ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. It is proved in [CHS] that any two CD-bases in a finite distributive lattice have the same number of elements. We investigate CD-bases in posets, semilattices and lattices. It is shown that their CD-bases can be characterized as maximal chains in a related poset or lattice. We point out two known lattice classes characterized by some ``$0$-conditions" whose CD-bases satisfy the mentioned property.

AMS Subject Classification (1991): 06A06, 06B99

Keyword(s): CD-base, disjoint system, distributive pair, $0$-modular lattice

Received September 18, 2010, and in final form March 25, 2011. (Registered under 67/2010.)

Abstract. We investigate the connection between the dualisability of a finite algebra and its graph---the relational structure obtained by replacing each fundamental operation by its graph. We show that if the graph of an algebra is dualisable, then the algebra is also dualisable. The two-element meet semilattice is shown to be a counterexample to the converse. We prove that the graph of every finite algebra with a single unary operation in its type is dualisable. We also show that a duality for each finite directed path, considered as a partial algebra, can be established from a duality for its graph.

AMS Subject Classification (1991): 06D50, 06A06

Keyword(s): natural duality, graph, unar, directed path

Received June 22, 2010, and in revised form March 12, 2011. (Registered under 41/2010.)

Abstract. In this paper, we prove that a bounded poset $P$ is a pseudocomplemented poset satisfying the Stone identity if the set of all semicomplements of every element of $P$ forms an $u$-ideal which is a dual modular direct factor of $P$. Further, we prove that a bounded pseudocomplemented poset $P$ in which every normal ideal is principal satisfies the Stone identity if and only if $Id(P)$ also satisfies the Stone identity.

AMS Subject Classification (1991): 06C15, 06A12

Keyword(s): Stone poset, dual modular poset, pseudocomplemented poset

Received June 3, 2011, and in final form September 2, 2011. (Registered under 28/2011.)

Abstract. Given a presentation of a finitely generated commutative monoid $M$, suitable algorithms compute an explicit completion, the corresponding congruence ${\cal K}$, and a subdirect decomposition of $M$.

AMS Subject Classification (1991): 20M14

Keyword(s): finitely generated commutative monoids, completion, subdirect decomposition, algorithms

Received May 4, 2010, and in revised form November 6, 2010. (Registered under 33/2010.)

Abstract. In the present paper we prove two embedding theorems. Both give necessary and sufficient conditions, herewith improving and unifying some previous results.

AMS Subject Classification (1991): 26A15, 42A10

Keyword(s): strong approximation, embedding theorems

Received November 5, 2010. (Registered under 78/2010.)

Abstract. For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus ${\msbm T}^m:=[0,2\pi )^m$, let $\hat f({\bf n})$ denote the Fourier coefficient of $f$, where ${\bf n}=(n^{(1)},\ldots,n^{(m)})\in{\msbm Z}^m$. Recently, in [{\it Acta Math. Hungar.}, \bf128 \rm(2010), 328--343], we have defined the notion of bounded $p$-variation ($p\ge1$) for a complex-valued function on a rectangle $[a_1,b_1]\times\cdots \times[a_m,b_m]$ and studied the order of magnitude of Fourier coefficients of such functions on $[0,2\pi ]^m$. In this paper, the order of magnitude of Fourier coefficients of a function of bounded $p$-variation ($p\ge1$) from $[0,2\pi ]^m$ to ${\msbm C}$ and having lacunary Fourier series with certain gaps is studied and a result analogous to Theorem 2 in [\it Acta Math. Hungar.\rm, \bf104 \rm(2004), 95--104] and Theorem 2 in [\it Acta Math. Hungar.\rm, \bf128 \rm(2010), 328--343] is proved.

AMS Subject Classification (1991): 42B05, 26B30, 26D15

Keyword(s): multiple Fourier coefficient, function of bounded $p$-variation in several variables, order of magnitude

Received April 1, 2011, and in revised form June 26, 2011. (Registered under 17/2011.)

Abstract. Let $F$ be a measurable $\kappa $-indefinite generalized Toeplitz kernel defined on a, finite or infinite, interval. We prove that $F = F^{(c)} + F^{(o)}$, where $F^{(c)}$ is a $\kappa $-indefinite generalized Toeplitz kernel given by four continuous functions and $F^{(o)}$ is a positive definite generalized Toeplitz kernel which vanishes almost everywhere. We also prove an extension result for measurable $\kappa $-indefinite generalized Toeplitz kernels defined on a finite interval.

AMS Subject Classification (1991): 47B50, 47D03, 46C20, 28A20

Keyword(s): indefinite kernel, indefinite metric space, measurable, reproducing kernel space, semigroups of operators, Toeplitz kernel

Received February 4, 2011, and in revised form March 10, 2011. (Registered under 9/2011.)

Abstract. Let $M$ be an invariant subspace of the Hardy space $H^2$ over the bidisk and $N=H^2\ominus M$. It is revisited the study of $M$ on which $R_zR^*_w=R^*_wR_z$ and of $N$ on which $S_zS^*_w=S^*_wS_z$.

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

Keyword(s): invariant subspace, backward shift invariant subspace, Hardy space, compression operator, two-variable Jordan blocks

Received January 18, 2011, and in revised form March 16, 2011. (Registered under 5/2011.)

Abstract. Although the characterization of ring derivations has an extensive literature, up to now, all of the characterizations have had the following form: additivity and another property imply that the function in question is a derivation. The aim of this note is to point out that derivations can be described via a single equation.

AMS Subject Classification (1991): 39B50, 13N15

Keyword(s): derivation, Cauchy difference, cocycle equation

Received July 22, 2010, and in revised form March 7, 2011. (Registered under 46/2010.)

Abstract. We investigate some properties of an algebraic operator $A$ on a general vector space $X$ and especially in the case when $X$ is a locally convex space. We prove that $A$ is always hyporeflexive and that it is reflexive if its minimal polynomial is simple. Moreover, we show that this condition is necessary and sufficient for the reflexivity of the commutant of $A$. We also show that the second commutant of $A$ is equal to the algebra generated by $A$ and the identity operator. In the last section we prove that every locally algebraic operator acting on a Fréchet space is algebraic, and that an operator which is a finite rank perturbation of an algebraic operator is again algebraic.

AMS Subject Classification (1991): 46A03, 46A04, 47A15, 47A99, 47L10

Keyword(s): locally convex space, algebraic operator, nilpotent operator, invariant subspace, reflexivity, hyporeflexivity

Received June 9, 2010, and in revised form October 14, 2010. (Registered under 38/2010.)

Abstract. We study the compactness of $b$-weakly compact operators. As a consequence, we characterize Banach lattices for which the class of $b$-AM-compact operators coincides with that of compact operators (resp. weakly compact operators).

AMS Subject Classification (1991): 46A40, 46B40, 46B42

Keyword(s): order continuous norm, Schur property, $b$-AM-compact operator, compact operator, $b$-weakly compact operator

Received November 15, 2010, and in revised form December 22, 2010. (Registered under 81/2010.)

Abstract. In this paper we consider thickness and thinness of the unit sphere for infinite dimensional Banach spaces, as proposed by Whitley in [Wh]. We compute the thickness of several outer sums of Banach spaces and derive conclusions about the almost Daugavet property for those spaces. We also find estimates of the thickness of unit sphere in $L_p$ spaces, which allows to decide whether those spaces are almost Daugavet. In the last section we also present some results on the thinness of the unit sphere of some Banach spaces.

AMS Subject Classification (1991): 46B03, 46B25, 46B04

Keyword(s): thickness and thinness of a Banach space, almost Daugavet property

Received March 8, 2011, and in final form July 21, 2011. (Registered under 15/2011.)

Abstract. Let $\varphi $ be a holomorphic map on the open unit disk ${\msbm D}$ such that $\varphi({\msbm D}) \subset{\msbm D}$ and $H({\msbm D})$ be the space of holomorphic functions on ${\msbm D}.$ For a non-negative integer $n,$ we define linear operators $I^n_{\varphi }$ and $J^n_{\varphi }$ as $I^n_{\varphi }f = (f^{(n)} \circ\varphi )$ and $J^n_{\varphi }f = (f^{(n)} \circ\varphi )', f \in H({\msbm D}),$ respectively, where $f^{(n)}$ denotes the $n$-th derivative of $f.$ In this paper, we characterize boundedness and compactness of $I^n_{\varphi }$ and $J^n_{\varphi }$ between Hardy and weighted Bergman spaces. We also compute the essential norms of $I^n_{\varphi }$ and $J^n_{\varphi }$ acting between these spaces.

AMS Subject Classification (1991): 47B33, 46E10, 30D55

Keyword(s): generalized composition operator, Hardy space, Bergman space, Nevanlinna counting function

Received June 29, 2010, and in revised form September 29, 2010. (Registered under 44/2010.)

Abstract. In this article we give some sufficient conditions for the boundedness and compactness of a weighted composition operator on the Hardy spaces $H^p$, $1\leq p< \infty $, in terms of an appropriate weighted counting function and a Volterra type integral operator.

AMS Subject Classification (1991): 47B33, 47B38; 46E15

Keyword(s): weighted composition operators, counting function

Received April 20, 2011, and in revised form May 11, 2011. (Registered under 21/2011.)

Abstract. If $T$ or $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator acting on an infinite-dimensional separable Hilbert space, then we prove that generalized Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$, where $H(\sigma(T))$ denotes the set of all analytic functions in a neighborhood of $\sigma(T)$. Moreover, if $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. Also, we prove that the spectrum, Weyl spectrum and Browder spectrum are continuous on the class of all quasi-class $(A, k)$ operators.

AMS Subject Classification (1991): 47A10, 47A53, 47B20

Keyword(s): algebraically quasi-class $(A, k)$ operator, generalized Weyl's theorem, generalized $a$-Weyl's theorem, continuity of the spectrum

Received September 24, 2010, and in final form January 26, 2011. (Registered under 69/2010.)

Abstract. A Banach space operator $T\in B({\cal X})$ is left polaroid if for each $\lambda\in \mathop{\rm iso} \sigma_a(T)$ there is an integer $d(\lambda )$ such that $\mathop{\rm asc} (T-\lambda )=d(\lambda )< \infty $ and $(T-\lambda )^{d(\lambda )+1}{\cal X}$ is closed; $T$ is finitely left polaroid if $\mathop{\rm asc} (T-\lambda )< \infty $, $(T-\lambda ){\cal X}$ is closed and $\dim(T-\lambda )^{-1}(0)< \infty $ at each $\lambda\in \mathop{\rm iso} \sigma_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $\tau_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$ if and only if $0\not\in\mathop{\rm iso} \sigma_a(A\otimes B)$; a similar result holds for $\tau_{AB}$ for finitely left polaroid $A$ and $B^*$.

AMS Subject Classification (1991): 47A80, 47A53, 47A10

Keyword(s): Banach space, left polaroid operator, finitely left polaroid operator, tensor product, left-right multiplication, generalized $a$-Weyl's theorem

Received December 6, 2010. (Registered under 84/2010.)

Abstract. In a purely multi-variable setting (i.e., the issues discussed in this note are not interesting in the single-variable operator theory setting), we show that the coincidence of two operator-valued Schur class multipliers of a certain kind on the Drury--Arveson space is characterized by the fact that the associated colligations (or a variant, obtained canonically) are `unitarily coincident' in a sense to be made precise in this article.

AMS Subject Classification (1991): 47A13, 47A56

Keyword(s): weakly coisometric realizations, Schur class, Drury--Arveson space, de Branges--Rovnyak space

Received June 9, 2010, and in revised form November 10, 2010. (Registered under 37/2010.)

Abstract. In [9] the shift index $\kappa(T)$ of a contraction $T$ acting on a Hilbert space is defined: $\kappa(T)$ is the supremum of $n$ such that $S_n$ can be injected into $T$, where $S_n$ is the unilateral shift of multiplicity $n$. In [11] the following question is posed: if $T$ is a $C_{10}$-contraction and its unitary asymptote is a reductive unitary operator, then $\kappa(T)=\infty $? In this paper, a positive answer to this question is given. A combination of the answer to this question with results of [11] gives that, for a $C_{10}$-contraction $T$, $\kappa(T) < \infty $ if and only if $T$ is a quasiaffine transform of $S_n$ for some finite $n$.

AMS Subject Classification (1991): 47A45

Keyword(s): contraction, unilateral shift, injection

Received January 19, 2011, and in final form January 20, 2012. (Registered under 6/2011.)

Abstract. We consider local symmetric semigroups of Hilbert space operators. For an open semigroup ${\eufm S}$ in some topological group and a dense subsemigroup ${\eufm S}'$ of ${\eufm S}$, these are semigroups of unbounded selfadjoint operators $(H(t))_{t \in{\eufm S}'}$ that admit local continuous extensions to open subsets of ${\eufm S}$. We study the possibility to continuously extend $H(\cdot )$ to a semigroup of selfadjoint operators defined for all $t \in{\eufm S}$ in several settings. Integral representation formulae for the extended semigroups $(H(t))_{t \in{\eufm S}}$ by means of real characters of ${\eufm S}$ are established. Our proofs rely on graph limits of selfadjoint operators, commutativity of unbounded operators and semigroup techniques, among others.

AMS Subject Classification (1991): 47D03, 47B15, 47B25

Keyword(s): local semigroups of operators, integral representation, selfadjoint operators

Received May 14, 2010, and in revised form December 17, 2010. (Registered under 35/2010.)

Abstract. We answer a question raised by Nordgren and Rosenthal about the Schatten--von Neumann class membership of operators in standard reproducing kernel Hilbert spaces in terms of their Berezin symbols.

AMS Subject Classification (1991): 47B38, 4B07; 47B35

Keyword(s): Berezin symbols, compact operators, Schatten--von Neumann classes, reproducing kernel Hilbert space, model spaces

Received September 1, 2011, and in revised form October 25, 2011. (Registered under 42/2011.)

Abstract. We prove asymptotic formulas for best approximations of planar spindle convex sets by inscribed and circumscribed convex disc-polygons with respect to the Hausdorff metric, the area deviation, and the perimeter deviation.

AMS Subject Classification (1991): 52A10

Keyword(s): approximations, disc-polygons, spindle convexity

Received March 7, 2011, and in final form August 4, 2011. (Registered under 14/2011.)

Abstract. The purpose of the present paper is to investigate the extension problem for the category of commutative hypergroups. In fact, we determine all extensions of the Golden hypergroup by locally compact abelian groups.

AMS Subject Classification (1991): 43A62, 20N20

Keyword(s): hypergroup, extension, group

Received October 14, 2010. (Registered under 71/2010.)