
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)
Abstract. In this note we will investigate some particular classes of ideals in Hilbert algebras with supremum. We shall study the relation between $\alpha $ideals and annihilator ideals in bounded Hilbert algebras with supremum. We shall introduce the class of $\sigma $ideals and we will see that this class is strongly connected with the deductive systems. We will also characterize the bounded Hilbert algebras with supremum satisfying the Stone identity.
DOI: 10.14232/actasm0122679
AMS Subject Classification
(1991): 05C38, 15A15; 05A15, 15A18
Keyword(s):
Hilbert algebras with supremum,
annihilators,
$\alpha $ideals,
$\sigma $ideals,
Stone identity
Received March 19, 2012, and in final form June 8, 2012. (Registered under 17/2012.)
Horst Alzer,
Man Kam Kwong

2126

Abstract. The inequality $\sum_{k=1}^n (nk+a)(nk+b) \sin(kx) \cos(ky)>0$ $(a,b\in\opr )$ is proved to hold for all $n\in\opn $ and $x,y\in\opr $ with $0<x+y< \pi $, $0<xy< \pi $ if and only if $0< ab\leq a+b+1$. This extends a theorem of Turán, who showed that our inequality is valid for $a=1$, $b=2$, $y=0$.
DOI: 10.14232/actasm0135626
AMS Subject Classification
(1991): 26D05, 42A05
Keyword(s):
trigonometric sums,
inequalities,
sine integral
Received September 12, 2013. (Registered under 62/2013.)
X. X. Xia,
S. P. Zhou

2729

Abstract. This short note gives a mending to a little but sensitive flaw in the original proof of an important and useful inequality established by Leinder.
DOI: 10.14232/actasm0122992
AMS Subject Classification
(1991): 26D15
Keyword(s):
inequality,
mending
Received June 20, 2012, and in revised form August 1, 2012. (Registered under 49/2012.)
Henrik Winkler,
Harald Woracek

3194

Abstract. We study twodimensional Hamiltonian systems of the form $(*)$ $y'(x)=zJH(x)y(x),\quad x\in[s_,s_+)$, where the Hamiltonian $H$ is locally integrable on $[s_,s_+)$ and nonnegative, and $J:=\big({0 1\atop1 0}\big )$. The spectral theory of the equation changes depending on the growth of $H$ towards the endpoint $s_+$; the classical distinction into the Weyl alternatives `limit point' or `limit circle' case. A refined measure for the growth of a limit point Hamiltonian $H$ can be obtained by comparing with $H$polynomials. This growth measure is concretised by a number $\Delta(H)\in\bb N_0\cup\{\infty\}$ and appeared first in connection with a Pontryagin space analogue of the equation $(*)$. It is known that the growth restriction `$\Delta(H)< \infty $' has some striking consequences on the spectral theory of the equation; in many respects, the case `limit point but still $\Delta(H)< \infty $' is similar to the limit circle case. In general, the number $\Delta(H)$ is given in a rather implicit way, difficult to handle and not suitable for concrete calculations. In the present paper we provide a more accessible way to compute $\Delta(H)$ for some particular classes of Hamiltonians which occur in connection with SturmLiouville equations and Kre\u?n strings.
DOI: 10.14232/actasm0120288
AMS Subject Classification
(1991): 34B20, 37J99, 34L40; 47B50, 47E05
Keyword(s):
Hamiltonian system,
Kre\u?n string,
growth towards singular endpoint
Received May 8, 2012, and in revised form August 27, 2013. (Registered under 28/2012.)
Abstract. In this paper, we will study a certain Markov coded system $S_G$ defined by a finite directed graph $G$. We will prove that if the transition matrix of $G$ is aperiodic, the associated $C^*$algebra ${\cal O}_{S_G}$ is unital, simple and purely infinite. We will compute its Kgroups and Extgroups and apply the results to classification of a certain class of symbolic dynamical systems under flow equivalence.
DOI: 10.14232/actasm0120246
AMS Subject Classification
(1991): 37B10; 46L35
Keyword(s):
subshift,
Markov code,
$C^*$algebra,
Ktheory,
BowenFranks group,
flow equivalence
Received April 12, 2012, and in final form January 16, 2014. (Registered under 24/2012.)
Abstract. We verify that in a divergence theorem proved by L.~Csernyák and the author the classical monotone decreasing assumption on the ``blocksequence'' can be weakened to locally almost monotone condition. Furthermore by means of the new result we improve a theorem pertaining to the divergence of the generalized de la ValléePoussin means.
DOI: 10.14232/actasm013287z
AMS Subject Classification
(1991): 42A20, 42A16
Keyword(s):
orthogonal series,
coefficient conditions,
locally almost monotonicity
Received June 14, 2013, and in revised form July 24, 2013. (Registered under 37/2013.)
Abstract. We consider a function space $\mathscr{QA}$ on the unit sphere of ${\msbm R}^3$, which contains $ L\log L\log\log \log L$, and prove the spherical harmonics expansions of functions in $\mathscr{QA}$ are summable a.e. with respect to the Ces?ro means of the critical order $1/2$. We also prove that a similar result holds for the BochnerRiesz means of multiple Fourier series of periodic functions on ${\msbm R}^d$, $d\geq2$.
DOI: 10.14232/actasm0122878
AMS Subject Classification
(1991): 42C10, 42B08
Keyword(s):
spherical harmonics expansion,
Ces?ro means,
BochnerRiesz means
Received May 21, 2012, and in revised form September 17, 2012. (Registered under 37/2012.)
Abstract. The paper deals with holomorphic functions of a bounded operator $A$ acting in a Banach complex lattice. A norm estimate for the considered operator valued functions is derived. Applications of the obtained bound to functions of integral operators, partial integral operators, infinite matrices and differential equations are also discussed.
DOI: 10.14232/actasm012052x
AMS Subject Classification
(1991): 46B42, 47A56, 47A60, 47G10, 35A23
Keyword(s):
Banach lattice,
operator functions,
integral operators,
infinite matrices,
partial integral operator,
Barbashin equation
Received July 13, 2012. (Registered under 52/2012.)
Bernard Aupetit,
Endre Makai, Jr.,
Mostafa Mbekhta,
Jaroslav Zemánek

149174

Abstract. Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda )\colon A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(\lambda )$ an analytic family of idempotents in~$B$. We want to find an analytic family $p(\lambda )$ of idempotents in~$A$, lifting $q(\lambda )$, i.e., such that $\pi(\lambda )p(\lambda ) = q(\lambda )$, under hypotheses of the type that the elements of $\Ker\pi (\lambda )$ have small spectra. For spectra which do not disconnect ${\msbm C}$ we obtain a local lifting theorem. For real analytic families of surjective $^*$homomorphisms (for continuous involutions) and selfadjoint idempotents we obtain a local lifting theorem, for totally disconnected spectra. We obtain a global lifting theorem if the spectra of the elements in $\Ker\pi (\lambda )$ are $\{0\}$, both in the analytic case, and, for $^*$algebras (with continuous involutions) and selfadjoint idempotents, in the real analytic case. Here even an at most countably infinite set of mutually orthogonal analytic families of idempotents can be lifted to mutually orthogonal analytic families of idempotents. In the proofs, spectral theory is combined with complex analysis and general topology, and even a connection with potential theory is mentioned.
DOI: 10.14232/actasm013765y
AMS Subject Classification
(1991): 46H05; 46T25, 47B48, 47L10
Keyword(s):
Banach algebras,
homomorphisms,
idempotents,
analytic families,
liftings
Received February 15, 2013, and in revised form April 29, 2013. (Registered under 15/2013.)
Dan Popovici,
Zoltán Sebestyén

175194

Abstract. Given two linear operators $S$ and $T$ acting between Hilbert spaces $\mathscr{H}$ and $\mathscr{K}$, respectively $\mathscr{K}$ and $\mathscr{H}$, which satisfy the relation $ \langle Sh, k\rangle =\langle h, Tk\rangle, \quad h\in\dom S, k\in\dom T, $ i.e., according to the classical terminology of M.$ $H. Stone, which are adjoint to each other, we provide necessary and sufficient conditions in order to ensure the equality between the closure of $S$ and the adjoint of $T$. A central role in our approach is played by the range of the operator matrix $M_{S, T}=\left({1_{\dom S} T\atop S 1_{\dom T}}\right )$. We obtain, as consequences, several results characterizing skewadjointness, selfadjointness and essential selfadjointness. We improve, in particular, the celebrated selfadjointness criterion of J. von Neumann.
DOI: 10.14232/actasm0128577
AMS Subject Classification
(1991): 47A05, 47A20, 47B25
Keyword(s):
unbounded operator,
operators which are adjoint to each other,
symmetric,
skewadjoint,
selfadjoint,
essentially selfadjoint,
closable
Received November 30, 2012, and in final form June 3, 2013. (Registered under 107/2012.)
S. Hassi,
H. S. V. de Snoo,
F. H. Szafraniec

195232

Abstract. Let $S$ be a closed symmetric relation in a Hilbert space. If $S$ is densely defined, then the normal extensions of $S$ are selfadjoint. In general, normal nonselfadjoint intermediate extensions may not exist. Necessary and sufficient conditions for the existence of normal nonselfadjoint intermediate extensions are developed and all such extensions are parametrized.
DOI: 10.14232/actasm0130080
AMS Subject Classification
(1991): 47A06, 47B15, 47B25
Keyword(s):
symmetric operator,
symmetric relation,
selfadjoint extension,
normal extension
Received January 23, 2013, and in revised form February 21, 2013. (Registered under 8/2013.)
KarlHeinz Förster,
Béla Nagy

233260

Abstract. We investigate whether variants of equivalence of (singular) matrix polynomials imply those of the first companion linearizations, and the converse question. We study a method of deciding whether two polynomials are strictly equivalent, and which are all the pairs of matrices effecting this equivalence. The corresponding problems for the (strict) similarity of square matrix polynomials are studied. We investigate the strict equivalence of a square polynomial to a polynomial whose all coefficient matrices are diagonal, and study also the singular case.
DOI: 10.14232/actasm012012z
AMS Subject Classification
(1991): 47A56, 15A22
Keyword(s):
singular matrix polynomial,
companion linearization,
equivalence,
strong and strict equivalence,
similarity,
strictly diagonizable,
(block) diagonally strict equivalence
Received February 22, 2012, and in revised form April 5, 2012. (Registered under 12/2012.)
Abstract. Selfadjoint Jacobi matrices with matrix entries and invertible blocks on the offdiagonals are considered. For three different classes of these matrices are given conditions which guarantee vanishing of their point spectra. Two of the conditions are extensions of the corresponding ones found by Damanik, Simon and Stolz for Jacobi matrices with scalar entries. The results are illustrated by two simple examples.
DOI: 10.14232/actasm0126102
AMS Subject Classification
(1991): 47B36, 47B39, 47B25
Keyword(s):
Jacobi matrix/operator,
spectral analysis,
point spectrum,
periodic sequences,
commutator
Received December 11, 2012, and in revised form March 8, 2013. (Registered under 110/2012.)
Wojciech Młocek,
Marek Ptak

275287

Abstract. The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space over a Jordan region  a simply connected region in the complex plane with analytic Jordan curve as its boundary  is investigated. The dichotomic behavior (either reflexivity or transitivity) of these subspaces is shown. It refers to the similar dichotomic behavior of subspaces of Toeplitz operators on the Hardy space over the unit disc.
DOI: 10.14232/actasm0123430
AMS Subject Classification
(1991): 47L80, 47L05, 47L45
Keyword(s):
reflexive subspace,
transitive subspace,
Toeplitz operator,
Hardy space,
simply connected region
Received November 14, 2012. (Registered under 93/2012.)
Abstract. Let $X$ be a Banach function space over a probability space. We consider the inequality $\norm{(v{\ast }f)_{\infty }}_X \le C\norm{f_{\infty }}_X$, where $f=(f_n)_{n \in\Z }$ is a uniformly integrable martingale, $v=(v_n)_{n \in\Z }$ is a predictable process such that $\sup_n \abs{v_n}\le1$ almost surely, and $v{\ast }f=((v{\ast }f)_n)_{n \in\Z }$ denotes the martingale transform of $f$ by $v$. The main result gives necessary and sufficient conditions on $X$ for this inequality to hold.
DOI: 10.14232/actasm0125423
AMS Subject Classification
(1991): 60G42, 46E30, 46N30
Keyword(s):
martingale,
martingale transform,
Banach function space,
rearrangementinvariant function space
Received May 30, 2012. (Registered under 42/2012.)
Ferenc Balogh,
Éva Krauczi

307326

Abstract. We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the locationscale family associated to the logistic distribution. Explicit representations of the limiting distribution are given in terms of integrals of weighted Brownian bridges or alternatively as infinite series of independent Gaussian random variables. The power of this test and the test for the location logistic family against some alternatives are demonstrated by numerical simulations.
DOI: 10.14232/actasm0138098
AMS Subject Classification
(1991): 62F05, 60G15, 62Q05
Keyword(s):
Correlation test,
KarhunenLo?ve expansion,
power study,
simulation,
test of Logistic distribution,
Wasserstein distance
Received September 3, 2013, and in final form April 2, 2014. (Registered under 59/2013.)
Abstract. We study the strong consistency of an autoregression parameter of a discrete time HeathJarrowMorton type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. In this paper the size of the subsamples corresponding to different time points is fixed: the observations of the forward rates are given for the same time to maturity values at each time point. We show the consistency in the stable case and in an unstable case and give empirical results based on simulations as well.
DOI: 10.14232/actasm014020z
AMS Subject Classification
(1991): 62F12, 91B84, 65C60
Keyword(s):
HJM forward interest rate models,
geometric spatial autoregression field,
strong consistency of ML estimators,
simulations with R
Received March 4, 2014, and in revised form March 26, 2014. (Registered under 20/2014.)

349362
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