ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We generalize results concerning the relation between semimodularity and irreducible elements.

AMS Subject Classification (1991): 06A06

Keyword(s): Semimodular lattice, ordered set, completely join-irreducible element

Received September 30, 1997 and in revised form March 16, 1998. (Registered under 3292/2009.)

Abstract. In this paper we study the existence of simultaneous representations of finite distributive lattices by congruences of finite atomistic lattices. We prove a 2-dimensional theorem that can be interpreted as an amalgamation property for the category of finite atomistic lattices with congruence preserving embeddings as morphisms.

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

Received January 20, 1998. (Registered under 3293/2009.)

Abstract. In the early eighties, A. Huhn proved that if $D$, $E$ are finite distributive lattices and $\psi\colon D \to E$ is a $\{0\} $-preserving join-embedding, then there are finite lattices $K$, $L$ and there is a lattice homomorphism $\varphi\colon K \to L$ such that $\mathop{\rm Con}K$ (the congruence lattice of $K$) is isomorphic to $D$, $\mathop{\rm Con}L$ (the congruence lattice of $L$) is isomorphic to $E$, and the natural induced mapping $\mathop{\rm ext}\varphi\colon \mathop{\rm Con}K \to\mathop{\rm Con}L$ represents $\psi $. The present authors with H. Lakser generalized this result to an arbitrary $\{0\} $-preserving join-homomorphism $\psi $. It was also A. Huhn who introduced the {\it $2$-distributive identity}: $$ x \wedge(y_1 \vee y_2 \vee y_3)= (x \wedge(y_1 \vee y_2)) \vee(x \wedge(y_1 \vee y_3)) \vee(x \wedge(y_2 \vee y_3)). $$ We shall call a lattice {\it doubly $2$-distributive}, if it satisfies the $2$-distributive identity and its dual. In this note, we prove that {\it the lattices $K$ and $L$ in the above result can be constructed as doubly $2$-distributive lattices}.

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

Keyword(s): Lattice, finite, congruence, distributive, join-homomorphism, 2, -distributive

Received May 7, 1998 and in revised form August 25, 1998. (Registered under 3294/2009.)

Abstract. Within the class of semimodular lattices of finite length we give a common approach to the proofs of modularity and strongness by means of forbidden sublattices.

AMS Subject Classification (1991): 06C10, 06C05

Received May 7, 1998 and in revised form July 14, 1998. (Registered under 3295/2009.)

Abstract. In this paper we thoroughly investigate several kinds of residuated ordered structures, connected with propositional logics. In particular we give ternary deduction terms for several classes of algebras, that are the equivalent algebraic semantics of deductive systems, coming from logics not necessarily satisfying the structural rules.

AMS Subject Classification (1991): 08A99, 06F05

Received December 27, 1996, in revised form April 20, 1998 and in final form May 18, 1998. (Registered under 3296/2009.)

Abstract. We characterize varieties of algebras for which the congruence kernel is determined by any congruence class.

AMS Subject Classification (1991): 08B05, 08A30

Keyword(s): Regularity, weak regularity, local regularity, Mal'cev condition

Received February 26, 1998 and in final form June 15, 1998. (Registered under 3297/2009.)

Abstract. The first part of the paper simplifies proofs of some basic theorems of the original paper [4]. The second part presents a general method for constructing dual discriminator varieties with $2^{\aleph_0}$ subvarieties, in contrast to the complicated single variety given in [1].

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

Keyword(s): Varieties, filtrality, dual discriminators, definable principal congruences

Received August 15, 1995 and in revised form June 22, 1998. (Registered under 3298/2009.)

Abstract. We associate a certain lattice with every relational set. We characterize finite irreducible relational sets by the property that their associated lattice leaves a lattice if its top element is removed. This characterization is somewhat dual to that of subdirectly irreducible algebras by their congruence lattices. As a corollary we prove that if the idempotent clone related to a finite relational set $P$ is trivial then $P$ is irreducible. A stronger version of irreducibility is also explored.

AMS Subject Classification (1991): 08C05, 08A05, 08B25

Keyword(s): relational sets, irreducible relational sets, retract, product

Received March 10, 1998 and in final form June 16, 1998. (Registered under 3299/2009.)

Abstract. (1) For any $\alpha < \omega_1$ there exists a Borel measurable function $g\colon{\msbm R}\to{\msbm R}$ such that $g+f$ is a Darboux function (is almost continuous in the sense of Stallings) for every $f\in B_{\alpha }$. This solves a problem of J. Ceder. (2) There is a function $g$ that is universally measurable and has the Baire property in restricted sense such that $g+f$ is Darboux for every Borel measurable function $f$. (3) There is $g\colon{\msbm R}\to{\msbm R}$ such that $f+g$ is extendable for each $f\colon{\msbm R}\to{\msbm R}$ that is Lebesgue measurable (has the Baire property). (4) For every $\alpha < \omega_1$, each $f\in B_{\alpha }$ is the sum of two extendable functions $f_1,f_2\in B_{\alpha }$. This answers a question of A. Maliszewski.

AMS Subject Classification (1991): 04A15, 26A15, 28A05, 28A20

Keyword(s): Darboux functions, almost continuous functions, extendable functions, universal set, universal function, universal summand

Received October 16, 1997 and in revised form May 8, 1998. (Registered under 3300/2009.)

Abstract. Levin--Carleman type inequalities are obtained for a class of weighted geometric means that generalize inequalities of Cochran and Lee and of Love.

AMS Subject Classification (1991): 26D15

Received April 3, 1998 and in revised form June 30, 1998. (Registered under 3301/2009.)

Abstract. We give a necessary and sufficient condition for a real trigonometric series to be the Fourier series of a function of bounded mean oscillation (BMO).

AMS Subject Classification (1991): 30D50, 42A24

Received October 16, 1997 and in revised form March 20, 1998. (Registered under 3302/2009.)

Abstract. If $\{M_{k}\} _{k=0}^{\infty }$ is a logarithmically convex sequence of positive real numbers such that for all $k,$ $\rho ^{-k}M_{k}\leq M_{k+1}\leq\rho ^{-k}M_{k}$ for some $\rho >1,$ then it is proved that a function $f$ of $n$ real variables which is separately ultradifferentiable, with an additional ambient hypothesis, of class $C\{M_{k}\} $ in each variable is necessarily jointly ultradifferentiable of class $C\{M_{k}\} $.

AMS Subject Classification (1991): 30D60, 46F05

Keyword(s): Ultradifferentiable functions, real analytic, quasi-analytic, separate analyticity

Received December 23, 1997. (Registered under 3303/2009.)

Abstract. As a common generalization of the formulas due to Minton (1970), Karlsson (1971) and Chu (1994), a well-poised bilateral hypergeometric identity is established by means of partial-fraction expansions. The basic counterpart is also obtained by the Cauchy residue method. Several terminating cases are considered and dualized, by using the inverse series relations of Chu (1993), which results in some strange hypergeometric evaluations.

AMS Subject Classification (1991): 33A30, 05A19

Keyword(s): Inverse series relations, Partial-fraction expansion, The meromorphic function, The very well-poised series, Bilateral hypergeometric series

Received December 15, 1997. (Registered under 3304/2009.)

Abstract. The classical Hardy spaces $H_p({\msbm R})$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p({\msbm R})$ to $L_p({\msbm R})$ $(1/2< p< \infty )$ and is of weak type (1,1). As a consequence we obtain that the Fejér means of a function $f \in L_1({\msbm R})$ converge a.e. to $f$. Moreover, we prove that the Fejér means are uniformly bounded on the spaces $H_p({\msbm R})$ whenever $1/2< p< \infty $. Thus, in case $f \in H_p({\msbm R})$, the Fejér means converge to $f$ in $H_p({\msbm R})$ norm $(1/2< p< \infty )$. The same results are proved for the conjugate Fejér means, too.

AMS Subject Classification (1991): 42A38, 42B30

Keyword(s): Hardy spaces, p-atom, interpolation, Fourier transforms, Fejér means

Received September 10, 1997 and in revised form April 3, 1998. (Registered under 3305/2009.)

Abstract. Making use of the methods elaborated by F. Weisz [2], we shall prove an extension of Theorem 1 in [3].

AMS Subject Classification (1991): 42A38, 42B30

Received March 19, 1998 and in revised form April 23, 1998. (Registered under 3306/2009.)

Abstract. In this paper we find the spaces of multipliers $(H^{p,q.\alpha },l^s)$, except when $1< p< 2$ and $0< s< p/(p-1)$, and, for certain values of parameters the spaces of multipliers $(H^{p,q,\alpha },H^{u,v,\beta })$, where $H^{p,q,\alpha }$ denotes the space of analytic functions on the unit disc such that $(1-r)^\alpha M_p(r,f)\in L^q(dr/(1-r))$. In particular, we calculate the multipliers from Bergman and Hardy spaces into Bloch space.

AMS Subject Classification (1991): 42A45

Received February 10, 1998 and in revised form July 23, 1998. (Registered under 3307/2009.)

Abstract. We consider finite subsets $\Lambda\subset {\msbm Z}^d$ which posses the extension property, namely that every collection $\{c_k\} _{k\in\Lambda -\Lambda }$ of complex numbers which is positive with respect to $\Lambda $ is the restriction to $\Lambda -\Lambda $ of the Fourier coefficients of some positive measure on ${\msbm T}^d$. Using matrix extension methods, we recover two recent results by J.P. Gabardo and introduce a new class of subsets which posses the extension property. The maximum entropy extensions are explicitly constructed for each class of finite index sets in ${\msbm Z}^2$ which posses the extension property.

AMS Subject Classification (1991): 42B99, 42A70, 47A57, 62M15, 94A17

Keyword(s): multidimensional trigonometric moment problem, positive definite extension, maximum entropy extension

Received November 3, 1997 and in revised form March 2, 1998. (Registered under 3308/2009.)

Abstract. We consider the summability of orthogonal series (OS) $\sum c_nf_n(x)$, $f_n\in L^2[0,1]$, $\{c_n\} \in l^2$, by the generalized methods of the summatorial function $(SF^*,\varphi,\lambda )$. We prove a theorem on a condition sufficient for the absolute summability of OS a.e. by these methods. This theorem is an analogue of the theorem on Ces?ro methods in \cite8 and also a generalization of one of the author's theorems in [11]. Applications to the particular methods of the form $(SF^*,\varphi,\lambda )$ are indicated, including the Riesz and the Bernstein--Rogosinski methods.

AMS Subject Classification (1991): 42C15

Received May 23, 1998. (Registered under 3309/2009.)

Abstract. We construct a new majorant for the consecutive partial sums of a finite sum $\sum ^n_{k=1}\xi_k$ whose terms are pairwise orthogonal vectors in a noncommutative $L_2(${\eufm A},$\phi )$ space. Here {\eufm A} is a $\sigma $-finite von Neumann algebra, and $\phi $ is a faithful and normal state defined on {\eufm A}. We extend a theorem of Tandori [15] on the convergence of the orthogonal series $\sum ^\infty_{k=1} \xi_k$ from the classical commutative case to the noncommutative one, in terms of bundle convergence. As it is known, hence almost sure convergence follows. The condition imposed on $\|\xi_k\|^2$ in our theorem is weaker than that in the noncommutative Rademacher-Menshov theorem proved by Hensz, Jajte and Paszkiewicz [5]. We also deduce an improved strong law of large numbers for an orthogonal sequence $(\xi_k)$ of vectors in $L_2(${\eufm A},$\phi )$ as well as new criteria for bundle convergence of a given subsequence of the partial sums of the series $\sum ^\infty_{k=1}\xi_k$. As a by-product, we improve a theorem of Hensz [3] by weakening the condition and strengthening the conclusion in it.

AMS Subject Classification (1991): 46L50, 60F15, 42C15

Keyword(s): von Neuman algebra, faithful and normal state, scalar product, prehilbert space, $L_2$-completion, Gelfand-Naimark-Segal representation theorem, cyclic vector, bundle convergence, almost sure convergence, orthogonal vectors in noncommutative setting, Rademacher-Menshov inequality and theorem, convergence of a given subsequence of partial sums, Cesàro average of partial sums, strong law of large numbers

Received September 25, 1997 and in revised form April 6, 1998. (Registered under 3310/2009.)

Abstract. A theorem of Fillmore and Williams implies that a bounded operator $A$ on a separable Hilbert space $H$ is compact if and only if it satisfies $\lim_n Ae_n=0$, or equivalently, $\lim_n (Ae_n|e_n)=0$ for each orthonormal basis $(e_n)$ for $H$. In the present note this theorem is reproved using the fact observed by Halmos that each sequence of unit vectors weakly converging to $0$ approximately contains an orthonormal subsequence. It is also noted that the stronger version of the theorem remains true, namely without the continuity assumption on $A$. In the second part of the note the bounded operators for which there exists an orthonormal basis such that either of the above equalities holds are exhibited and completely described.

AMS Subject Classification (1991): 47A10, 47B07

Received February 16, 1998 and in revised form July 27, 1998. (Registered under 3311/2009.)

Abstract. We generalize the concept of the Littlewood-Richardson sequence associated with an invariant subspace of a nilpotent operator on a finite dimensional vector space to the context of $C_0$-contractions. The similarity invariants of nilpotent operators (decreasing sequences of sizes of the Jordan blocks) are replaced by the quasisimilarity invariants of $C_0$-contractions (sequences of inner functions).

AMS Subject Classification (1991): 47A45, 47A20, 15A23

Keyword(s): Littlewood-Richardson sequences, similarity, Jordan models, $C_0$-invariant subspaces of operators in the class

Received November 3, 1997. (Registered under 3312/2009.)

Abstract. For a class of closed symmetric, not necessarily semibounded, operators with defect numbers $(1,1)$, we introduce a generalization of the Kreĭn-von Neumann extension. By a formal inversion of the graphs of the underlying linear operators or relations this extension is related to a generalization of the Friedrichs extension. We characterize the Kreĭn--von Neumann extension in different ways, by the so-called $Q$-functions and by certain quadratic forms. Models involving triplet spaces or the spectral measure are also presented.

AMS Subject Classification (1991): 47A70, 47B15, 47B25, 47A55, 47A57

Keyword(s): Symmetric operator, selfadjoint extension, Friedrichs extension, Kreĭn-von Neumann extension, Q-function, Nevanlinna function, operator model, domain perturbation

Received October 28, 1997 and in revised form July 31, 1998. (Registered under 3313/2009.)

Abstract. In this paper we discuss various generalizations of the Fock space. We will prove that for a bounded radial function $f$ on ${\msbm C}^n$ the Toeplitz operator $T_f$ is compact on these generalized Fock spaces if and only if its Berezin transform has zero limit.

AMS Subject Classification (1991): 47B35, 47B38

Received February 26, 1998. (Registered under 3314/2009.)

Abstract. The aim of this paper is to show that the automorphism and isometry groups of the infinite direct sum $\ell_\infty({\msbm N}, {\cal B}({\cal H}))$ of operator algebras are topologically reflexive which, as one of our former results shows, is not the case with the ``scalar algebra" $\ell_\infty $.

AMS Subject Classification (1991): 47B49, 46L40, 47D25 54D35

Keyword(s): Reflexivity, automorphism group, isometry group, operator algebra, Stone-Čech compactification

Received October 13, 1997 and in revised form March 16, 1998. (Registered under 3315/2009.)

Abstract. A pseudo-resolvent on a Banach space, indexed by positive numbers and tempered at infinity, gives rise to a bounded strongly continuous one-parameter semigroup $S$ on a closed subspace of the ambient Banach space. We prove that the range space of the pseudo-resolvent contains the domain of the generator of $S$, and is contained in the Favard class of $S$, which consists of all uniformly Lipschitz vectors for $S$. We explore when some or all of these three spaces coincide.

AMS Subject Classification (1991): 47D03, 47A10, 46J25

Keyword(s): Favard class, one-parameter semigroup, pseudo-resolvent, uniformly Lipschitz vector

Received January 16, 1998. (Registered under 3316/2009.)

Abstract. It is known that the infinitesimal generator of every $C_0$-group of operators in a hereditarily indecomposable Banach space is necessarily a bounded operator [20]. For $C_0$-semigroups this is not the case in general. We present certain classes of $C_0$-semigroups whose infinitesimal generators are always bounded operators. It is also shown that the infinitesimal generator of any strongly continuous, non-quasianalytic cosine family in a hereditarily indecomposable Banach space is necessarily a bounded operator.

AMS Subject Classification (1991): 47D03, 47D09

Received July 16, 1997. (Registered under 3317/2009.)

Abstract. In this note, we discuss the problem of membership in the classes ${\msbm A}_{m,n}$ (subclasses of the set of absolutely continuous contractions on Hilbert space with isometric Sz. Nagy--Foias functional calculus) through concrete examples as well as related topics. In particular we prove that $S\oplus S^*\not\in{\msbm A}_{2,2}$, where $S$ denotes the unilateral shift of multiplicity one. Besides considering a series of examples, we introduce intermediate additional subclasses which make the characterization of the classes ${\msbm A}_{n,n}$ somewhat clearer. We also study the question of the effect on the membership in ${\msbm A}_{m,n}$ when certain direct summands are ``removed''. In particular we prove that if $A\in C_0$, then $T\oplus A\in{\msbm A}_{m,n}$ implies that $T\in{\msbm A}_{m,n}$.

AMS Subject Classification (1991): 47D27, 47A20, 47A15

Received August 4, 1997 and in revised form May 29, 1998. (Registered under 3318/2009.)

Abstract. We find the maximal ideal spaces of a class of C*-algebras of pseudodifferential operators, called comparison algebras in [2], related to Schrödinger type expressions. This is achieved by constructing some multiplicative linear functionals explicitly, made possible by an improvement of a method of H. H. Sohrab. These algebras are useful in studying the Fredholm and spectral properties of partial differential operators (see [2]).

AMS Subject Classification (1991): 47F05, 35P99

Received November 5, 1997. (Registered under 3319/2009.)