ACTA SCIENTIARUM MATHEMATICARUM最新文献

We study operator means more general than the Kubo-Ando means. They are given as minima of geodesically convex functions and allow uniquely defined extensions to any number of variables. Multivariate hyper-means is a class of means of this type bounded from below by the arithmetic mean. We extend the definition of a hyper-man in the bivariate case and discover bivariate means that are not restrictions of the multivariate means studied earlier. We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer.

Inspired by generalized invertibility, Drazin invertibility and some recent works (Lahmar and Skhiri in Pseudo-generalized inverse I, 2022; Pseudo-generalized inverse II, 2022; Results Math 78:24, 2023; Acta Sci. Math. (Szeged) 89:389–411, 2023), this paper explores a novel class of operators called (Delta _{{{varvec{i}}}_{g}})-invertible operators including all the mentioned concepts. Due to this extension, we successfully introduce a new concept of inverse that extends various inverse concepts, such as Drazin inverse, group inverse, Moore-Penrose inverse and DPG inverses, establishing a unified framework for all these concepts. We discuss several properties of this new inverse such as its uniqueness and outer inverse nature. In the context of Hilbert space, we present a specific case in which several properties of the Moore-Penrose inverse remain valid. As an application of our new concept, we prove various properties and perturbation results related to the pseudo-generalized invertibility introduced in (Lahmar and Skhiri in Pseudo-generalized inverse I, 2022).

The seminal work of Kubo and Ando (Math Ann 246:205–224, 1979/80) provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavour. On the other hand, it is highly natural to take the geomeric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.

In this paper, we first give two new upper bounds for the numerical radius of the product of two Hilbert space operators. The obtained bounds are compared numerically with previously known bounds. After that, the Hilbert–Schmidt numerical radius is studied for the generalized Aluthge transform and a pair of commuting operators. In the end, off-diagonal, tridiagonal, and anti-tridiagonal operator matrices are treated, where many upper bounds that extend some existing results are shown.

This paper investigates an important functional representation of the cone of bounded positive semidefinite operators. It is known that the representation by strength functions turns the Löwner order into the pointwise order. However, very little is known about the structure of strength functions. Our main result says that the representation behaves naturally with the infimum and supremum operations. More precisely, we show that the pointwise minimum of two strength functions (f_A) and (f_B) is a strength function if and only if the infimum of A and B exists. This complements a recent result of L. Molnár stating that the pointwise maximum of (f_A) and (f_B) exists if and only if A and B are comparable, as this latter statement is equivalent to the existence of the supremum. The cornerstone of each argument in this paper is a fact that was discovered recently, namely that the strength function of the parallel sum A : B (which is half of the harmonic mean) equals the parallel sum of the strength functions (f_A) and (f_B). We provide a new proof for this statement, and as a byproduct, in some special cases, we describe the strength function of the so-called (generalized) short.

In this paper we consider two new variants of the classical Weyl ’s theorem for operators defined on Banach spaces. These variants called S-Weyl’s theorem and property (t) are stronger than the more classical variants of Weyl’s theorem, as a-Weyl’s theorem and property ((omega )) studied by several authors. In particular, we explore these two new variants for operators that commute with an injective quasi-nilpotent operators.

In 2008, László Zádori proved that the lattice (Sub (V) ) of all subspaces of a vector space V of finite dimension at least 3 over a finite field F has a 5-element generating set; in other words, (Sub (V) ) is 5-generated. We prove that the same holds over every 1- or 2-generated field; in particular, over every field that is a finite degree extension of its prime field. Furthermore, let F, t, V, (dge 3), (lfloor d/2rfloor ), and m denote an arbitrary field, the minimum cardinality of a generating set of F, a finite dimensional vector space over F, the dimension (assumed to be at least 3) of V, the integer part of d/2, and the least cardinal such that (mlfloor d^2/4rfloor ) is at least t, respectively. We prove that (Sub (V) ) is ((4+m))-generated but none of its generating sets is of size less than m. Moreover, the kth direct power of (Sub (V) ) is ((5+m))-generated for many positive integers k; for all positive integers k if F is infinite. Finally, let n be a positive integer. For (i=1,dots , n), let (p_i) be a prime number or 0, and let (V_i) be the 3-dimensional vector space over the prime field of characteristic (p_i). We prove that the direct product of the lattices (Sub (V_1) ), ..., (Sub (V_n) ) is 4-generated if and only if each of the numbers (p_1), ..., (p_n) occurs at most four times in the sequence (p_1), ..., (p_n). Neither this direct product nor any of the subspace lattices (Sub (V) ) above is 3-generated.

The purpose of this paper is to extend symmetric means of multi-variable positive matrices to those of multi-variable positive operators on an infinite dimensional Hilbert space. We also consider the situations where symmetric means of operators A, B, C become linear combinations of them.

In this paper, we consider two generalized matrix equations that involve an arbitrary Kubo–Ando mean. We also study the multi-step stationary iterative method for these equations and prove the corresponding convergences.

We are concerned with log-majorization for matrices in connection with the multivariate Golden–Thompson trace inequality and the Karcher mean (i.e., a multivariate extension of the weighted geometric mean). We show an extension of Araki’s log-majorization and apply it to the (alpha )-z-Rényi divergence in quantum information. We discuss the equality cases in the multivariate trace inequality of Golden–Thompson type and in the norm inequality for the Karcher mean. The paper includes an appendix to correct the proof of the author’s old result on the equality case in the norm inequality for the weighted geometric mean.