisfies the following conditions: $$ \widehat{\phi }(k)=\widehat{\phi }(-k), k\in{\msbm Z}, \hbox{ and } \sum_{k\in{\msbm Z}}|\widehat{\phi }(k)|< \infty. $$ A PBF $\phi $ is said to be strictly positive definite if every Fourier coefficient of $\phi $ is positive. It is known that if $\phi $ is strictly positive definite, then given any continuous $24$-periodic function $f$ and any triangular array $\{\theta_{j,\mu } : 1\le j\le\mu, \mu\in {\msbm N}\} $ of distinct points in $[-4,4)$, there exists a unique PBF interpolant $ I(\theta ):= \sum_{j=1}^\mu a_j\phi(\theta -\theta_{j,\mu })$, $a_j\in{\msbm R}$, such that $ I(\theta_{k,\mu })=f(\theta_{k,\mu })$, $1\le k\le\mu $. This paper studies the uniform convergence of these PBF interpolants to the approximand $f$. Even though there is a rather well-developed theory which supplies various results of this nature, it also has the shortcoming that if $\phi $ is very smooth, then the class of functions $f$ which can be simultaneously approximated and interpolated by PBF interpolants is highly restricted. The primary objective of this paper is to suggest an oversampling strategy to overcome this problem. Specifically, it is shown that by increasing the dimension of the underlying space of approximants/interpolants judiciously, one can construct PBF interpolants (based on very smooth $\phi $) that converge to approximands which are only assumed to be continuous. The main tool in the analysis is a periodic version of a result of Szabados on algebraic polynomials, the proof of which relies on the trigonometric version of a fundamental theorem due to Erdős.

AMS Subject Classification (1991): 41A05, 41A30, 42A08, 42A12

Received November 21, 2000, and in revised form July 17, 2001. (Registered under 2834/2009.)

Abstract. We prove Bernstein type inequalities for algebraic polynomials on the interval $I:=[-1,1]$ and for trigonometric polynomials on {\msbm R} when the roots of the polynomials are outside of a certain domain of the complex plane. The cases of real vs. complex coefficients are handled separately. In case of trigonometric polynomials with real coefficients and root restriction, the $L_p$-situation will also be considered. In most cases, the sharpness of the estimates will be shown.

AMS Subject Classification (1991): 41A17

Received February 26, 2001, and in revised form August 8, 2001. (Registered under 2835/2009.)

Abstract. We study a new sequence of Bernstein-type operators on the $d$-dimensional simplex. The binomial coefficients considered in the classical definition are substituted with more general ones satisfying a similar recursive formula and this produces a first-order perturbation term in the Voronovskaja formula. As a consequence of the Trotter's theorem, we may approximate the solutions of suitable second-order degenerate parabolic problems, which are of particular interest as gene frequency models in population genetics. With respect to the classical Bernstein operators, these new sequences of operators allow to consider new factors as mutation, migration and selection in the associated diffusion processes.

AMS Subject Classification (1991): 41A36, 60J70, 34A45, 92D15

Keyword(s): Bernstein-type Operators, Positive Approximation, Population Genetics

Received November 15, 2000. (Registered under 2836/2009.)

Abstract. In the one- and two-dimensional cases it has been proved (see [1], [2]) that the dyadic Cesàro operator is bounded on the spaces $L^p[0,1)$, $L^p([0,1) \times[0,1))$ $(1\le p< \infty )$ and on the dyadic Hardy spaces $H^1[0,1)$, $H^1([0,1)\times[0,1)$), but it is not bounded on the spaces VMO$[0,1)$, $L^\infty[0,1)$ and $L^\infty([0,1)\times[0,1)$). It is also proved that the continuous variant of the dyadic Cesàro operator is bounded on $L^p[0,\infty )$ (see [3]). In this paper we prove that the operator is bounded on the Hardy spaces $H^p[0,1)$ ($1/2< p\le1$) wich gives a new proof for boundedness of the Cesàro operator on the Hardy space $H^1[0,1)$.

AMS Subject Classification (1991): 42B30, 42C10, 42B05

Received November 6, 2000, and in revised form February 12, 2001. (Registered under 2837/2009.)

Abstract. Let $\{f_n\} $ be an orthonormal system in $L^2[0,1]$ (ONS). It is called a system of convergence if the orthogonal series in $L^2$ (OS) $\sum c_nf_n(x), x\in[0,1], \{c_n\} \in l^2$, is convergent a.e. for any $c_n$. The following Kolmogorov--Men'shov problem is classical: for an arbitrary ONS $\{f_n\} $, does there exist a rearrangement $\{f_{\tau_n}\} $ that is a system of convergence? The answer is not known. In this note we consider a similar problem in which the convergence of OS after a rearrangement is replaced by the summability by methods of the class $\Phi\Lambda $. This class contains a number of well-known special summability methods. We find conditions on a method $(\varphi,\lambda )\in $$\Phi\Lambda $ sufficient for the existence, for any ONS, of a rearrangement $\{f_{\tau_n}\} $ such that the OS $\sum c_nf_{\tau_n}(x)$ is $(\varphi,\lambda )$-summable a.e. for any $c_n$.

AMS Subject Classification (1991): 42C15, 40A30

Received January 29, 2001. (Registered under 2838/2009.)

Abstract. The sharp bounds of exponents, for which inclusions of Muckenhoupt classes in Gehring classes are valid, are obtained for the one-dimensional case.

AMS Subject Classification (1991): 46E30, 26A48

Keyword(s): Muckenhoupt classes, Gehring classes, reverse Hölder inequality

Received September 5, 2000, and in revised form November 12, 2001. (Registered under 2839/2009.)

Abstract. A concept of an orthonormal basis for Hilbert $C^*$-modules is discussed. It is proved that each Hilbert $C^*$-module $W$ over an arbitrary $C^*$-algebra ${\cal A}$ of (not necessarily all) compact operators on a Hilbert space possesses an orthonormal basis. The $C^*$-algebra of all adjointable operators on $W$ is naturally represented on a Hilbert space contained in $W$. Also, ``compact" operators on $W$ are characterized.

AMS Subject Classification (1991): 46L05, 46C50

Keyword(s): C^*, -algebra, C^*, Hilbert-module, orthonormal basis, compact operator, adjointable operator

Received September 26, 2000, and in revised form December 20, 2000. (Registered under 2840/2009.)

Abstract. We give a complete classification of homotopy classes of partial isometries in von Neumann algebras (Theorem 3.1). This classification is particularly explicit in the case of factors (Section 4) and generalizes and explains the results obtained in [3] for type I factors.

AMS Subject Classification (1991): 46L10, 46L35, 47A05

Received November 3, 2000, and in revised form December 5, 2001. (Registered under 2841/2009.)

Abstract. The class of polynomially bornological topological vector spaces over a topological field is introduced and studied.

AMS Subject Classification (1991): 46S10

Keyword(s): topological vector spaces, polynomials, bounded sets, equicontinuous sets

Received November 24, 2000. (Registered under 2842/2009.)

Abstract. A bounded linear operator in a Banach space is called Koliha--Drazin invertible (generalized Drazin invertible) if ${0}$ is not an accumulation point of its spectrum. In this paper the main result is the stability of the Koliha--Drazin invertible operators with finite nullity under commuting Riesz operator perturbations. We also generalize some recent results of Castro, Koliha and Wei, and characterize the perturbation of the Koliha--Drazin invertible operators with essentialy equal eigenprojections at zero.

AMS Subject Classification (1991): 47A05, 47A53, 15A09

Keyword(s): generalized Drazin inverse, perturbation, Riesz operator

Received January 2, 2001, and in revised form March 26, 2001. (Registered under 2843/2009.)

Abstract. We give a sufficient condition involving local spectra for an operator on a separable Banach space to be hypercyclic. Similar conditions are given for supercyclicity. These spectral conditions allow us to characterize the hyponormal operators with hypercyclic adjoints and those with supercyclic adjoints.

AMS Subject Classification (1991): 47A10, 47A11, 47A16, 47B20, 47B40

Keyword(s): Hypercyclic, supercyclic, hyponormal, (\beta ), (\delta ), propertiesand

Received February 7, 2001, and in final form October 2, 2001. (Registered under 2844/2009.)

Abstract. The paper continues and completes the study of pure bi-isometries begun in [Po2]. Characterizations for different parts of a double commuting bi-isometry in terms of the Berger-Coburn-Lebow model [BCL] are given. Finally, as consequences, similar results are obtained for dual double commuting bi-isometries.

AMS Subject Classification (1991): 47A45

Keyword(s): bi-isometry, Wold-type decomposition, double commuting, bi-shift, unitary, completely non-unitary

Received January 2, 2001. (Registered under 2845/2009.)

Abstract. A generalization of a result of L. A. Simakova on $j_{pq}$-contractive matrix valued functions is given in this note. L. A. Simakova was able to show that a matrix valued function is $j_{pq}$-contractive matrix valued if certain linear fractional transformations defined by such a matrix function maps the class of Schur functions into the class of Schur functions. We consider an indefinite analogue of these matrix functions where the corresponding matrix kernel has $\kappa $ negative squares. These indefinite analogues can then be characterized by the fact that the mentioned linear fractional transformations defined by such a matrix function maps the class of Schur functions into a certain class of indefinite Schur functions.

AMS Subject Classification (1991): 30D50, 47A56, 47A57, 47B50

Keyword(s): J, -contractive matrix functions, linear fractional transformation, Pontryagin space, indefinite kernel

Received February 12, 2001, and in revised form May 9, 2001. (Registered under 2846/2009.)

Abstract. The Hilbert space operator $C$ is called $T$-Toeplitz if the equation $T^*CT=r(T)^2C$ holds, where $r(T)$ denotes the spectral radius of $T$. The set of all $T$-Toeplitz operators is studied, for an arbitrary bounded, linear operator $T$. It turns out that a satisfactory symbolic calculus can be given, if $T$ has a regular norm-sequence $\{\|T^n\|\} _{n=1}^\infty $. A projection mapping onto the set of $T$-Toeplitz operators is constructed, the spectral properties of the symbolic calculus are examined, and invariant subspace theorems for $T$ are derived from the study of $T$-Toeplitz operators. These investigations are also extended to the case when $T$ is replaced by a representation $\rho $ of an abelian semigroup.

AMS Subject Classification (1991): 47A62, 47A10, 47A15, 47B35

Received September 6, 2001. (Registered under 2847/2009.)

Abstract. A necessary condition will be given for the hyponormality of $C^*_\phi $ when $\phi $ is a linear fractional transformation mapping the unit disk into itself. This necessary condition is an improvement on a condition given by Sadraoui and partially answers a question raised by Cowen and MacCluer.

AMS Subject Classification (1991): 47B20, 47B33

Keyword(s): Composition Operators, Hyponormality

Received October 27, 2000, and in revised form July 5, 2001. (Registered under 2848/2009.)

Abstract. The commutants of analytic Toeplitz operators have been studied in depth. One of the main results in this area is that certain Toeplitz operators have commutants generated by the Toeplitz operators and the composition operators that commute with it. This result gives rise to a question about the commutant of a composition operator: Is the commutant of a composition operator generated by those composition operators and multiplication operators that commute with it? Certain examples suggest a modified question: Is the commutant of a composition operator generated by the multiplication operators that commute with it and the composition operator itself? We will give examples where this is not the case by constructing composition operators with commutants larger than this.

AMS Subject Classification (1991): 47B38

Received November 28, 2000, and in revised form July 3, 2001. (Registered under 2849/2009.)

Abstract. Let ${\cal A}$ be a reflexive operator algebra acting on a Hilbert space ${\cal H}$, ${\cal U}$ be a reflexive ${\cal A}$-module, and ${\cal U}_\perp $ be the preannihilator of ${\cal U}$. Suppose $\phi $ is an order homomorphism of Lat${\cal A}$ determining ${\cal U}$, that is ${\cal U}={\cal U}_\phi := \{T\in{\cal B} ({\cal H}): \phi(E)^\perp TE = 0, \forall E\in\mathop{\rm Lat}{\cal A}\} $. In this paper, it is proved that $\phi_\sim(E) = \left[{\cal U}_\perp E\right ] =\left[{\cal U}_{\phi_\sim } E\right ]$ for each $E\in\mathop{\rm Lat}{\cal A}$, where $\phi_\sim(E)$ is defined as $\vee\{F\in\mathop{\rm Lat}{\cal A}: \phi(F)\not\supseteq E \} $. We also characterize the invariant subspaces of ${\cal U}_\perp $ in terms of order homomorphisms of Lat${\cal A}$, and show that if Lat${\cal A}$ is a nest then Lat${\cal A}_\perp =\mathop{\rm Lat}{\cal A}$ if and only if Lat${\cal A}$ is maximal. Moreover, we investigate the relationships between reflexive modules and their preannihilators. If ${\cal A}$ is additionally $\sigma $--weakly generated by rank one operators, a necessary and sufficient condition for an order homomorphism to be the least one determining ${\cal U}$ is given.

AMS Subject Classification (1991): 47L05, 47L35, 47L75

Keyword(s): Reflexivity, module, preannihilator, invariant subspace, order homomorphism

Received September 26, 2000, and in final form July 13, 2001. (Registered under 2850/2009.)

Abstract. This note is to point out a misprint in the paper [1].

AMS Subject Classification (1991): 43A35 (1991): 44A60

Received March 18, 2002. (Registered under 2851/2009.)