ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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/actasm-018-538-y

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.)

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/actasm-018-311-5

AMS Subject Classification (1991): 20M10, 20M17

Keyword(s): regular semigroups, E-solid locally inverse semigroups, $\lambda $-semidirect product, extensions

Received July 3, 2018. (Registered under 61/2018.)

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/actasm-018-080-z

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.)

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/actasm-018-594-6

AMS Subject Classification (1991): 47B47; 16N60

Keyword(s): derivation, $\sigma $-derivation, 2-torsion free prime ring, two-variable $\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 Herglotz--Nevanlinna functions in two variables. Based on this, we then proceed to solve the convex combination problem for Herglotz--Nevanlinna functions in several variables.

DOI: 10.14232/actasm-018-040-1

AMS Subject Classification (1991): 32A26, 32A10, 32A99

Keyword(s): integral representation, Herglotz--Nevanlinna function, several complex variables, convex combination

Received April 24, 2018 and in final form April 11, 2019. (Registered under 40/2018.)

Abstract. A Toeplitz type operator $ T_\phi $ with co-analytic 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, self-adjoint 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 non-negative integer $ k $ is a Fredholm operator and its point spectrum is the closed unit disk.

DOI: 10.14232/actasm-018-805-0

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.)

Abstract. For non-reflexive 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 Birkhoff--James orthogonality in spaces of operators.

DOI: 10.14232/actasm-018-809-2

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.)

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 skew-symmetric operators, which is analogous to the finite-dimensional 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 infinite-dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states.

DOI: 10.14232/actasm-018-570-5

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.)

Abstract. We study the relative position of three subspaces in an infinite-dimensional Hilbert space. In the finite-dimensional 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 infinite-dimensional Hilbert space over the complex numbers.

DOI: 10.14232/actasm-018-821-x

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 non-automorphic with a fixed point in the disk, and an affirmative answer to a conjecture of MacCluer and Saxe.

DOI: 10.14232/actasm-018-072-7

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 half-plane, 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/actasm-018-578-9

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/actasm-018-582-z

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 finite-dimensional 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 finite-dimensional joint cokernel, which are not essential spherical unitaries. We also discuss some strictly higher-dimensional obstructions in representing an essential spherical isometry as a compact perturbation of a spherical isometry.

DOI: 10.14232/actasm-018-335-6

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/actasm-018-088-3

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.)

Abstract. Surjective, not necessarily linear isometries $T\colon {\rm AC}(X,E) \to {\rm AC}(Y,F)$ between vector-valued 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 vector-valued functions, (little) Lipschitz functions and also continuously differentiable functions.

DOI: 10.14232/actasm-018-092-6

AMS Subject Classification (1991): 47B38, 47B33; 46J10

Keyword(s): real-linear isometries, vector-valued function spaces, T-sets

Received October 29, 2018 and in final form March 12, 2019. (Registered under 92/2018.)

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/actasm-018-343-y

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.)

Abstract. The paper is devoted to counterexamples involving the triviality of domains of products and/or adjoints of densely defined operators.

DOI: 10.14232/actasm-018-857-5

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.)

Abstract. We discuss the order-theoretic 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/actasm-019-259-6

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.)

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, injection-similarity and complete injection-similarity, introduced long ago by Sz.-Nagy and Foias in \cite {SzNF}.

DOI: 10.14232/actasm-019-765-9

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.)

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/actasm-019-020-6

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.)