ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In this paper we give a characterization of the following three clones of operations on binary relations: the clone of primitive-positive Tarski operations, the clone of positive Tarski operations and the clone of all Tarski operations (or the classical clone). Operations from each of the three clones can be represented by special first-order formulas; to each such formula we assign a labelled multigraph and show that an operation belongs to the respective clone if and only if the suitably transformed graph of its formula does not contain a subgraph homeomorphic to $K_4$.

AMS Subject Classification (1991): 03G15, 08A40

Keyword(s): Tarski relation algebra, clone

Received September 16, 2003, and in final form March 31, 2004. (Registered under 5825/2009.)

Abstract. We call a lattice $L$ {\it isoform}, if for any congruence relation $\Theta $ of $L$, all congruence classes of $\Theta $ are isomorphic sublattices. In an earlier paper, we proved that for every finite distributive lattice $D$, there exists a finite isoform lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$. In this paper, we prove a much stronger result: {\it Every finite lattice has a congruence-preserving extension to a finite isoform lattice}.

AMS Subject Classification (1991): 06B10, 06B15

Keyword(s): Congruence lattice, congruence-preserving extension, isoform, uniform

Received February 26, 2004. (Registered under 5826/2009.)

Abstract. It is shown that certain cubic polynomials with only real roots belong to the set of CNS polynomials. The results support a conjecture of B. Kovács and A. Pethő (1991) on bases of canonical number systems in real cubic number fields.

AMS Subject Classification (1991): 11A63, 11C08, 11R04, 11R16, 12D99

Received December 30, 2003, and in final form April 18, 2004. (Registered under 5827/2009.)

Abstract. We investigate sequences of real numbers and $p$-adic numbers with the property that sums of pairs are always far apart.

AMS Subject Classification (1991): 11B05, 11B50, 11B75

Received May 21, 2003, and in revised form December 3, 2003. (Registered under 5828/2009.)

Abstract. The problem of nonnegative realizations of a transfer function (i.e. of a rational matrix function vanishing at infinity) is an important question from the engineering and a highly nontrivial one from the mathematical point of view. We give an upper estimate for the dimension of a nonnegative realization of a primitive scalar transfer function with simple pole at the spectral radius. We show how information on nonnegative realizations of the entries can be used for the upper estimation of the dimension of such realizations of a matrix transfer function.

AMS Subject Classification (1991): 15A48, 15A60, 93B15

Received November 11, 2003, and in revised form June 10, 2004. (Registered under 5829/2009.)

Abstract. A congruence $\rho $ on a regular semigroup $S$ is completely determined by its kernel and its trace. These two parameters induce the kernel relation $K$ and the trace relation $T$ on the lattice ${\cal C}(S)$ of congruences on $S$. The former is a complete $\wedge $-congruence and the latter is a complete congruence on ${\cal C}(S)$. The relation $K$ is generally not a $\vee $-congruence. We provide a number of necessary and sufficient conditions on $S$ for $K$ to be a congruence in terms of conditions on congruences, their kernels and their traces. When $K$ is a congruence, we consider conditions on $S$ which ensure that ${\cal C}(S)/K$ be modular. We conclude by examining the closure properties of the class of all regular semigroups $S$ for which $K$ is a congruence on ${\cal C}(S)$.

AMS Subject Classification (1991): 20M10, 20M17

Keyword(s): regular semigroup, congruence, kernel, trace, lattice congruence, modular, closure properties

Received April 1, 2003, and in final form April 14, 2004. (Registered under 5830/2009.)

Abstract. No finite commutative semigroup has a congruence preserving extension.

AMS Subject Classification (1991): 20M14

Keyword(s): congruence preserving extension, finite commutative semigroup

Received April 2, 2004, and in revised form June 27, 2004. (Registered under 5831/2009.)

Abstract. A commutative semigroup whose congruence lattice is finite is itself finite.

AMS Subject Classification (1991): 20M14

Received April 2, 2004, and in revised form June 27, 2004. (Registered under 5832/2009.)

Abstract. Wiener type characterizations are given for Hölder continuity of Green's functions at boundary points of the underlying domains. For Hölder continuity with some positive exponent it is shown that if $0$ is a boundary point of a domain $G\subset{\overline{\bf C}}$ and $G$ satisfies the cone condition at $0$, then Green's function $g_G(\cdot,a)$ is Hölder continuous at $0$ if and only if the sequence ${\cal N}_{\partial G}(\varepsilon )$ of those $n\in{\bf N}$ for which $\mathop{\rm cap} (\partial G\cap D_{2^{-n}}(0))\ge\varepsilon 2^{-n}$, is of positive lower density in ${\bf N}$. For $G=\overline{\bf C}\setminus E$ with $E\subseteq[0,1]$ the optimal Hölder 1/2 smoothness holds at 0 if and only if $\sum_k 2^k(\mathop{\rm cap} (I_k)-\mathop{\rm cap} (E_k))< \infty $ where $I_k=[0,2^{-k}]$, and $E_k$ is the union of $I_k\cap E$ with $[0,\varepsilon2^{-k}]\cup[(1-\varepsilon )2^{-k},2^{-k}]$ for some $\varepsilon < 1/3$. The corresponding uniform results are also true, and similar statements hold in higher dimensions.

AMS Subject Classification (1991): 30C85, 31A15

Keyword(s): Green's functions, Hölder continuity, logarithmic capacity, harmonic measure, cone condition, Cantor sets, Wiener type characterization

Received September 16, 2003, and in final form September 17, 2004. (Registered under 5833/2009.)

Abstract. We obtain a necessary and sufficient condition for a noninjective Toeplitz operator on $H^2$ of the unit disk to be surjective. The condition involves the extremal function for the kernel of the operator. The canonical right inverse of a surjective Toeplitz operator is shown to be a product of three Toeplitz operators.

AMS Subject Classification (1991): 30D55, 46C07, 46E22, 47B32, 47B35

Keyword(s): {Toeplitz operators, Devinatz--Widom theorem, de Branges--Rovnyak spaces, reproducing kernels, Helson--Szegő weight, extremal function}

Received May 3, 2004. (Registered under 5834/2009.)

Abstract. We study the analytic classification and the continuity properties of the holomorphic invariants of bounded continuous symmetric Reinhardt domains.

AMS Subject Classification (1991): 32M15, 58B12, 46G20

Keyword(s): Reinhardt domains, Bounded symmetric domains, ^*, JB-triples

Received April 15, 2004. (Registered under 5835/2009.)

Abstract. In the paper we deal with strong summability in families of summability methods which depend on a continuous parameter and where two different methods are connected either by a Cesàro-type or Euler--Knopp-type or Riesz-type method. We prove various results for strong summability based on these families. As particular cases, the families of generalized Nörlund methods both in matrix and in integral form, the families of Cesàro, Euler--Knopp and Riesz methods and the family of Borel-type methods are considered. This paper extends the authors investigations on ordinary summability in the families mentioned above started in [21] and [22]. Our interest in strong summability was supported by different recent papers on strong summability of orthogonal, in particular Fourier series (e.g. [18], [13], [15]).

AMS Subject Classification (1991): 40F05, 40G05

Received February 12, 2003. (Registered under 5836/2009.)

Abstract. Let $m\in{\msbm N}$ and $u$ be a sequence of nonzero terms. If $x=(x_{k})_{k=0}^{\infty }$ is any sequence of complex numbers we write $\Delta ^{(m)}x$ for the sequence of the $m$--th order differences of $x$ and $\Delta ^{(m)}_{u}X=\{x=(x)_{k=0}^{\infty }\colon u\Delta ^{(m)}x\in X\} $ for any set $X$ of sequences. We determine the $\beta $--duals of the sets $\Delta ^{(m)}_{u}X$ for $X=c_{0},c,\ell_{\infty }$, and characterize some matrix transformations between these spaces $\Delta ^{(m)} X$. Furthermore, we deal with similar problems on matrix transformations, using a well chosen Banach algebra $S_{\tau }$, and consider perturbed matrices.

AMS Subject Classification (1991): 40H05, 46A45

Keyword(s): Sequence spaces, difference sequences, matrix transformations

Received June 26, 2004. (Registered under 5837/2009.)

Abstract. The authors introduce a class of summability methods of the inverse Fourier integral on ${\msbm R}^d$, which grew out of investigations known as $\ell $-1 summability. There is a basic kernel $F_d(\rho; \cdot )$, $\rho >0$, $\rho\to \infty $, on ${\msbm R}^d$: Starting with $F_1(\rho; \cdot )=2\cos\rho (\cdot )$, on ${\msbm R}$, the kernel $F_d$ is defined inductively as Laplace convolution w.r.t. the parameter $\rho $; i.e., $F_d(\cdot;{\bf x})=F_{d-1}(\cdot;{\bf x}')*_L F_1(\cdot;x_d)$, ${\bf x}=({\bf x}',x_d)=(x_1,\ldots,x_d)$ in ${\msbm R}^d$. It defines the Dirichlet kernel of the inverse Fourier integral w.r.t. the $l_1$-norm on ${\msbm R}^d$. Here we investigate an extension of the kernel $F_d$ by introducing a set of parameters $({\bf a};{\bf b})\in{\msbm R}_+^d\times{\msbm R}^d$ and by further taking means, see the definitions below. We are interested in basic properties of the summability kernels, especially in convergence of the methods. The paper extends and, in a way, completes previous work of Y. Xu, Zh.-K. Li and the authors.

AMS Subject Classification (1991): 42B08, 26A33, 40A10

Keyword(s): summability, radial functions, fractional derivatives, Ces?ro means, Riesz means, Abel means, convolution kernels

Received July 21, 2003, and in revised form April 20, 2004. (Registered under 5838/2009.)

Abstract. Two theorems verifying the necessity of certain conditions, which imply the exact order of magnitude of partial sums and Riesz-means for orthogonal series with strict monotone sequences, are generalized such that the necessity of the same conditions is proved for almost monotone sequences.

AMS Subject Classification (1991): 42C15, 40G99

Received April 25, 2003, and in revised form June 2, 2004. (Registered under 5839/2009.)

Abstract. Let $T$ be a bounded linear operator on a Banach space $X$. Let $\bf N$ be the generalized null space of $T$ (see the terminology in the first part of the Introduction). If $\bf N$ is dense in $X$, then we call $T$ a general backward shift (a GBS). We show that when $T$ is a GBS, then $T$ has many spectral and Fredholm properties in common with classical weighted backward shifts on $l^2.$ A similar study is made of a related dual concept involving general shifts. Important examples are given of bounded linear operators to which these results apply (see the last section).

AMS Subject Classification (1991): 47A05, 47A10

Received June 16, 2004. (Registered under 5840/2009.)

Abstract. We present a von Neumann--Wold decomposition of a two-isometric operator on a general Hilbert space. A pure two-isometry is shown to be unitarily equivalent to a shift operator (multiplication by the independent variable) on a Dirichlet space $D(\mu )$ corresponding to a positive operator measure $\mu $ on the unit circle. Our result contains a previous result by S. Richter [Richter2] as well as the classical von Neumann--Wold decomposition of an isometry.

AMS Subject Classification (1991): 47A15, 31C25

Keyword(s): Two-isometry, von Neumann--Wold decomposition, Dirichlet space

Received February 26, 2004. (Registered under 5841/2009.)

Abstract. In the previous paper, we gave a characterization of backward shift invariant subspaces of the Hardy space in the bidisc which satisfy the doubly commuting condition $S_zS^*_w = S^*_wS_z$ for the compression operators $S_z$ and $S_w$. In this paper, we give a characterization of backward shift invariant subspaces satisfying $S^2_zS^*_w = S^*_wS^2_z$.

AMS Subject Classification (1991): 47A15, 32A35

Keyword(s): Hardy space, backward shift, invariant subspace

Received March 24, 2004, and in revised form June 1, 2004. (Registered under 5842/2009.)

Abstract. It is well-known (see [1], [9]) that the invariant vectors for a contraction $T$ on a Hilbert space ${\cal H}$ and for its adjoint $T^*$ coincide. As an immediate consequence one obtains the orthogonal decomposition $$ {\cal H}= \overline{{\cal R}(I_{\cal H}-T)} \oplus{\cal N}(I_{\cal H}-T), $$ where ${\cal N}(S)$ and ${\cal R}(S)$ denote the kernel, respectively the range, of an operator $S$ on ${\cal H}$. In this paper we obtain such a decomposition for a class of generalized contractions, more specifically for $A$-contractions, i.e. the operators $T \in{\cal B(H)}$ which satisfy $T^*AT \le A$, where $A$ is a positive operator on ${\cal H}$. We also derive several applications involving some decompositions for quasinormal operators and for oblique projections.

AMS Subject Classification (1991): 47A20, 47B20

Received January 5, 2004, and in revised form April 5, 2004. (Registered under 5843/2009.)

Abstract. In the present paper we establish that a contraction on a Hilbert space is a partial isometry if and only if it has a contractive generalized inverse. An equivalent characterization is that its reduced minimum modulus is greater than or equal to $1$. The first equivalence enables us to define a partial isometry in the more general context of Banach spaces.

AMS Subject Classification (1991): 47A53, 47A68, 46B04

Keyword(s): partial isometry, generalized inverse, minimum modulus, reduced minimum modulus

Received February 24, 2004, and in final form June 8, 2004. (Registered under 5844/2009.)

Abstract. We derive an index formula for band-dominated operators on $l^p({\msbm Z})$ when $1 < p < \infty $ in terms of local indices of their limit operators. This formula is applied to verify the stability of the finite section method for invertible band-dominated operators with slowly oscillating coefficients. Hilbert space versions of these results (partially under further restrictions) were obtained in [9] and [6], respectively.

AMS Subject Classification (1991): 47A53, 65J10

Received March 23, 2004, and in revised form May 3, 2004. (Registered under 5845/2009.)

Abstract. We determine which composition operators with linear-fractional symbol are essentially normal on the Bergman spaces $A^2_{\alpha }({\msbm D})$.

AMS Subject Classification (1991): 47B33

Received September 24, 2003, and in revised form March 9, 2004. (Registered under 5846/2009.)

Abstract. In this paper, we estimate the essential norm of a weighted composition operator on the ball algebra $A(B)$ and the space $H^{\infty } (B)$ of bounded holomorphic functions.

AMS Subject Classification (1991): 47B33, 46J15

Keyword(s): essential norm, weighted composition operators, ball algebra

Received March 30, 2004, and in revised form June 16, 2004. (Registered under 5847/2009.)

Abstract. In this paper, we give a correction to the main result of Zhou [16], we prove some new convergence theorems of the Ishikawa iterative process with errors for strongly pseudocontractive operators in arbitrary real Banach spaces. The results presented in this paper improve and extend important known results in [1], [3], [5], [13], [15], [16] and others.

AMS Subject Classification (1991): 47H10, 47H06, 47H17

Keyword(s): strongly pseudocontractive operator, strongly accretive operator, Ishikawa iterative process with errors

Received April 25, 2003. (Registered under 5848/2009.)

Abstract. The main result contained in this paper is the following: If $X$ is a uniform space with at least two points and ${\cal A}$ is a family of continuous selfmappings of $X$ such that ${\cal A}$ is equicontinuous on some nonempty open subset of $X$, then its second-order graph $\{(x, \varphi(x), \varphi ^2(x)) \in X^3: x \in X, \varphi\in {\cal A}\} $ is not dense in $X^3$. This improves strongly a recent result by C. Pe?a and applies to get results on infinite-dimensional holomorphy and on universal operators. For instance, we prove that if $G$ is a bounded domain in a complex Banach space then the second-order graph of the family of holomorphic selfmappings on $G$ is not dense in $G^3$. We also show that if $E$ is an infinite-dimensional separable Fréchet space then the $\infty $-graph of the family of all hypercyclic operators on $E$ is dense in $E^{{\msbm N}_0}$.

AMS Subject Classification (1991): 54H20, 30F45, 47A16, 54E15, 54H11

Keyword(s): N, -graph, equicontinuity, uniformizable space, universal operator, holomorphic selfmapping, orbit, Helly space

Received November 26, 2003, and in revised form March 18, 2004. (Registered under 5849/2009.)

Abstract. We consider a generalization $\star_\alpha $ of the fundamental star product built by Kontsevich $\star_\alpha^K$ for linear structures $\alpha$. We study closeness and relativity of these star products and we prove that the only Kontsevich star product strict and relative is $\star_\alpha^K$.

AMS Subject Classification (1991): 16S80, 16E40, 17B35

Keyword(s): Deformation quantization, star products

Received November 4, 2003. (Registered under 5850/2009.)