ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. This paper discusses completeness problems in $r$-valued set logic, which is the logic of functions mapping $n$-tuples of subsets into subsets over $r$ values. Boolean functions are convenient choices as building blocks in the design of set logic functions. A set of functions $F$ is Boolean complete if any set logic function can be composed from $F$ once all Boolean functions (where constants are excluded) are added to $F$. This paper proves that there are $2^r+w(r)-3$ Boolean maximal sets in $r$-valued set logic (here $w(r)$ is the number of equivalence relations on an $r$-element set) and gives a description of them using equivalence- and unary central relations. A set of functions $F$ is then Boolean complete iff it is not a subset of any of these Boolean maximal sets. This is a completeness criterion in many-valued set logic under compositions with Boolean functions. The lattice of clones containing all Boolean functions is also studied.

AMS Subject Classification (1991): 03B50, 68Q05

Received February 27, 1996. (Registered under 6119/2009.)

Abstract. We prove that if ${\cal V}$ is a variety with factorable congruences in which every member can be represented as a Boolean product of directly indecomposable algebras, then ${\cal V}$ is a discriminator variety.

AMS Subject Classification (1991): 08A05, 08A30, 08A40, 08B10; 06E15

Keyword(s): Boolean product, discriminator variety, factorable congruences, Boolean factor congruences, Pierce sheaf, principal congruence formula

Received January 5, 1994 and in revised form May 23, 1996. (Registered under 6120/2009.)

Abstract. For any numbers $m$, $n$ with $1< m\le n$ let $\rho_m (n) = \max\sum _{i=1}^r 1/a_i$, where the maximum is taken over all integers $a_1, a_2,\ldots, a_r$ $(r>1)$ which satisfy $1\le a_1 < a_2 < \ldots < a_r\le m$ and $[a_i, a_j]> n$ $(1\le i < j\le r)$. In this paper we derive some properties of $\rho_m (n)$. For example one of our results implies that $\rho_{\pi(n)} (n) \to1$ ($n\to\infty $).

AMS Subject Classification (1991): 11A99

Received March 20, 1996 and in revised form June 26, 1996. (Registered under 6121/2009.)

Abstract. We obtain a formula which expresses the square of the number of solutions of certain equations of the form $Ax^2+Cy^2=N$ in terms of the number of solutions of the equations $Ax^2+Cy^2=d$ and of the number of primitive solutions of the equations $ACx^2+y^2=d$, where $d$ runs through all the divisors of $4ACN$.

AMS Subject Classification (1991): 11D09

Received April 20, 1995 and in revised form March 9, 1996. (Registered under 6122/2009.)

Abstract. This article gives a description of all congruences with finitely many classes on a finitely generated free commutative semigroup.

AMS Subject Classification (1991): 20M14

Received August 15, 1995. (Registered under 6123/2009.)

Abstract. Some reverse Hölder type inequalities yielding for monotone or quasimonotone functions of one variable have recently been obtained and applied (see e.g. [2-5], [14]). In this paper we prove a sharp multidimensional integral inequality of this type. It is pointed out that this inequality may be regarded as a unification and extension of many inequalities of this type including some recent results in [1] and [11].

AMS Subject Classification (1991): 26D15, 26D07, 26D20

Keyword(s): Inequalities, integral inequalities, reverse Hölder inequalities, monotone functions, several variables, best constants

Received October 10, 1995 and in revised form April 10, 1996. (Registered under 6124/2009.)

Abstract. We study Minkowski's inequality $$S_{ab}(x_1+x_2, y_1+y_2)\le S_{ab}(x_1, y_1)+S_{ab}(x_2,y_2) \quad(x_1,x_2, y_1,y_2\in{\msbm R}_+)$$ and its reverse where $S_{ab}$ is the two variable Gini mean defined by $$S_{ab}(x,y)=\cases{ \left(\frac{x^a+y^a}{x^b+y^b}\right)^{\frac1{a-b}} & if a-b\ne0,\cr\exp\left(\frac{x^a\ln x+y^a\ln y}{x^a+y^a}\right) & if a-b=0. }$$ Generalizing results of Beckenbach~[\B], Dresher~[\Dr], Danskin~[\Da], Losonczi~[\Ltwo], we give necessary and sufficient conditions (concerning the parameters $a,b$) for the inequality above to hold. For the reverse inequality we give necessary conditions (which are not sufficient), sufficient conditions (which are not necessary) and also a conjecture concerning the necessary and sufficient condition.

AMS Subject Classification (1991): 26D15, 26D07

Keyword(s): Gini means, Minkowski's inequality

Received March 22, 1996. (Registered under 6125/2009.)

Abstract. We give a necessary and sufficient condition for a $J$--outer measure $\mu ^*$ on a topological space under which a set $M$ is Caratheodory $\mu ^*$--measurable iff $\mu ^*(\partial M)=0$. Under this condition we can translate the criterions of Riemann integrability in terms of $\mu ^*$--measurability. Particularly, under this condition the Lebesgue's criterion depending on the additive property of the measure is determined by the lack of the measurement precision described by the equality $\mu ^*(E)=\mu ^*(\bar E)$.

AMS Subject Classification (1991): 28C15

Received February 22, 1996 and in revised form June 14, 1996. (Registered under 6126/2009.)

Abstract. We prove that if $\lambda(x,y)$ is an odd multiplier for $L({\msbm R}^2)$, then its Ces?ro mean defined by $$\sigma\lambda(u,v) : = (uv)^{-1} \int^u_0 \int^v_0 \lambda(x,y) dx dy$$ is the double Fourier transform of some function in ${\cal H}({\msbm R}\times{\msbm R})$, the Hardy space on the plane defined by R. Fefferman. As a corollary, we obtain that if $\lambda(x,y)$ is an arbitrary multiplier for $L({\msbm R}^2)$, then we have necessarily $$\int^\infty_0 \int^\infty_0 \Bigl|{1\over uv} \int^u_0 \int^v_0 \{\lambda(x,y) - \lambda(x, -y) -\lambda(-x,y) + \lambda(-x,-y)\} dx dy\Bigr| {du\over u} {dv\over v} < \infty.$$ We also present analogous results in the case of odd multipliers for $L({\cal T}^2)$ involving double Fourier series.

AMS Subject Classification (1991): 42B30

Keyword(s): double Fourier transform, double Hilbert transforms, Hardy space on product domain, Hardy inequality, Ces?ro mean, L({\msbm R}^2), {\cal H}({\msbm R}\times{\msbm R}), multiplier forand, double Fourier series, conjugate functions, arithmetic mean, L({\msbm T}^2), {\cal H}({\msbm T}\times{\msbm T}), multiplier forand

Received June 20, 1995 and in revised form April 30, 1996. (Registered under 6127/2009.)

Abstract. Let $X_1$ and $X_2$ be real Banach spaces. Let $K$ be a weakly compact subset in $X_2$, $A\colon X_1\to X_2$ be a closed linear map and $\phi $ be a bounded linear functional on $X_1$. We consider the following linear programming problem: $$\hbox{Maximize }\phi(x) \hbox{ subject to }Ax\in K.$$ Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of $A$ and an optimal solution of a linear program in the dual space $X_2$. These results are then applied to linear programs with equality constraints and explicitly constrained feasible sets.

AMS Subject Classification (1991): 46B, 90C

Keyword(s): Banach spaces, Linear programming problems, generalised inverses, explicit solutions

Received April 12, 1995 and in revised form December 13, 1995. (Registered under 6128/2009.)

Abstract. In this paper an attempt is made to generalize the classical spectral theorem for normal elements in $C^*$-algebras. Three types of $C^*$-algebras will be introduced which are characterised by a particular spectral property. A relatively simple and clear line of reasoning leads to a very strong form of the spectral theorem for ultraspectral $C^*$-algebras. The essential part of the proof is based on the measure theoretical representation theorem of Riesz and an extension theorem for weakly continuous linear operators in locally convex spaces. The proof is independent of the classical spectral theorem of Hilbert for bounded normal operators in Hilbert spaces. With regard to the simplicity of the conditions defining ultraspectrality, the result throws new light upon the classical spectral theorem revealing its proper reasons. Moreover, it is observed that the spectral property imposed on an ultraspectral $C^*$-algebra ensures that its set of projection is a $\sigma $-complete lattice with respect to the natural ordering. This is a new remarkable relation between spectrality and general probabilistic aspects of $C^*$-algebras.

AMS Subject Classification (1991): 46L05, 46A22

Received September 4, 1995. (Registered under 6129/2009.)

Abstract. We give characterization for the range and the domain of the square root of smallest (Krein--von Neumann) and largest (Friedrichs) positive self-adjoint extension of an operator. In the last theorem we give range characterization for the square root of the largest positive extension of smallest possible norm, if bounded positive extension exists.

AMS Subject Classification (1991): 47A05, 47A20, 47B25

Keyword(s): Krein--von Neumann, Friedrichs extension

Received May 6, 1996 and in revised form September 11, 1996. (Registered under 6130/2009.)

Abstract. For a bounded linear operator $T$ on a Banach space, the $\beta $-spectrum $\sigma_{\beta }(T)$ is defined and then is used to determine Nagy's spectral residuum. Duality theory for the $\beta $-spectrum is established and is used to prove the spectral mapping theorem for $\sigma_{\beta }(T)$. The relation between property ($\beta $) and the local spectral approximation property is also discussed. In particular, it is shown that, among other things, each operator has the local spectral approximation property in the complement of its spectral residuum. A few examples to illustrate various spectral properties of operators are also given.

AMS Subject Classification (1991): 47A10, 47B40

Received November 30, 1995 and in revised form August 22, 1996. (Registered under 6131/2009.)

Abstract. If $A$ and $B$ are compact operators on a Hilbert space, with singular values satisfying $s_j(A)=s_j(B)=s_j((A+B)/2)$, for all $j=1, 2,...$, then $A=B$. Two proofs, geometric and analytic, are given.

AMS Subject Classification (1991): 47B06, 47B07, 31A05

Received June 11, 1996. (Registered under 6132/2009.)

Abstract. The purpose of this paper is to show some spectral mapping theorems for $p$-hyponormal operators, using the concepts of angular cutting and singular integral model.

AMS Subject Classification (1991): 47B20

Keyword(s): Hilbert Space, p, -hyponormal, spectral mapping theorem

Received February 19, 1996. (Registered under 6133/2009.)

Abstract. We prove that the maximal Fejér operator is bounded from the (real) Hardy space $H^1({\msbm R})$ into $L^1({\msbm R})$, and is also bounded from $L^1({\msbm R})$ into {\it weak}-$L^1({\msbm R})$. We introduce the (hybrid) Hardy spaces $H^{(1,0)} ({\msbm R}^2)$, $H^{(0,1)} ({\msbm R}^2)$, and $H^{(1,1)} ({\msbm R}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,1)} ({\msbm R}^2)$ into $L^1({\msbm R}^2)$, and is also bounded from $H^{(1,0)} ({\msbm R}^2)$ or $H^{(0,1)}({\msbm R}^2)$ into {\it{\rm weak}}-$L^1({\msbm R}^2)$. We establish analogous results for the maximal conjugate Fejér operators, too.

AMS Subject Classification (1991): 47B28

Received February 29, 1996. (Registered under 6134/2009.)

Abstract. We consider the Riemann means of Fourier series of functions belonging to the Hardy space $H^1({\msbm T})$ or $L^1({\msbm T})$, respectively. We prove that the maximal Riemann operator as well as the maximal conjugate Riemann operator are bounded from $H^1({\msbm T})$ into $L^1({\msbm T})$. It is also true that this operator is bounded from $L^1({\msbm T})$ into weak-$L^1({\msbm T})$. On closing, we formulate a conjecture about the maximal conjugate Riemann operator of a function in $L^1({\msbm T})$.

AMS Subject Classification (1991): 47B38

Received February 29, 1996 and in revised form July 11, 1996. (Registered under 6135/2009.)

Abstract. Let $(X,{\cal A},\mu )$ be a complete finite measure space and let $T\colon X\to X$ be a measurable transformation for which $\mu\circ T^{-1}\ll\mu $ and ${d\mu\circ T^{-1}\over d\mu } \in L^\infty(X,{\cal A},\mu )$. Then $C\colon f\mapstochar\rightarrow f\circ T$ is a bounded linear operator on $L^2$. We examine the von Neumann algebra $\cal W$ generated by $C$. Two sigma algebras $\cal M$ and ${\cal A}^\prime $ play a key role in the investigation. The orthogonal projection $E^{{\cal A}^\prime }$ onto the space of ${\cal A}^\prime $ measurable functions is given by the conditional expectation with respect to the sigma algebra ${\cal A}^\prime $. This projection is always abelian in $\cal W$ and it is minimal in $\cal W$ whenever ${\cal M}\cap{\cal A}^\prime ={\cal T}$ ($\cal T$ is the sigma algebra generated by $\emptyset $ and $X$). Similarly, the orthogonal projection $E^{\cal M}$ onto the space of $\cal M$ measurable functions is always abelian in ${\cal W}^\prime $ and minimal whenever ${\cal M}\cap{\cal A}^\prime ={\cal T}$. Moreover, both $E^{\cal M}$ and $E^{{\cal A}^\prime }$ have central carrier $E^{{\cal M}\vee{\cal A}^\prime }$ (${\cal M}\vee{\cal A}^\prime $ denotes the smallest $\sigma $-algebra containing both $\cal M$ and ${\cal A}^\prime $). If ${\cal M}\vee{\cal A}^\prime ={\cal A}$ then $E^{{\cal A}^\prime }$ and $E^{\cal M}$ are faithful abelian projections and $\cal W$ is a type I von Neumann algebra. In this case, the center of $\cal W$ consists of all $L^\infty({\cal M} \cap{\cal A}^\prime )$ multiplication operators, hence $\cal W$ is a factor if and only if ${\cal M}\cap{\cal A}^\prime ={\cal T}$. Furthermore, there is an apparent symmetry involving ${\cal W},{\cal W}^\prime $ (the commutant of $\cal W$), $\cal M$ and ${\cal A}^\prime $.

AMS Subject Classification (1991): 47C15, 47B38

Received January 9, 1996 and in revised form July 15, 1996. (Registered under 6136/2009.)

Abstract. Let $(M,g)$ be a pseudo--Riemannian manifold and $(TM,g^C)$ its tangent bundle equipped with the complete lift metric $g^C$. Using the notion of a harmonic almost product structure on $(TM,g^C)$, as introduced in [11], we define and study harmonic connections. This notion is used to introduce harmonic tensor fields of type $(1,2)$. We illustrate the theory by treating harmonic Ambrose--Singer, almost symplectic, almost complex and almost product connections, harmonic foliations and minimal plane fields. Finally, we construct examples on $TM$ by means of the lifting procedure.

AMS Subject Classification (1991): 53C05

Keyword(s): Tangent bundles, endomorphism fields, harmonic maps, harmonic connections, harmonic tensor fields

Received July 15, 1996. (Registered under 6137/2009.)