ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We discuss two possible ways of representing tolerances: first, as a homomorphic image of some congruence; second, as the relational composition of some compatible relation with its converse. The second way is independent from the variety under consideration, while the first way is variety-dependent. The relationships between these two kinds of representations are clarified. As an application, we show that any tolerance on some lattice ${\eufm{L}} $ is the image of some congruence on a subalgebra of ${\eufm{L}} \times{\eufm{L}}$. This is related to recent results by G. Czédli, G. Grätzer and E. W. Kiss.

AMS Subject Classification (1991): 08A30, 06B10; 08B99, 08C15, 06B20, 06B75

Keyword(s): tolerance, representable, image of a congruence, lattice, \hbox{(quasi-)}variety

Received June 12, 2012, and in revised form October 15, 2012. (Registered under 47/2012.)

Abstract. Consider a torsion module $G$ over a discrete valuation ring ${\eufm O}$, and a submodule $G'\subset G$. It is known that the partitions describing the structure of the modules $G,G',$ and $G/G'$ satisfy the Littlewood--Richardson rule. In particular, these partitions must also satisfy all the Horn inequalities. We show that these inequalities can be obtained directly from the intersection theory of Grassmannians. Moreover, when one of these inequalities is saturated, there is a direct summand $H$ of $G$ such that $H\cap G'$ and $(H+G')/G'$ are direct summands of $G'$ and $G/G'$, respectively. The partitions describing these direct summands correspond precisely to the summands appearing in the saturated Horn inequality. These results apply to those Horn inequalities for which the corresponding Littlewood--Richardson coefficient is 1, and these are sufficient to imply all the others.

AMS Subject Classification (1991): 13F10; 14M15,15A23, 20K01

Keyword(s): Horn inequalities, submodules

Received October 9, 2012, and in revised form February 4, 2013. (Registered under 81/2012.)

Abstract. A covering system of a finite group $G$ is a set ${\eufm S}$ of pairs of its subgroups, ${\eufm S}= \{ (L_1,M_1), \ldots, (L_n,M_n) \},$ which satisfies the following axioms: $M_i < L_i$ for every $i \in\{1,\ldots,n\},$ $\bigcup_{i=1}^n(L_i \setminus M_i) = G \setminus\{e\},$ and $|G| = \prod_{i=1}^n |L_i : M_i|,$ where $e$ is the identity element of $G$. The covering system ${\eufm S}$ is said to be regular if $L_i=G$ for some $i \in\{1,\ldots,n\} $. In this paper we show that every covering system of every direct product of elementary $p$-groups is regular.

AMS Subject Classification (1991): 20K01, 20K27

Keyword(s): abelian group, covering system, group character

Received August 23, 2011, and in revised form January 9, 2013. (Registered under 41/2011.)

Abstract. We show the existence of sequences of rest bounded variation which cannot be decomposed into a monotonic sequence and a sequence having terms whose sum is absolute convergent. Furthermore we give an additional condition wherewith such a decomposition is possible.

AMS Subject Classification (1991): 26A15, 42A32

Keyword(s): uniform convergence, monotone sequences

Received May 30, 2012. (Registered under 43/2012.)

Abstract. Given a double sequence $\{c_{m,n}: (m,n) \in{\msbm Z}^2\} $ of complex numbers, we consider the double trigonometric series $(*)$ $\sum_{m\in{\msbm Z}}^{\strut } \sum_{n\in{\msbm Z}} c_{m,n} e^{i(mx+ny)}_{\strut },$ which converges absolutely and uniformly, thus its sum $f(x,y)$ is continuous. We give sufficient conditions in terms of certain means of $\{c_{m,n}\} $ to guarantee that $f(x,y)$ belongs to one of the Zygmund classes ${\rm Zyg}(\alpha, \beta )$ and ${\rm zyg}(\alpha, \beta )$ for some $0< \alpha, \beta\le 2$. The present theorems extend those in [3] from single to double trigonometric series, the latter ones in turn were the generalizations of the corresponding theorem of Zygmund in [5]. Our method of proof is essentially different from that of Zygmund. We establish four lemmas, which reveal interrelations between the order of magnitude of certain initial means and that of certain tail means of the double sequence $\{c_{m,n}\} $.

AMS Subject Classification (1991): 26A16, 26B05, 42A16, 42B05

Keyword(s): double trigonometric series, Lipschitz classes ${\rm Lip}(\alpha, \beta )$ and ${\rm lip}(\alpha, \beta )$ for $0< \alpha, \beta\le 1$, Zygmund classes ${\rm Zyg}(\alpha, \beta )$ and ${\rm zyg}(\alpha, \beta\le 2$

Received August 27, 2012. (Registered under 69/2012.)

Abstract. We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.

AMS Subject Classification (1991): 30E05, 32E30

Keyword(s): Nevanlinna--Pick interpolation, Takagi problem, rational interpolation, degenerate Pick problem

Received April 13, 2012, and in revised form December 12, 2012. (Registered under 61/2012.)

Abstract. We introduce the concept of logarithmic convex hull for Reinhardt domains of continuous functions, and show that the holomorphic hull of a complete Reinhardt domain in ${\cal C}_0(\Omega )$ over a locally compact topological space contains the logarithmic convex hull in a natural manner.

AMS Subject Classification (1991): 32D26; 32D10, 46E15

Keyword(s): continuous Reinhardt domain, log-convexity, holomorphic hull

Received August 1, 2012, and in revised form November 23, 2012. (Registered under 56/2012.)

Abstract. A geometric method is presented to describe the dynamics of the linear second order differential equation with step function coefficient $ x'' + a^2(t) x =0, a(t):= a_k \hbox{ if } t_{k-1} \le t< t_k (k\in{\msbm N}), $ where $a_k > 0$, $t_0 =0$, $t_k\nearrow\infty $ as $k\to\infty $. We rewrite this equation into a discrete dynamical system on the plane. The method is applied to the Meissner equation $ x'' + \lambda ^2 Q (t) x = 0, $ where $\lambda > 0$ is a real parameter; $Q$ is a $2L$-periodic real function which is $1$ on $[0,2)$ and $a^2$ on $[2,2L)$; $a$, $L$ ($0< a\not=1$, $L>1$) are given constants. We give a complete elementary proof for the classical oscillation theorem on the $2L$-periodic and $4L$-periodic solutions of this equation not using even Floquet's theorem from the theory of differential equations.

AMS Subject Classification (1991): 34B24, 34B30, 34C10; 74H45

Keyword(s): second order linear differential equations, non-autonomous equations, equations with periodic coefficients, Hill's equation, step function coefficients, eigenvalues of first and second type, oscillation

Received March 4, 2013, and in revised form April 23, 2013. (Registered under 17/2013.)

Abstract. We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and apparent singularities at a finite set $Q$ (disjoint from $P$) determine a linear system of maximal rank. In addition, we show that certain auxiliary parameters can also be fixed. This enables us to conclude that the family of such differential equations is of the expected dimension and to define a birational map between an open subset of the moduli space of logarithmic connections with fixed logarithmic points and regular semi-simple residues and the Hilbert scheme of points on a quasi-projective surface.

AMS Subject Classification (1991): 34M03, 34M35

Keyword(s): second-order linear ordinary differential equation, regular singularity, apparent singularity

Received November 4, 2011, and in final form December 20, 2012. (Registered under 59/2011.)

Abstract. The classical Bernstein inequality estimates the derivative of a polynomial at a fixed point with the supremum norm and a factor depending on the point only. Recently, this classical inequality was generalized to arbitrary compact subsets on the real line. That generalization is sharp and naturally introduces potential theoretical quantities. It also gives a hint how a sharp $L^\alpha $ Bernstein inequality should look like. In this paper we prove this conjectured $L^\alpha $ Bernstein type inequality and we also prove its sharpness.

AMS Subject Classification (1991): 41A17, 26D05, 30C85

Keyword(s): polynomial inequalities, Bernstein inequality, potential theory, equilibrium measure

Received August 30, 2012, and in final form February 19, 2013. (Registered under 64/2012.)

Abstract. We consider complex-valued functions $f\in L^p({\msbm R}^2)$ for some $1< p\le2$ and give sufficient conditions for its Fourier transform $\hat f$ to belong to $L^r (\{(u,v) \in{\msbm R}^2: |u|\ge1$ and $|v|\ge1\} )$, where $0< r< q$ and $1/p+1/q=1$. Under additional conditions, we also give sufficient conditions, under which we have $\hat f\in L^r({\msbm R}^2)$. These sufficient conditions are in terms of the $L^p$-integral modulus or the ordinary modulus of continuity of $f$. Our theorems apply for functions in the Lipschitz classes $\mathop{\rm Lip}(\alpha_1, \alpha_2)$, where $0< \alpha_1, \alpha_2 \le1$ as well as for functions of bounded $s$-variation on ${\msbm R}^2$, where $0< s< p$. The results of this paper can be considered to be the nonperiodic versions of those results proved in [5] for double Fourier series, and the latter ones were in turn the two-dimensional extensions of the classical theorems of Bernstein, Szász and Zygmund on the absolute convergence of single Fourier series.

AMS Subject Classification (1991): 42A38, 42B10; 28A35

Keyword(s): double Fourier transform of functions $f\in L^p ({\msbm R}^2)$, $1\le p\le2$, inversion formula, Hausdorff--Young inequality, Lebesgue integrability of $\hat f$, $L^p$-integral modulus of continuity, integral Lipschitz classes $\mathop{\rm Lip}(\alpha_1, \alpha_2)_p$ ($0< \alpha_1$, $\alpha_2\le1$), functions of bounded $s$-variation, $s>0$

Received August 1, 2012, and in revised form November 23, 2012. (Registered under 85/2012.)

Abstract. We introduce completely bounded kernels taking values in ${\cal L}({\cal A}, {\cal B})$ where ${\cal A}$ and ${\cal B}$ are $C^*$-algebras. We show that if ${\cal B}$ is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and that this is automatic when the index set is countable.

AMS Subject Classification (1991): 46L07; 46L08, 46E22, 46B20

Keyword(s): completely bounded kernels, hermitian kernels, Kolmogorov decomposition

Received November 29, 2012, and in revised form December 14, 2012. (Registered under 103/2012.)

Abstract. In the present paper we offer an operator theoretic proof of the Lebesgue decomposition theorem among nonnegative forms on (real or complex) vector spaces. The basic tool in our treatment is the embedding operator between two auxiliary Hilbert spaces associated to the forms in question. As an application of our approach, we also provide the Lebesgue decomposition theorem among finitely additive bounded set functions on rings of sets.

AMS Subject Classification (1991): 47A07; 46N99

Keyword(s): Lebesgue decomposition, nonnegative forms, absolute continuity, singularity, Hilbert space methods

Received November 29, 2012. (Registered under 102/2012.)

Abstract. In the first part of the paper we study the decompositions of a (bounded linear) operator similar to a normal operator in Hilbert space into the orthogonal sum of a normal (self-adjoint, unitary) part and of a part free of the given property, respectively. In the second part we investigate in a finite dimensional Hilbert space the minimal unitary power dilations (till the exponent $k$) of a contraction. We determine the general form of such dilations, examine their spectra, and the question of their isomorphy. The first step of the study here is also the decomposition of the contraction into unitary and completely non-unitary parts.

AMS Subject Classification (1991): 47A10, 47A20, 47A30

Keyword(s): operator similar to a normal, spectral operator of scalar type, unitary and completely non-unitary parts, bounded Boolean algebras of idempotents in Hilbert space, equivalent scalar product, minimal unitary power dilations (till the exponent $k$) of a contraction, characteristic function, matrix polynomial, isomorphy of dilations

Received November 16, 2012. (Registered under 91/2012.)

Abstract. Giving a detailed analysis of intertwining pairs, the categorical approach of unitary asymptotes of contractions is provided. The projection-functions connected with unitary asymptotes in the functional model are characterized. Finally, properties of quasianalytic $C_{10}$-contractions are examined in the cases, when the defect index $d_T$ is finite or the asymptotic spectral multiplicity function $\delta_*\le2$.

AMS Subject Classification (1991): 47A15, 47A45

Keyword(s): unitary asymptote, $C_{10}$-contraction, quasianalycity

Received December 22, 2011, and in revised form March 30, 2012. (Registered under 68/2011.)

Abstract. In this paper we characterize those positive operators which are asymptotic limits of contractions in strong operator topology or uniform topology. We examine the problem when the asymptotic limits of two contractions coincide.

AMS Subject Classification (1991): 47A45, 47B15, 47B65

Keyword(s): asymptotic limit, contraction, positive operator

Received December 4, 2012. (Registered under 104/2012.)

Abstract. In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille--Yosida and Gomilko--Shi--Feng generation theorems, Trotter--Kato approximation theorem, Euler approximation formula, Gearhart--Prüss theorem) can be elegantly obtained in this framework. Then we discuss the notions of bounded ${\rm H}^\infty $-calculus and $m$-bounded calculus on half-planes and their relation to weak bounded variation conditions over vertical lines for powers of the resolvent. Finally we discuss the Hilbert space case, where semigroup generation is characterised by the operator having a strong $m$-bounded calculus on a half-plane.

AMS Subject Classification (1991): 47A60; 34G10 47D06 47N20

Keyword(s): functional calculus, half-plane, semigroup generator

Received June 4, 2012, and in revised form September 28, 2012. (Registered under 45/2012.)

Abstract. We characterize those sequences $\left\{h_n \right\} _{n=1}^{\infty }$ of bounded analytic functions, which have the property that an absolutely continuous contraction $T$ is stable (that is the powers $T^n$ converge to zero) exactly when the operators $h_n(T)$ converge to zero in the strong operator topology. Our result is extended to polynomially bounded operators too.

AMS Subject Classification (1991): 47A60, 47A45

Keyword(s): stability, contraction, polynomially bounded operator, functional calculus

Received May 29, 2012, and in revised form December 22, 2012. (Registered under 41/2012.)

Abstract. In the space ${\msbm M}_m({\msbm M}_n)$ of $m\times m$ block-matrices with entries in ${\msbm M}_n$ three natural cones related to positivity are introduced. In this paper several known results related to positivity: dilation theorem, truncated moment theorems and the Fejér--Riesz theorem on factorization of positive polynomials, are applied to derive interrelationships among those three cones. At the end of the paper, some form of matrix Schwarz inequalities is presented. The paper is largely of expository character.

AMS Subject Classification (1991): 47L07, 47A20, 47A68, 47B35, 81P16

Keyword(s): cones of matrices, duality of cones, separability and decomposability, dilation theorem, matrix truncated moment theorems, matrix Fejér--Riesz theorem, matrix Schwarz inequality

Received November 23, 2012, and in revised form March 15, 2013. (Registered under 9/2013.)

Abstract. We improve on Jarník's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta $ for a system of $ n\ge2$ real linear forms in two integer variables. Jarník (1949, 1954) proved that $\beta\ge \alpha(\alpha -1)$. In the present paper we give a better bound in the case $\alpha >1$.

AMS Subject Classification (1991): 11J13

Keyword(s): Diophantine exponents, linear forms, best approximations

Received September 11, 2012, and in revised form February 6, 2012. (Registered under 68/2012.)