Baksalary and Hauke introduced the diamond partial order in 1990, which we revisit in this paper. This order was defined on the set of rectangular matrices and is the same as the star and minus partial orders for partial isometries. New ways of describing and studying the diamond partial order are being looked into in this paper. Particularly, we present a new characterization by using an additivity property of the column spaces. Additionally, we also study the relationship between the left (resp., right) star and diamond partial orders. Specifically, we obtain conditions in which the diamond partial order means the left (resp., right) star partial order. The reverse order law for the Moore-Penrose inverse is characterized when A is below B under the diamond partial order. Finally, an interesting way of describing bi-dagger matrices is found.

We investigate two one-sided orthogonalities of matrices, the first of which is left (right) $*$-orthogonality for rectangular matrices and the other is left (right) core-orthogonality of index $1$ matrices. We obtain some basic results for these matrices, their canonical forms, and characterizations. Also, relations between left (right) orthogonal matrices and parallel sums are investigated. Finally under these one-sided orthogonalities we explore the conditions of additivity of the Moore-Penrose inverse and the core inverse.

We correct an error in the statement of Levis et al. (2023, Theorem 4.5).

The concept of star-dagger matrices was introduced in 1984 by Hartwig and Spindelböck. While they completely characterized the star-dagger matrices by using a block decomposition of the form PQ00 , they also proposed the following open problem: "Can the triangular form PQ0Rbe used to obtain further results on the star-dagger matrices?" In this paper, we have attempted this open problem by using an upper-triangularization of Schur's type for a square matrix, namely, the core-EP decomposition. Furthermore, similar problems regarding bi-dagger and bi-EP matrices are investigated.

Rao and Mitra in 1972 introduced two different types of constraints to extend the concept of Bott-Duffin inverse and defined a new constrained inverse. Mary in 2011 defined the inverse along an element that generalizes the Moore-Penrose and Drazin inverses in a semigroup. Drazin in 2012 introduced the (b,c)-inverse generalizing the Mary inverse. In 2017, Rakić noted that the Rao-Mitra inverse is a direct precursor of the (b,c)-inverse. In this paper, we introduce the notion of EF-inverse as a unified approach to the aforementioned generalized inverses. Moreover, we show that the recently introduced generalized bilateral inverses that in turn contain the OMP, MPO, and MPOMP inverses can also be considered as special cases of the EF-inverse.

It is well known that a square complex matrix is called EP if it commuteswith its Moore-Penrose inverse. In this paper, new classes of matrices whichextend this concept are characterized. For that, we consider commutativeequalities given by matrices of arbitrary index and generalized inversesrecently investigated in the literature. More specifically, these classes arecharacterized by expressions of type A^mX=XA^m, where X is an outer inverseof a given complex square matrix A and m is an arbitrary positive integer.The relationships between the different classes of matrices are also analyzed.Finally, a picture presents an overview of the overall studied classes.

In this paper, we give a strong uniqueness characterization theorem for the Chebyshev center of a set of infinitely many functions relative to a finite-dimensional linear space on a compact Hausdorff space. Additionally, we derive an alternation theorem for Chebyshev centers relative to a weak Chebyshev space on any compact set of the real line. Furthermore, we show an intrinsic characterization of those linear spaces where an alternation theorem holds.

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz-Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial L-phi-approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients.

In this paper, we consider the best polynomial approximation operator defined on an Orlicz-Lorentz space ?w,phi$\Lambda _{w,\phi }$, and its extension to ?w,phi '$\Lambda _{w,\phi <^>{\prime }}$, where w is a non-negative continuous weight function and phi '$\phi <^>{\prime }$ is the derivative of phi, which is not required to be an Orlicz function. Our work generalizes a recent result in this field on an Orlicz-Lorentz space generated by an Orlicz function. In addition, we establish some properties and estimates for any extended best polynomial approximation.

A Correction to this paper has been published: 10.1007/s13348-021-00331-8

In this article, we consider the best polynomial approximation operator defined on an Orlicz-Lorentz space ?w,phi$\Lambda _{w,\phi }$ generated by an Orlicz function phi and a non-negative continuous weight function w. Then we extend the best polynomial approximation operator from ?w,phi$\Lambda _{w,\phi }$ to ?w,phi '$\Lambda _{w,\phi <^>{\prime }}$, where phi '$\phi <^>{\prime }$ is the derivative function of phi. In addition, we establish some properties of the extended best polynomial approximation operator.

In this paper we prove that the extended best Lp polynomial approximation to a function f in Lq, 0<p−1≤q<p, is near-best approximation in Lq. Besides, we provide an analog result for the case p=1.

Davison in 2015 used the famous Banach Fixed Point Theorem to prove that a certain class of iterated function systems generated counterparts of the Hutchinson measure in the space of projection-valued measures. In this paper, we generalize this result by considering iterated function systems with infinitely many maps.

In this paper, we present necessary and sufficient conditions for the k-commutative equality , where X is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations.

In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681–697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu.

In a recent paper of Cuenya and Ferreyra, a condition, namely, the C-p-condition in L-p-spaces was introduced that is weaker than the notion of L-p-derivative given by Calderon-Zygmund. In the present article, we define the Legendre derivative for functions in L-2 generalizing both notions, the C-p-condition and L-p-derivative in the case p = 2. As a consequence, we give a necessary and sufficient condition for the existence of the best local approximation in L-2 by using this new concept of derivative. In addition, we study the convexity of the set of cluster points of the set of best L-2 approximations to a function on a interval when their measures tends to zero.

We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Zo and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193-213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials.

G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: if A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.

In this short note, we study inequalities for algebraic polynomials on measurable sets in Lorentz spaces and discuss their applications to best approximation.