ACTA SCIENTIARUM MATHEMATICARUM最新文献

We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields and imaginary quadratic fields with coprime discriminants. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.

Slim planar semimodular lattices (SPS lattices or slim semimodular lattices for short) were introduced by G. Grätzer and E. Knapp in 2007. More than four dozen papers have been devoted to these (necessarily finite) lattices and their congruence lattices since then. In addition to distributivity, the congruence lattices of SPS lattices satisfy seven known properties. Out of these seven properties, the first two were published by G. Grätzer in 2016 and 2020, the next four by the present author in 2021, and the seventh jointly by G. Grätzer and the present author in 2022. Here we give two infinite families of new properties of the congruence lattices of SPS lattices. These properties are independent. We also present stronger versions of these properties but not all of them are independent, and improve three out of the seven previously known properties. The approach is based on lamps, which we introduced in a 2021 paper. In addition to using the 2021 results, we need to prove some easy new lemmas on lamps.

It is well known that the lattice ({{,mathrm{Id_c},}}{G}) of all principal (ell )-ideals of any Abelian (ell )-group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some ({{,mathrm{Id_c},}}{G}), via a counterexample of cardinality (aleph _2). We prove that every completely normal distributive 0-lattice with at most (aleph _1) elements is a homomorphic image of some ({{,mathrm{Id_c},}}{G}). By Stone duality, this means that every completely normal generalized spectral space with at most (aleph _1) compact open sets is homeomorphic to a spectral subspace of the (ell )-spectrum of some Abelian (ell )-group.

Let (varphi ) be an analytic self map of the open unit disc (mathbb {D}). Assume that (psi ) is an analytic map of (mathbb {D}). Suppose that f is in the Hardy space of the open unit disc (H^p). The operator that takes f into (psi cdot f circ varphi ) is a weighted composition operator, and is denoted by (C_{psi ,varphi }). The operator that takes f into (psi cdot f^prime circ varphi ) is a weighted composition-differentiation operator. We prove that some weighted composition-differentiation operators belong to the closed algebra generated by weighted composition operators in the uniform operator topology.

In this paper, we introduce the Fibonomial sequence spaces (b_{0}^{r,s,F}) and (b_{c}^{r,s,F}) and show that these are linearly isomorphic to the spaces (c_{0}) and c,  respectively. In addition, we present (alpha -)dual, (beta -)dual and (gamma -)dual for those spaces and characterize certain matrix classes. In the final section, we obtain some criteria for the compactness of certain matrix operators via Hausdorff measure of noncompactness on the space (b_{0}^{r,s,F}.)

In the first part of this paper, we present some investigations on the class of almost (L) limited operators. We show that an operator (T:X rightarrow E), from a Banach space X to a Banach lattice E, is almost (L) limited iff its adjoint carries disjoint almost L-sequences to norm null ones. In addition, we improve several results obtained by Oughajji et al. In its second part, we study the relationship between the class of weakly precompact operators and that of order weakly compact (resp. b-weakly compact) operators. Among other things, we show that for a Banach lattice E and a Banach space X the following statements are equivalent:

1. (1)

   Every order weakly compact (resp. b-weakly compact) operator (T:E rightarrow X) is weakly precompact;

2. (2)

   The norm of (E') is order continuous or X does not contain any isomorphic copy of (ell ^ 1).

The paper contains some results on weakly sequentially precompact sets and operators. In particular, we establish some relationships between weakly sequentially precompat operators and those whose the adjoint map (L) sets into relatively norm compact ones. Besides, we characterize the class of weak* Dunford-Pettis operators through weakly sequentially precompact operators and deduce in the sequel a new characterization of Dunford-Pettis* property. Moreover, we generalize [9, Theorem 2.5.9] and show that order weakly compact operators carry almost order Dunford-Pettis sets into weakly sequentially precompact ones. Furthermore, we prove that the product of order weakly compact operators and b-weakly compact ones maps weakly sequentially precompact sets into relatively weakly compact ones. Finally, we present some results about the positive Schur property.

M. Embry and A. Lambert initiated the study of a weighted translation semigroup ({S_t}) in ({mathcal B}(L^2({{mathbb R}_+})),) with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic model for the left invertible weighted translation semigroup. The present paper deals with the generalization of the weighted translation semigroup in multi-variable set up. We develop the toral analogue of the analytic model and also describe the spectral picture. We provide many examples of weighted translation semigroups in multi-variable case. Further, we replace the space (L^2({{mathbb R}_+})) by (L^2({{mathbb R}_+^d})) and explore the properties of weighted translation semigroup ({S_{overline{t}}}) in ({mathcal B}(L^2({{mathbb R}_+^d})),) in both one and multi variable cases.

For an (ntimes n) complex matrix C, the C-numerical range of a bounded linear operator T acting on a Hilbert space of dimension at least n is the set of complex numbers (textrm{tr},(CX,^*,TX)), where X is a partial isometry satisfying (X^*X = I_n). It is shown that

for any contraction T if and only if C is a rank one normal matrix.