
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

363363
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Gábor Czédli,
Claudia Murełan

363380

Abstract. Let $\lambda $ and $\kappa $ be cardinal numbers such that $\kappa $ is infinite and either $2\leq \lambda \leq \kappa $, or $\lambda =2^\kappa $. We prove that there exists a lattice $L$ with exactly $\lambda $ many congruences, $2^\kappa $ many ideals, but only $\kappa $ many filters. Furthermore, if $\lambda \geq 2$ is an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this $L$ is even relatively complemented for $\lambda =2$. Related to some earlier results of George Gr\"atzer and the first author, we also prove that if $P$ is a bounded ordered set (in other words, a bounded poset) with at least two elements, $G$ is a group, and $\kappa $ is an infinite cardinal such that $\kappa \geq P$ and $\kappa \geq G$, then there exists a lattice $L$ of cardinality $\kappa $ such that (i) the principal congruences of $L$ form an ordered set isomorphic to $P$, (ii) the automorphism group of $L$ is isomorphic to $G$, (iii) $L$ has $2^\kappa $ many ideals, but (iv) $L$ has only $\kappa $ many filters.
DOI: 10.14232/actasm018538y
AMS Subject Classification
(1991): 06B10
Keyword(s):
lattice ideal,
lattice filter,
simple lattice,
more ideals than filters,
number of ideals,
cardinality,
lattice congruence,
principal congruence
Received April 14, 2018 and in final form February 13, 2019. (Registered under 38/2018.)
Tamás Dékány,
Mária B. Szendrei,
István Szittyai

381411

Abstract. We prove that if $S$ is an $E$solid locally inverse semigroup, and $\rho $ is an inverse semigroup congruence on $S$ such that the idempotent classes of $\rho $ are completely simple semigroups then $S$ is embeddable into a $\lambda $semidirect product of a completely simple semigroup by $S/\rho $. Consequently, the $E$solid locally inverse semigroups turn out to be, up to isomorphism, the regular subsemigroups of $\lambda $semidirect products of completely simple semigroups by inverse semigroups.
DOI: 10.14232/actasm0183115
AMS Subject Classification
(1991): 20M10, 20M17
Keyword(s):
regular semigroups,
Esolid locally inverse semigroups,
$\lambda $semidirect product,
extensions
Received July 3, 2018. (Registered under 61/2018.)
István Gaál,
László Remete

413429

Abstract. Investigations of monogenity and power integral bases were recently extended from the absolute case (over $\mathbb Q $) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded in calculating generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of cubic and quartic fields with real quadratic fields. In case the quartic field is totally complex, we present an especially simple algorithm. \par We illustrate our method with two detailed examples.
DOI: 10.14232/actasm018080z
AMS Subject Classification
(1991): 11R04; 11D59,11Y50
Keyword(s):
monogenic fields,
composites of number fields,
relative cubic and relative quartic extensions,
relative Thue equations
Received September 4, 2018 and in final form February 20, 2019. (Registered under 80/2018.)
Amin Hosseini,
Mehdi Mohammadzadeh Karizaki

431440

Abstract. The main purpose of this article is to prove the following result: For integers $m$, $n$ with $m \geq 0$, $n \geq 0$, and $m + n \neq 0$, let $\mathcal {R}$ be an $(m + n + 2)!$torsion free prime ring with the identity element $\textbf {e}$. Suppose that $d, \sigma \colon \mathcal {R} \rightarrow \mathcal {R}$ are two additive mappings such that $\sigma $ is a monomorphism with $\sigma (\textbf {e}) = \textbf {e}$, and $d(\mathcal {R}) \subseteq \sigma (\mathcal {R})$. If $d$ and $\sigma $ satisfy both of the equations \[d(xy)(\sigma (z)z)d(x)(\sigma (yz)\sigma (y) z)+\sigma (xy) d(z)\sigma (x)(d(yz)d(y)z) = 0\] and \[d(x^{m + n + 1}) = (m + n + 1)\sigma (x^m) d(x) \sigma (x^n)\]for all $x, y, z \in \mathcal {R}$, then $d$ is a $\sigma $derivation.
DOI: 10.14232/actasm0185946
AMS Subject Classification
(1991): 47B47; 16N60
Keyword(s):
derivation,
$\sigma $derivation,
2torsion free prime ring,
twovariable $\sigma $derivation,
commutativity of rings
Received November 15, 2018 and in final form May 11, 2019. (Registered under 94/2018.)
Abstract. In this paper, we begin by investigating a particular subclass of boundary measures of HerglotzNevanlinna functions in two variables. Based on this, we then proceed to solve the convex combination problem for HerglotzNevanlinna functions in several variables.
DOI: 10.14232/actasm0180401
AMS Subject Classification
(1991): 32A26, 32A10, 32A99
Keyword(s):
integral representation,
HerglotzNevanlinna function,
several complex variables,
convex combination
Received April 24, 2018 and in final form April 11, 2019. (Registered under 40/2018.)
Anuradha Gupta,
Shivam Kumar Singh

473493

Abstract. A Toeplitz type operator $ T_\phi $ with coanalytic symbol $ \phi $ which can be seen as the adjoint of the multiplication operator on $ S^2(\mathbb {D}) $ is introduced and studied on the derivative Hardy space $ S^2(\mathbb {D}) $. The characterizations for the operator $ T_\phi $ to be normal, selfadjoint and isometric on $ S^2(\mathbb {D}) $ have been obtained. In addition, it has been shown that the operator $ T_{\bar {z}^k} $ for a fixed nonnegative integer $ k $ is a Fredholm operator and its point spectrum is the closed unit disk.
DOI: 10.14232/actasm0188050
AMS Subject Classification
(1991): 47B35; 47B32
Keyword(s):
Toeplitz operator,
multiplication operator,
derivative Hardy space
Received June 9, 2018 and in final form July 27, 2018. (Registered under 55/2018.)
T. S. S. R. K. Rao

495505

Abstract. For nonreflexive Banach spaces $X,Y$, for a very smooth point in the space of compact linear operators ${\mathcal K}(X,Y)$, we give several sufficient conditions for the adjoint to be a very smooth point in ${\mathcal K}(Y^\ast , X^\ast )$. We exhibit a new class of extreme points in the dual unit ball of injective product spaces. These ideas are also related to BirkhoffJames orthogonality in spaces of operators.
DOI: 10.14232/actasm0188092
AMS Subject Classification
(1991): 47L05, 46B28, 46B25
Keyword(s):
smooth points,
very smooth points,
adjoints of operators,
spaces of operators,
essential norm,
injective and projective tensor product spaces
Received June 23, 2018 and in final form January 21, 2019. (Registered under 59/2018.)
B. V. Rajarama Bhat,
Tiju Cherian John

507518

Abstract. A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose (adjoint). A structure theorem for invertible skewsymmetric operators, which is analogous to the finitedimensional situation, is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson's normal form for bounded positive operators on infinitedimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states.
DOI: 10.14232/actasm0185705
AMS Subject Classification
(1991): 47B15
Keyword(s):
spectral theorem,
real normal operator,
Williamson's normal form,
infinite mode quantum systems
Received August 3, 2018 and in final form November 22, 2018. (Registered under 70/2018.)
Masatoshi Enomoto,
Yasuo Watatani

519537

Abstract. We study the relative position of three subspaces in an infinitedimensional Hilbert space. In the finitedimensional case over an arbitrary field, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinitedimensional Hilbert space over the complex numbers.
DOI: 10.14232/actasm018821x
AMS Subject Classification
(1991): 46C07, 47A15, 15A21, 16G20, 16G60
Keyword(s):
three subspaces,
Hilbert space
Received August 6, 2018 and in final form November 25, 2018. (Registered under 71/2018.)
Abstract. We extend known results on the spectra of composition operators to the weighted Bergman spaces. Our results include a study of the essential spectral radius, a determination of the spectrum when the symbol of the composition operator is univalent and nonautomorphic with a fixed point in the disk, and an affirmative answer to a conjecture of MacCluer and Saxe.
DOI: 10.14232/actasm0180727
AMS Subject Classification
(1991): 47B33
Keyword(s):
composition operator,
spectrum,
essential spectral radius,
weighted Bergman space,
Bloch space
Received August 6, 2018 and in final form January 20, 2019. (Registered under 72/2018.)
Abstract. Operators of type $f\to f\circ \varphi $ acting on function spaces are called composition operators. We consider composition operators acting on the Hilbert Hardy space on the open unit disc or the right halfplane, study when they are similar to contractions, and obtain results interesting from the point of view of dilation theory of contractions and function theory.
DOI: 10.14232/actasm0185789
AMS Subject Classification
(1991): 47B33, 47A45
Keyword(s):
composition operators,
contractions
Received August 24, 2018 and in final form December 14, 2018. (Registered under 78/2018.)
Abstract. Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta $ be a singular inner function. It is shown that if $\theta (T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial hyperinvariant subspaces.
DOI: 10.14232/actasm018582z
AMS Subject Classification
(1991): 47A15; 47A60, 47A10
Keyword(s):
hyperinvariant subspace,
polynomially bounded operator
Received September 12, 2018 and in final form April 4, 2019. (Registered under 82/2018.)
Abstract. A theorem of Fillmore, Stampfli and Williams asserts that a bounded linear Hilbert space operator is an essential isometry if and only if it is a compact perturbation of either an isometry or a coisometry with finitedimensional kernel. In this note, we discuss the spherical analog of this result. It turns out that the spherical analog of this result does not hold verbatim, and this failure may be attributed to the fact that in dimension $d \geqslant 2$, there exist spherical isometries with finitedimensional joint cokernel, which are not essential spherical unitaries. We also discuss some strictly higherdimensional obstructions in representing an essential spherical isometry as a compact perturbation of a spherical isometry.
DOI: 10.14232/actasm0183356
AMS Subject Classification
(1991): 47A13; 47B20
Keyword(s):
spherical isometry,
essential spherical isometry
Received September 21, 2018 and in final form February 11, 2019. (Registered under 85/2018.)
Abstract. Let $B$ be a proper open subset in \rn \ and $C$ be an open convex cone in \rn . We define a generalization of the spaces of Hardy functions, \gb , $1 \leq p < \infty ,$ and extended tempered distributions, \Swp , of Beurling's tempered distributions, \swp . We obtain the analytical and topological properties of \Swp \ and show that the functions in \gc , $1 < p \leq 2$, have distributional boundary values in the weak topology of \swp \ using the analytical properties of \Swp .
DOI: 10.14232/actasm0180883
AMS Subject Classification
(1991): 32A40, 42B30, 46F20
Keyword(s):
Beurling distributions,
generalized Hardy functions,
distributional boundary values
Received October 7, 2018 and in final form January 13, 2019. (Registered under 88/2018.)
Mojtaba Mojahedi,
Fereshteh Sady

613627

Abstract. Surjective, not necessarily linear isometries $T\colon {\rm AC}(X,E) \to {\rm AC}(Y,F)$ between vectorvalued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line have recently been described as generalized weighted composition operators. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T\colon (A,\\cdot \_A) \to (B,\\cdot \_B)$ between certain normed subspaces $A$ and $B$ of $C(X,E)$ and $C(Y,F)$, respectively. We consider three cases for $F$ with some mild conditions. The first case, in particular, provides a short proof for the above result, without assuming that the target spaces are strictly convex. The other cases give some generalizations in this topic. As a consequence, the results can be applied, for isometries (not necessarily linear) between spaces of absolutely continuous vectorvalued functions, (little) Lipschitz functions and also continuously differentiable functions.
DOI: 10.14232/actasm0180926
AMS Subject Classification
(1991): 47B38, 47B33; 46J10
Keyword(s):
reallinear isometries,
vectorvalued function spaces,
Tsets
Received October 29, 2018 and in final form March 12, 2019. (Registered under 92/2018.)
Klaus Schiefermayr

629649

Abstract. In this paper, we give an \emph {explicit} representation of the complex Chebyshev polynomials on a given arc of the unit circle (in the complex plane) in terms of real Chebyshev polynomials on two symmetric intervals (on the real line). The real Chebyshev polynomials, for their part, can be expressed via a conformal mapping with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's.
DOI: 10.14232/actasm018343y
AMS Subject Classification
(1991): 30E10, 30C10, 33E05, 41A50
Keyword(s):
Chebyshev polynomials,
circular arc,
Jacobian elliptic function,
Jacobian theta function
Received October 29, 2018 and in final form January 16, 2019. (Registered under 93/2018.)
Mohammed Hichem Mortad

651658

Abstract. The paper is devoted to counterexamples involving the triviality of domains of products and/or adjoints of densely defined operators.
DOI: 10.14232/actasm0188575
AMS Subject Classification
(1991): 47A05
Keyword(s):
closed operators,
trivial domain
Received December 25, 2018 and in final form March 12, 2019. (Registered under 107/2018.)
Anindya Ghatak,
Anil Kumar Karn

659679

Abstract. We discuss the ordertheoretic properties of $CM$ideals in matricially order smooth $\infty $normed spaces. We study the relation between $CM$ideals and $CL$summands in the matrix duality setup. We introduce the notion of $L^{1}$matricial split faces in an $L^{1}$matricially normed space and characterize $CM$ideals in a matricially order smooth $\infty $normed space $V$ in terms of the $L^{1}$matricial split face of the $L^{1}$matrix convex set $\{Q_{n}(V)\}$.
DOI: 10.14232/actasm0192596
AMS Subject Classification
(1991): 46B40; 46L05, 46L30
Keyword(s):
operator spaces,
operator systems,
$CM$ideals,
split faces
Received January 25, 2019 and in final form May 29, 2019. (Registered under 9/2019.)
H. Bercovici,
I. B. Jung,
E. Ko,
C. Pearcy

681691

Abstract. In this note we first briefly review the progress on the hyperinvariant subspace problem for operators on Hilbert space made possible by the equivalence relation of ampliation quasisimilarity recently introduced in \cite {FP}. Then we introduce another equivalence relation, which we call \emph {pluquasisimilarity}, with bigger equivalence classes than ampliation quasisimilarity but very different in appearance, which preserves the existence of hyperinvariant subspaces for operators, and thus may be useful in the future. We also compare these with two other equivalence relations, injectionsimilarity and complete injectionsimilarity, introduced long ago by Sz.Nagy and Foias in \cite {SzNF}.
DOI: 10.14232/actasm0197659
AMS Subject Classification
(1991): 47A15; 47A65
Keyword(s):
hyperinvariant subspace,
quasisimilarity,
ampliation quasisimilarity,
quasiaffinity,
quasitriangular operator
Received February 23, 2019 and in final form March 28, 2019. (Registered under 15/2019.)
Miklós Horváth,
Orsolya Sáfár

693708

Abstract. In this note we continue the study of the properties of phase shifts arising in the 3D inverse scattering with spherically symmetrical potential. The main result of our paper improves Regge's inequality for a class of potentials with compact support.
DOI: 10.14232/actasm0190206
AMS Subject Classification
(1991): 35P25
Keyword(s):
inverse scattering,
fixed energy phase shifts
Received March 21, 2019 and in final form October 5, 2019. (Registered under 20/2019.)

709710
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