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/actasm-012-267-9

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.)

Abstract. The inequality $\sum_{k=1}^n (n-k+a)(n-k+b) \sin(kx) \cos(ky)>0$ $(a,b\in\opr )$ is proved to hold for all $n\in\opn $ and $x,y\in\opr $ with $0<x+y< \pi $, $0<x-y< \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/actasm-013-562-6

AMS Subject Classification (1991): 26D05, 42A05

Keyword(s): trigonometric sums, inequalities, sine integral

Received September 12, 2013. (Registered under 62/2013.)

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/actasm-012-299-2

AMS Subject Classification (1991): 26D15

Keyword(s): inequality, mending

Received June 20, 2012, and in revised form August 1, 2012. (Registered under 49/2012.)

Abstract. We study two-dimensional 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 Sturm--Liouville equations and Kre\u?n strings.

DOI: 10.14232/actasm-012-028-8

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 K-groups and Ext-groups and apply the results to classification of a certain class of symbolic dynamical systems under flow equivalence.

DOI: 10.14232/actasm-012-024-6

AMS Subject Classification (1991): 37B10; 46L35

Keyword(s): subshift, Markov code, $C^*$-algebra, K-theory, Bowen--Franks 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 Vallée-Poussin means.

DOI: 10.14232/actasm-013-287-z

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 Bochner--Riesz means of multiple Fourier series of periodic functions on ${\msbm R}^d$, $d\geq2$.

DOI: 10.14232/actasm-012-287-8

AMS Subject Classification (1991): 42C10, 42B08

Keyword(s): spherical harmonics expansion, Ces?ro means, Bochner--Riesz 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/actasm-012-052-x

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.)

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 self-adjoint 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 self-adjoint 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/actasm-013-765-y

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.)

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/actasm-012-857-7

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.)

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/actasm-013-008-0

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.)

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/actasm-012-012-z

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 off-diagonals 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/actasm-012-610-2

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.)

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/actasm-012-343-0

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/actasm-012-542-3

AMS Subject Classification (1991): 60G42, 46E30, 46N30

Keyword(s): martingale, martingale transform, Banach function space, rearrangement-invariant function space

Received May 30, 2012. (Registered under 42/2012.)

Abstract. We summarize the results of investigating the asymptotic behavior of the weighted quantile correlation tests for the location-scale 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/actasm-013-809-8

AMS Subject Classification (1991): 62F05, 60G15, 62Q05

Keyword(s): Correlation test, Karhunen--Lo?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 Heath--Jarrow--Morton 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/actasm-014-020-z

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.)