In this paper, we give a new characterization for the perspective of a continuous function under certain assumptions. This result generalizes a non-commutative analogue of the arithmetic–geometric mean inequality. Our main result extends and strengthens the existing result concerning the characterization of the geometric operator mean. We provide the characterization of other operator means, in particular

ABSTRACT We deal with the Roberts numerical radius orthogonality. In the case of 2 × 2 complex matrices, we give some necessary and sufficient conditions for the numerical range to be symmetric by employing the Roberts orthogonality with respect to the numerical radius. In addition, we present an interrelation between the Roberts orthogonality and the Birkhoff–James orthogonality with respect to the

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let Λ = d i a g ( λ 1 , … , λ p ) ∈ R p × p and Φ = [ ϕ 1 , … , ϕ p ] ∈ R n × p be the measured eigenvalue

ABSTRACT Motivated by the Araki-Lieb-Thirring inequality t r ( A 1 / 2 B A 1 / 2 ) r p ≤ t r ( A r / 2 B r A r / 2 ) p for p ≥ 0 ,   r ≥ 1 ( t r ( A 1 / 2 B A 1 / 2 ) r p ≥ t r ( A r / 2 B r A r / 2 ) p for p ≥ 0 ,   0 ≤ r ≤ 1 ) for Hermitian positive semidefinite matrices and the Golden-Thompson inequality t r ( exp ⁡ ( A + B ) ) ≤ t r ( exp ⁡ ( A ) exp ⁡ ( B ) ) for Hermitian matrices, in this paper

A square matrix A is completely positive if A = B B T , where B is a (not necessarily square) nonnegative matrix. In general, a completely positive matrix may have many, even infinitely many, such CP factorizations. But in some cases a unique CP factorization exists. We prove a simple necessary and sufficient condition for a completely positive matrix whose graph is triangle free to have a unique CP

ABSTRACT In the present paper, we will characterize linear maps of M n ( F ) , the ring of all n × n matrices over an algebraically closed field F of characteristic zero, which preserve the set of all algebraic elements of degree 2.

ABSTRACT A graph G is called triangle-free if G does not contain a triangle as an induced subgraph. Let H n be the set of triangle-free graphs of order n with six non-zero eigenvalues. In this paper, we find 19 graphs of H n , and we show that the other graphs of H n can be constructed from these 19 graphs by adding some congruent vertices. Hence we completely characterize the triangle-free graphs

ABSTRACT Let n>1 be a positive integer, { H 1 , … , H n } be a finite collection of complex Hilbert spaces with dim ⁡ ( H k ) ≥ 2 , and P 1 ( H k ) be the set of all rank-1 self-adjoint projections on H k , k = 1 , … , n . Set D P 1 ⨂ k = 1 n H k = { A 1 ⊗ ⋯ ⊗ A n : A k ∈ P 1 ( H k ) , k = 1 , … , n } . We characterize the maps ϕ from D ( P 1 ( ⨂ k = 1 n H k ) ) to P 1 ( ⨂ k = 1 n H k ) preserving

ABSTRACT Let L ( X ) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let X T ( { λ } ) denote the local spectral subspace of an operator T ∈ L ( X ) associated with a singleton { λ } ⊂ C , and fix a scalar λ 0 ∈ C . We characterize maps ϕ on L ( X ) which satisfy X ϕ ( T ) − ϕ ( S ) ( { λ 0 } ) = X T − S ( { λ 0 } ) for all  T ,   S ∈ L ( X ) . Furthermore, we give

ABSTRACT We consider an evolution algebra identifying the coefficients of SIS–SIR worm propagation models as the structure constants of the algebra. The basic properties of this algebra are studied. We prove that it is a commutative (and hence flexible), not associative and baric algebra. We describe the full set of idempotent elements and the full set of absolute nilpotent elements. We find all the

ABSTRACT A fractional matching of a graph G is a function f : E ( G ) → [ 0 , 1 ] such that for any v ∈ V ( G ) , ∑ e ∈ E G ( v ) f ( e ) ≤ 1 where E G ( v ) = { e ∈ E ( G ) : e is incident with v in G } . The fractional matching number of G is μ f ( G ) = max { ∑ e ∈ E ( G ) f ( e ) : f is a fractional matching of G } . For any real numbers a ≥ 0 and k ∈ ( 0 , n ) , it is observed that if n = | V

In this paper, we show that P O ( 2 , 1 ) can be represented by invertible split quaternions. We introduce the concepts of t-similarity and t-pseudosimilarity of split quaternions. By such two similarities, we obtain several equivalence relations in split quaternions. We also show that a and b are semisimilar if and only if a 2 and b 2 are similar.

Given a linear map T on a Euclidean Jordan algebra of rank n, we consider the set of all nonnegative vectors q in R n with decreasing components that satisfy the pointwise weak-majorization inequality λ ( | T ( x ) | ) ≺ w ⁡ q ∗ λ ( | x | ) , where λ is the eigenvalue map and ∗ denotes the componentwise product in R n . With respect to the weak-majorization ordering, we show the existence of the least

ABSTRACT The primary purpose of this paper is to discuss phase-retrievable g-frames and exact phase-retrievable g-frames. Firstly, some characterizations of phase-retrievable g-frames are discussed. And some necessary and sufficient conditions to make g-frames phase-retrievable g-frames can be obtained. Furthermore, we find that for a phase-retrievable g-frame, its canonical dual g-frame is also phase-retrievable

As is well known, the generalized inverse, Moore–Penrose inverse and group inverse are not continuous, i.e. for θ = { 1 , 2 } , { 1 , 2 , 3 , 4 } and { 1 , 2 , 5 } , a linear bounded operator T has a θ-inverse T θ , the perturbed operator T ¯ = T + δ T is not necessary θ-invertible and even if it is θ-invertible, lim δ T → 0 T ¯ θ = T θ may not be true. In this paper, we prove that T + T T θ δ T T

ABSTRACT In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a nilpotent matrix of order less than or equal to two. In addition, each 2 × 2 matrix over any field admits such a representation. We, moreover, show that, for all natural

A connected graph is called a multi-block graph if each of its blocks is a complete multipartite graph. We consider the distance matrix of multi-block graphs with blocks whose distance matrices have nonzero cofactor. In this case, if the distance matrix of a multi-block graph is invertible, we find the inverse as a rank one perturbation of a multiple of a Laplacian-like matrix. We also provide the

ABSTRACT It is shown in this paper that the Schur complements and the diagonal-Schur complements of DZ-type matrices are DZ-type matrices under some conditions. A numerical example for solving the linear equations with the coefficient matrix being a DZ-type matrix is given to show that the Schur-based Gauss–Seidel iteration method and the Schur-based conjugate gradient method can compute out the solution

In this paper, we study the properties of extreme points of slice-stochastic tensors, and present an analogue of the Birkhoff–von Neumann theorem for slice-stochastic tensors. Some equivalent characterizations for a slice-stochastic tensor to be an extreme point are presented. The number of permutation tensors is also discussed.

ABSTRACT We consider oriented and also signed graphs which are cospectral, i.e. they are not switching isomorphic but share the same spectrum. We prove that there is a bijective correspondence between cospectral bipartite oriented graphs and cospectral bipartite signed graphs. We also give certain constructions of cospectral oriented (signed) graphs; for example, we provide infinite families of cospectral

ABSTRACT A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices that are close enough to the identity matrix. In this paper we exhibit

ABSTRACT We give an algebraic classification of complex 4-dimensional nilpotent C D -algebras.

ABSTRACT In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations.

ABSTRACT The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the α k -generalized derivation algebra G D e r ( J ) of a ternary Hom-Jordan algebras. In particular, we prove that G D e r ( J ) = Q D e r ( J ) + Q Γ ( J ) , the sum of the α k -quasiderivation algebra and the α k -quasicentroid. We

ABSTRACT Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of

ABSTRACT In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solution set of a system of polynomial equalities and inequalities, the state-of-art results in error bounds for polynomial systems can be applied directly. Starting from this, a much better error bound

ABSTRACT Von Neumann regularity within a ring of upper triangular matrices is characterized in terms of how such a matrix and an upper triangular generalized inverse can be constructed. In the case of a finite ring, this yields a method for implicitly counting all upper triangular matrices that admit upper triangular generalized inverses.

ABSTRACT This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations

A complete description of 4-by-4 matrices α I C D β I , with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which C − D ∗ also is a scalar multiple of the identity. In particular cases when in addition α − β is real or pure imaginary, the results take an especially

ABSTRACT The power graph of a group Ω is a graph, whose node set is Ω and two distinct elements are adjacent if and only if one is an integral power of the other. A metric dimension of a graph Γ, denoted by ψ ( Γ ) is the minimum cardinality of the resolving set of Γ. In this context, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric

ABSTRACT We determine the tree which maximizes the distance between characteristic set and subtree core over all trees on n vertices. The asymptotic nature of this distance is also discussed. The problem of extremizing the distance between different central parts of trees on n vertices with fixed diameter is studied.

Embedding discrete Markov chains into continuous ones is a famous open problem in probability theory with many applications. Inspired by recent progress, we study the closely related questions of embeddability of real and positive operators into real or positive C 0 -semigroups, respectively, on finite and infinite-dimensional separable sequence spaces. For the real case we give both sufficient and

ABSTRACT We introduce the notion of level numbers of a bounded linear operator between normed linear spaces, as a generalization of the singular values of an operator between inner product spaces. We study the geometric and the analytic properties of the level numbers, in connection with Birkhoff–James orthogonality and norm optimization problems. We also illustrate the similarities and the differences

ABSTRACT Let V be an n-dimensional linear space over an algebraically closed base field. We provide a general method for classifying, up to equivalence, all bilinear maps f : V × V → V such that d i m ( r a d ( f ) ) = n − m , in case a classification of bilinear maps f : W × W → W when d i m ( W ) = m is known. This is equivalent to giving a procedure to classify (up to isomorphism) all n-dimensional

In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed

We prove multilinear variants of the Maurey factorization theorem. Let n be a natural number, π = A 1 , … , A k a partition of the set 1 , … , n , 0 < q 1 , … , q k < ∞ , 1 / v k = 1 / q 1 + ⋯ + 1 / q k and 1 ≤ p , r < ∞ such that 1 / p = 1 / v k + 1 / r . Let U : X 1 × ⋯ × X n → L p μ , Y be a positive homogeneous operator in each variable and V 1 : ∏ i ∈ A 1 X i → 0 , ∞ , … , V k : ∏ i ∈ A k X i

ABSTRACT In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant ( i 2 + d j 2 ) i 2 + d j 2 n 0 ≤ i , j ≤ ( n − 1 ) / 2 , where d is any integer and ( ⋅ n ) is the Jacobi symbol. Then we prove some divisibility results concerning | ( i + d j ) n | 0 ≤ i , j ≤ n − 1 and | ( i 2 + d j 2 ) n | 0 ≤ i , j ≤ n − 1 , where d

ABSTRACT In this paper, we analyse the matrix structures of various generalized inverses, suggest their applicable scopes and build their relationships with particular projections. As applications, several equivalent conditions for core–EP (Range-Hermitian) relation, star order, sharp order and MP–Core–EP (MPCEP) relation are presented. The relationships among these orders or relations are built. In

In this paper, we extend the operations of parallel addition A : B and parallel subtraction A ÷ B from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations A † : B † = ( A + B ) † , B = ( A : B ) ÷ A , ( A C ) ÷ ( B C ) = ( A ÷ B ) C , A ÷ B = ( P ( A † − B † ) P ) † , B = A : ( B ÷ A ) , ( C A ) : ( C B ) = C (

ABSTRACT The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given.

ABSTRACT Let U T m be the algebra of all m × m upper triangular matrices over a field F whose characteristic is different from 2. Given an involution ∗ of the first kind on U T m , we will obtain the minimal integer t such that the Lie polynomial [ [ z 1 , z 2 ] , … , [ z 2 t − 1 , z 2 t ] ] is an identity for K ( m , ∗ ) = { U ∈ U T m ∣ U ∗ = − U } . Afterwards, under a mild technical restriction

We give a classification of irreducible representations of generalized Heisenberg groups K n ( F q ) ,   n ≥ 5 , which is the pattern group associated to the closed set { ( 1 , i ) ,   ( 2 , j ) ,   ( s , n − 1 ) ,   ( t , n ) | 2 ≤ i ≤ n ,   3 ≤ j ≤ n ,   3 ≤ s < n − 1 ,   3 ≤ t < n } . In light of the conjectures of Higman, Lehrer and Isaacs for unitriangular groups, this result shows that the number

ABSTRACT Characterizing which graphs are determined by their (signless) Laplacian permanental polynomials is an interesting problem. In this paper, we prove that paths, cycles and lollipop graphs are determined by their Laplacian permanental polynomials as well as their signless Laplacian permanental polynomials, respectively.

ABSTRACT A quadratic Lie conformal algebra corresponds to a Hamiltonian pair in Gel'fend and Dorfman (Hamiltonian operators and algebraic structures related to them. Funkts Anal Prilozhen. 1979;13:13–30), which plays fundamental roles in completely integrable systems. Moreover, it also corresponds to a certain compatible pair of a Lie algebra and a Novikov algebra which was called Gel'fand–Dorfman

ABSTRACT The recently introduced concept of closed unit segment for rings allows to define convexity in modules. In this manuscript, we construct a new class of closed unit segments in totally ordered rings, which allows to define convex functions on the category of modules over totally ordered rings. Among other results, it is proved that the boundary of a proper open convex subset of a topological

In the present paper, we introduce and investigate the notion of 2-local linear maps on vector spaces. A sufficient condition is obtained for the linearity of a 2-local linear map on finite-dimensional vector spaces. Based on this result, we prove that every 2-local derivation on a finite-dimensional semisimple Jordan algebra over an algebraically closed field of characteristic different from 2 is

Necessary and sufficient conditions are obtained for a pair of matrix equations AX = C and XB = D to have common Re-nnd and Re-pd solutions and the explicit representations of the general common Re-nnd and Re-pd solutions are given when the stated conditions hold.

ABSTRACT Besides, unfortunately I can't make a tick on the check boxes below. Here is our new suggestion for an abstract: The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. Recently, Zhao et al. [A geometric nonlinear conjugate gradient method for stochastic inverse eigenvalue problems. SIAM J Numer Anal. 2016;54(4):2015–2035] proposed a constrained

ABSTRACT A bounded adjointable operator T ∈ L ( H ) in Hilbert C*-modules is called EP if ranges of T and T ∗ have the same closure. This definition is employed to achieve a new characterization of EP operators. We show that the anticommutator of EP operators is again an EP operator. It follows that the product of commuting EP operators is an EP operator. Some other conditions implying the product

ABSTRACT We characterize k-smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize k-smoothness of operators on some particular spaces, namely L ( X , ℓ ∞ n ) , L ( ℓ ∞ 3 , Y ) , where X is a finite-dimensional Banach space and Y is a two-dimensional Banach

ABSTRACT Let D be a digraph of order n and let A ( D ) be the adjacency matrix of D. Let D e g ( D ) be the diagonal matrix of vertex out-degrees of D. For any real α ∈ [ 0 , 1 ] , the generalized adjacency matrix A α ( D ) of the digraph D is defined as A α ( D ) = α D e g ( D ) + ( 1 − α ) A ( D ) . The largest modulus of the eigenvalues of A α ( D ) is called the generalized adjacency spectral radius

ABSTRACT The current research investigates the core partial order for its further properties. We introduce a new property for matrices of index at most 1, namely ( B − A ) ◯ # = B ◯ # − A ◯ # and is called as ‘the core-subtractivity’. This property then is used to explore relations between the powers and subtractivity properties of the core partial order. In particular, we prove that A ≤ ◯ # B is equivalent

ABSTRACT In this paper, we give an alternative characterization of complex symmetry of Toeplitz operators and block Toeplitz operators. In particular, we prove that a block Toeplitz operator T Φ is complex symmetric with the conjugation C on the vector-valued Hardy space H C 2 2 if and only if M Φ = C M Φ ∗ C , where M Φ denotes the multiplication operator on L C 2 2 with symbol Φ. As some applications

Let W n denote the wheel graph having n-vertices. If i and j are any two vertices of W n , define d i j := 0 if  i = j 1 if  i  and  j  are adjacent 2 else . Let D be the n × n matrix with ( i , j ) t h entry equal to d i j . The matrix D is called the distance matrix of W n . Suppose n ≥   5 is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely

ABSTRACT First, we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an associative algebra such that the algebra is graded-simple and satisfies the DCC on graded left ideals. We give necessary and sufficient conditions for the grading

ABSTRACT Let T be a tree with vertex set V ( T ) = { v 1 , v 2 , … , v n } . The adjacency matrix A ( T ) of T is an n × n matrix ( a i j ) , where a i j = a j i = 1 if v i is adjacent to v j and a i j = 0 if otherwise. In this paper, we consider the multiplicity of − 1 as an eigenvalue of A ( T ) , which is written as m ( T , − 1 ) . It is proved that among all trees T with p ≥   2 pendant vertices

ABSTRACT For a simple connected graph G of order n, we obtain the distance signless Laplacian spectrum of the joined union of regular graphs G 1 , G 2 , … , G n in terms of their adjacency spectrum and the spectrum of an auxiliary matrix. As a consequence, we obtain the distance signless Laplacian spectrum of the zero divisor graphs of finite commutative rings Z n for some values of n. We show that

The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rectangular matrices. Solvability of matrix equation systems as AXA = AEA and BAEAX = B; or AXA = AEA and XAEAD = D, where A ∈ C m × n , B ∈ C p × m , D ∈ C n × q and E ∈ C n × m , is studied by means of left

ABSTRACT In this paper, we study the properties ( b z ) and ( w π 00 a ) , which we had introduced in [Ben Ouidren K, Zariouh H. New approach to a-Weyl's theorem and some preservation results. Rend Circ Mat Palermo. doi: 10.1007/s12215-020-00525-2], for an operator having the SVEP on the complementary of distinguished parts of its spectrum. We prove in particular, that a bounded linear operator T acting

ABSTRACT By using a q-analogue of the ‘magic’ matrix introduced by H.Huang in his elegant solution of the sensitivity conjecture, we give a direct generalization of his result, replacing a hypercube graph by a Cartesian power of a directed l-cycle.