We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic ≠ 2 . It is proved that the algebras of upper triangular 3 × 3 or 4 × 4 matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding 6 contains a non-commutative local centrally essential subalgebra.

ABSTRACT Kemeny's constant κ ( G ) of a connected graph G is a measure of the expected transit time for the random walk associated with G. In the current work, we consider the case when G is a tree and, in this setting, we provide lower and upper bounds for κ ( G ) in terms of the order n and diameter δ of G by using two different techniques. The lower bound is given as Kemeny's constant of a particular

We give several norm and numerical radius inequalities for Hilbert space operators. These inequalities improve on some earlier related inequalities. As an application of one of our results, we give a necessary condition for the equality case in the triangle inequality for the usual operator norm.

In this paper, two formulas, which were studied by Wang and Liu (2015 WangHX, LiuXJ. Characterizations of the core inverse and the core partial ordering. Linear Multilinear Algebra. 2015;63:1829–1836. doi: 10.1080/03081087.2014.975702 [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Ma and Li (2019), respectively, for the core inverse, are simplified. Then two methods for computing

Convolutional neural network is an important model in deep learning, where a convolution operation can be represented by a tensor. To avoid exploding/vanishing gradient problems and to improve the generalizability of a neural network, it is desirable to have a convolution operation that nearly preserves the norm, or to have the singular values of the transformation matrix corresponding to the tensor

We introduce the notion of a tractable pair of operators as well as that of the generalized parallel sum in the setting of adjointable operators on Hilbert C ∗ -modules. Some significant results about the parallel sum known for matrices and Hilbert space operators are extended to the case of the generalized parallel sum. In particular, a factorization theorem on the parallel sum is proved, and a common

ABSTRACT Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not

A Hilbert space operator A ∈ B ( H ) is left ( X , m ) -invertible by B ∈ B ( H ) (resp., B ∈ B ( H ) is an ( X , m ) -adjoint of A ∈ B ( H ) ) for some operator X ∈ B ( H ) if △ B , A m ( X ) = ∑ j = 0 m ( − 1 ) j ( m j ) B m − j X A m − j = 0 (resp., δ B , A m ( X ) = ∑ j = 0 m ( − 1 ) j ( m j ) B ( m − j ) X A j = 0 ). No Drazin invertible operator A ∈ B ( H ) , with Drazin inverse A d , can be

Let G be a digraph and A ( G ) be the adjacency matrix of G. Let D ( G ) be the diagonal matrix with outdegrees of vertices of G. For any real α ∈ [ 0 , 1 ] , define the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . The largest modulus of the eigenvalues of A α ( G ) is called the A α spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum)

Let g ( G ) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k ( k ≥ 8 ) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g ( G ) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least

The present paper deals with new modification of Gamma operators preserving polynomials in Bohman-Korovkin sense and study their approximation properties: Voronovskaya type theorems, weighted approximation and rate of convergence are captured. The effectiveness of the newly modified operators according to classical ones are presented in certain senses as well. Numerical examples are also presented

The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix.

In this paper we prove among others that f y z − z f x ≤ 1 2 π y z − z x ∫ γ f ξ R y ξ R x ξ d ξ ≤ 1 2 π y z − z x ∫ γ f ξ d ξ ξ − y ξ − x , where f : D ⊂ C → C is an analytic function on the domain D, x, y, z ∈ B with σ x , σ y ⊂ D , R y ⋅ , R x ⋅ are the resolvent functions for the elements y and x, and γ is a closed rectifiable path in D and such that σ x , σ y ⊂ i n s   γ . Applications for the

Let Π be an equitable partition of a graph G with k cells. If G has no equitable partition with fewer cells than Π, then Π is called a minimal partition of G and G is called a k-equitable graph. It is clear that 1-equitable graphs are regular graphs. In this paper, we investigate 2-equitable graphs with four distinct eigenvalues.

Let X and Y be compact Hausdorff spaces, E and F be real or complex normed spaces and A ( X , E ) be a subspace of C ( X , E ) . For a function f ∈ C ( X , E ) , let coz ( f ) be the cozero set of f. A pair of additive maps S , T : A ( X , E ) ⟶ C ( Y , F ) is said to be jointly separating if coz ( T f ) ∩ coz ( S g ) = ∅ whenever coz ( f ) ∩ coz ( g ) = ∅ . In this paper, first, we give a partial

Given a connected graph G = ( V G , E G ) with x , y ∈ V G , the hitting time H G ( x , y ) is defined as the expected number of steps that a simple random walk takes to go from x to y. A hitting-time-based invariant, called the ZZ index, was first proposed by Zhu and Zhang [The hitting time of random walk on unicyclic graphs. Linear Multilinear Algebra. doi:10.1080/03081087.2019.1611732] recently

Let H be an infinite dimensional complex Hilbert space and let B ( H ) be the algebra of all bounded linear operators on H . For every positive integer k and A ∈ B ( H ) , the k-numerical range of A is the set W k ( A ) = ∑ j = 1 k ⟨ A x j , x j ⟩ : { x 1 , … , x k }   is an orthonormal set in  H . In this note, we show that the closure of W k ( A ) can be written as the convex hull of sets involving

In this paper, we study the relationship between numerical index of a Banach space E and the numerical index of a class of subspaces of C ( Ω , E ) , the space of all continuous E-valued functions defined on a compact Hausdorff topological space Ω. The subspaces considered, satisfy an invariance condition and a norm attaining condition proposed by J. Jamison. As a consequence, we prove that the numerical

In this paper, we consider a generalization of ( A , m ) -isometric Hilbert space operators to the multivariable setting. Inspired by the work [Sid Ahmed OAM, Chō M, Lee JE. On (m,C)-isometric commuting tuples of operators on a Hilbert space. Res Math. 2018;73:51. Doi:10.1007/s00025-018-0810-0], we introduce the class of ( A , m ) -isometric tuples of commuting operators. A d-tuple T = ( T 1 , … ,

We introduce the class of Riesz-representable multilinear mappings on products of L 1 ( μ ) spaces for finite measures μ and give their main properties. We prove Grothendieck type theorems for composition with the natural operators L ∞ → L 1 and C ( Ω ) → L 1 . Unlike the linear case, the composition is not nuclear but is characterized by what we call the left integral ℓ 1 -factorization property.

In this paper, we discuss a version of Birkhoff–James orthogonality in the C ∗ -algebra of all bounded linear operators on a finite-dimensional Hilbert space. We obtain a characterization of this relation and use it to describe some classes of operators. We also characterize linear preservers of this type of orthogonality.

In this paper, we introduce the notions of the spherical p-hyponormality and the spherical log-hyponormality for commuting pairs of operators. We first prove that for a commuting pair T = ( T 1 , T 2 ) of operators, if T is jointly hyponormal, then T is spherically p-hyponormal for 0 < p ≤   1 . Second, we show that for 0 < p , q ≤   1 and q ≤ p , a spherically p-hyponormal operator is spherically

Let G be a finite group and Γ be a (di)graph. Then Γ is called an n-Cayley (di)graph over G if Aut ( Γ ) admits a semiregular subgroup isomorphic to G with n orbits on V ( Γ ) . In this paper, we determine the normalized Laplacian polynomial of n-Cayley (di)graphs over a group G in terms of irreducible representations of G. We give exact formulas for the normalized Laplacian eigenvalues of 2-Cayley

For an n × n doubly substochastic matrix A, the sub-defect of A is the smallest integer k so that by adding k rows and columns, A becomes a doubly stochastic matrix. In this paper, we introduce the notion of sub-defect for an arbitrary I × I doubly substochastic matrix A = [ a i j ] i , j ∈ I by using cardinal numbers and then show that any doubly substochastic matrix can be turned into a doubly stochastic

As a first step of the study of the relation between the distribution of prime, reduced cycles of a graph and the limit theorem for the probability of a quantum walk on a graph, we present a trace formula with respect to the Grover matrix that is the transition matrix of the Grover walk on a graph. We define a zeta function with respect to the Grover matrix of a graph, and differentiate the logarithm

If G is a graph with n vertices, A G is its adjacency matrix and b is a binary vector of length n, then the pair ( A G , b ) is said to be controllable (or G is said to be controllable for the vector b ) if A G has no eigenvector orthogonal to b . In particular, if b is the all-1 vector j , then we simply say that G is controllable. In this paper, we consider the controllability of non-complete extended

Let R be a finite commutative ring with non-zero identity. Let R × and J ( R ) be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by G ( R ) , is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if x + y ∈ R × . If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit

The paper investigates the class of square matrices which have idempotent Moore–Penrose inverse. A number of original characteristics of the class are derived and new properties identified. Additionally, some isolated results scattered in the literature are recaptured and rederived in an extended or generalized form. The research was inspired by some matrix equations (occurring, e.g. in physics) which

We obtain upper bounds for the numerical radius of a product of Hilbert space operators which improve on the existing upper bounds. We generalize the numerical radius inequalities of n × n operator matrices by using non-negative continuous functions on [ 0 , ∞ ) . We also obtain some upper and lower bounds for the B-numerical radius of operator matrices, where B is the diagonal operator matrix whose

This paper mainly focuses on spectral conditions for the Hamiltonicity of balanced bipartite graphs and nearly balanced bipartite graphs. Let ρ ( G ) and q ( G ) be the spectral radius and signless Laplacian spectral radius of a graph G, respectively. One of our main results is the following theorem: Let G be a balanced bipartite graph of order 2 n and of minimum degree δ ( G ) ≥ k ≥ s ≥ 0 . If n ≥

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic

In this short note, we investigate the degree-inverting involutions on the full square matrices, assuming the base field is algebraically closed of characteristic not 2. We obtain a partial result if the grading group is arbitrary, and a complete answer for abelian groups. We also classify the degree-inverting involutions on the triangular matrices, for arbitrary groups, and arbitrary fields of characteristic

Let G ϕ be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of G ϕ , written as r ( G ϕ ) , is the rank of its adjacency matrix and denote by α ′ ( G ) the matching number of the underlying graph G. In this contribution, based on combinatorial interpretation of all the coefficients of the characteristic polynomial of G ϕ , we determine sharp upper and lower

In this work, we study the parametric Sylvester matrix equation A ( p ) X + X B ( p ) = C ( p ) whose elements are linear functions of uncertain parameters varying within some intervals. We first explore some properties of its solution set and then present some sufficient conditions for boundedness of the solution set. The main work is developing a modified variant of the parametric Krawczyk operator

In this paper, we investigate the least-squares solution with the least norm to the following system of tensor equations over quaternions A 1 ∗ N X = D 1 , Y ∗ N B 2 = D 2 , A 3 ∗ N X ∗ N B 3 = D 3 , A 4 ∗ N Y ∗ N B 4 = D 4 , A 5 ∗ N X + Y ∗ N B 5 = D 5 , where X , Y are unknown quaternion tensors and the others are given quaternion tensors. Using the expressions of the Moore-Penrose inverses of partitioned

The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm-based ergodicity coefficients { τ p } . If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using {

In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the M -tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order

Let N be a nest on a complex separable Hilbert space H and A l g N be the associated nest algebra. In this paper, we prove that every bilocal Lie derivation from A l g N into itself is of the form A → [ A , T ] + λ A + f ( A ) , where T ∈ A l g N , λ ∈ C and f : A l g N → C I is a linear map vanishing on each commutator. Moreover, we show that every bilocal Lie derivation from A l g N into itself is

Given two complex n × n matrices A and B, necessary and sufficient conditions are presented in order for the null space of A−B to be equal to the total strong common multiplicity of A and B. Using this characterization, when r a n k ( A − B ) = n − 1 , a simple characterization for A and B to have a strong common eigenvalue is also obtained.

A T -gain graph is a triple Φ = ( G , T , ϕ ) consisting of an underlying graph G = ( V ( G ) , E ( G ) ) , the circle group T = { z ∈ C : | z | = 1 } and a gain function ϕ : E → → T , such that ϕ ( e i j ) = ϕ ( e j i ) − 1 = ϕ ( e j i ) ¯ . In this paper, we focus our attention on the relations between the inertia indices of T -gain graph Φ and the inertia indices of its underlying graph G. We obtain

We give a lower bound for the numerical index of two-dimensional real spaces with absolute and symmetric norm. This allows us to compute the numerical index of the two-dimensional real L p -space for 3 / 2 ≤ p ≤ 3 .

In this article, we obtain some upper bounds for A-numerical radius of operators in semi-Hilbertian spaces which improve the existing ones. Furthermore, we discuss some generalizations of earlier A-numerical radius inequalities. We also establish several inequalities for B-numerical radius of n × n operator matrices, where B is an n × n diagonal matrix whose diagonal entries are positive operator A

In this paper, the Laplacian matrix of an n-prism network U g n and its applications are studied. Firstly, the relation between the Laplacian matrix of U g n and that of U 0 n is established. Secondly, the analytical expression for the product of all the nonzero Laplacian eigenvalues of U g n is obtained. At the same time, the sum of the reciprocals of all these nonzero Laplacian eigenvalues is deduced

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R and f ( x 1 , … , x n ) a multilinear polynomial over C. Suppose that f ( x 1 , … , x n ) is not central valued on R, and R does not embed in M 2 ( K ) , the algebra of 2 × 2 matrices over a field K. We describe all possible forms of F

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose i j t h entry (for i ≠ j ) is nonzero whenever ( i , j ) is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices in G. This article compares the minimum rank with the clique covering

A characterization of idempotency is given for triangular matrices with entries from an arbitrary ring. This yields an elementary approach to constructing such matrices. It also reveals structure allowing for the explicit enumeration or implicit counting of all idempotent triangular matrices over a given finite ring.

In this paper, we study the probability distribution of eigenvalues of the n×n Euclidean random matrix whose (i,j)th entry is of the form f(∥xi−xj∥2)−unδij∑kf(∥xi−xk∥2) for some real function f. Random points xi's are independently distributed in the lp ellipsoid or on its surface in RN including the unit sphere SN−1, the simplex, the ordinary ellipsoid and the hyper-cube. Here un is allowed to be

We present a characterization of eigenvalue inequalities between two Hermitian matrices by means of inertia indices. As applications, we deal with some classical eigenvalue inequalities for Hermitian matrices, including the Cauchy interlacing theorem and the Weyl inequality, in a simple and unified approach. We also give a common generalization of eigenvalue inequalities for (Hermitian) normalized

Let G be an r-uniform hypergraph of order n. For each p≥1, the p-spectral radius λ(p)(G) is defined as λ(p)(G):=max|x1|p+⋯+|xn|p=1r∑{i1,…,ir}∈E(G)xi1⋯xir. The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and λ(r)(G) is called the spectral radius of G. The α-normal labelling method, which was

We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms.

Let μ1 be a complex number in the numerical range W(A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W(A), we identify the smallest convex region containing all possible complex numbers μ2 for which μ1∗0μ2 is a 2-by-2 compression of A.

Let H be a separable complex Hilbert space with dim H ≥ 3, Bs(H) be the Lie algebra of all bounded self-adjoint operators on H, and let F:iBs(H)→[d,∞] with d≥ 0 be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on Bs(H) satisfying F(φ(A)φ(B)−φ(B)φ(A))=F(AB−BA) for all A,B∈Bs(H). As applications, we show that, for a surjective map φ on Bs(H)

In this paper, all connected graphs with three distinct signless Laplacian eigenvalues of which the second one is at most five are determined.

A Lie superalgebra H satisfying H2=Z(H) is called generalized Heisenberg Lie superalgebra. If d(H):=(m∣n) is the minimum number of generators required to describe H, then in this article we intend to find the structure of H when precisely dim⁡(H2)=12[(m+n)2+(n−m)]. Further, we give some results about the capability, and the Schur multiplier of H. Moreover, we find multiplier M(L) for any nilpotent

Let H and K be infinite dimensional separable Hilbert spaces. For A∈LR(H), B∈LR(K) and C∈LR(K,H), we denote by MC=AC0B the upper triangular relation matrix. In this paper, the sets ⋂C∈CM(K,H)σ(MC), ⋂C∈LR(K,H)σ1(MC) and ⋂C∈B(K,H)σ2(MC) are characterized, where σ1∈{σp,σr,σc} and σ2∈{σ,σp,σr,σc}. Moreover, the relationship between σ(MC), σ(A) and σ(B) is described for given relations A∈BCR(H), B∈BCR(K)

A signed graph is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph S is called sign-regular if d−(v) is same for all v∈V and d+(v) is same for all v∈V. The problems of embedding (i,j)-sign-regular signed graphs in (i+k,j+l)-sign-regular signed graphs is one of the fascinating problem from application point of view which is dealt in this paper with insertion of least number of vertices

Let L be a Lie superalgebra over a field of characteristic different from 2, 3 and write ID∗(L) for the Lie superalgebra consisting of superderivations mapping L to L2 and the central elements to zero. In this paper we first give an upper bound for the superdimension of ID∗(L) by means of linear vector space decompositions. Then we characterize the ID∗-superderivation superalgebras for the nilpotent

We give a complete classification of real and complex Lie superalgebras based on the Heisenberg Lie algebra h2n+1. This classification generalizes previous classifications.

Given disjoint non-empty subsets S and T of {1,…,n−1}, a digraph D with the vertex set {1,2,…,n} is called a directed Toeplitz graph provided the arc i→j occurs if and only if i−j∈S or j−i∈T. We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with n≥6 vertices has exponent at least 3 and for each n≥6, there is a primitive

In practice, the time variable cannot be negative. The square integrable function space L2(R+) defined on the half real line R+=(0,∞) models the causal signal space. Given a>1, this paper addresses Fa-frames in the setting of subspaces of L2(R+). We establish the connection between subspace Fa-frames and orthogonal projections; characterize all bounded linear operators on L2(Z×Ta) that transform Fa-frames