Eigenvalues and degree deviation in graphs

Let G be a graph with n vertices and m edges and let μ(G) = μ₁(G) ? ? ? μ_n(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑_u∈V(G)∣d(u)-2m/n∣. We prove that

     

Schatten p-norm inequalities related to an extended operator parallelogram law

Let C_p be the Schatten p-class for p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A₁,A₂,…,A_n} and B={B₁,B₂,…,B_n} are two sets of operators in, then C₂

     

On some isometries on the Bergman-Privalov class on the unit ball

Bergman-Privalov class AN_α(B) consists of all holomorphic functions on the unit ball B⊂Cⁿ such that

‖f‖_ANα:=∫_Bln(1+∣f(z)∣)dV_α(z)<∞,     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Numerical ranges of nilpotent operators

For any operator A on a Hilbert space, let W(A), w(A) and w₀(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ=0, then w(A)?(n-1)w₀(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w₀(A), (2) A is unitarily equivalent to an operator of the form aA_n⊕A^′, where a is a scalar satisfying |a|=2w₀(A), A_n is the n-by-n matrix

     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)

     

The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that

sup{∥U^*AU+V^*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the form

T(F)=∫log∫h(x,y)F(dx)F(dy)     

Converse theorem for the Minkowski inequality

Let (Ω,Σ,μ) a measure space such that 0<μ(A)<1<μ(B)<∞ for some A,B∈Σ. Under some natural conditions on the bijective functions φ,φ₁,φ₂,ψ,ψ₁,ψ₂:(0,∞)→(0,∞) we prove that if

     

A difference counterpart to a matrix Hölder inequality

We point out a sharp reverse Cauchy-Schwarz/Hölder matrix inequality. The Cauchy-Schwarz version involves the usual matrix geometric mean: Let A_i and B_i be positive definite matrices such that 0<mA_i?B_i?MA_i for some scalars 0<m?M and i=1,2,?,n. Then

     

Lower bounds of Copson type for Hausdorff matrices

Let 1 ? p ? ∞, 0 < q ? p, and A = (a_n,k)_n,k?0 ? 0. Denote by L_p,q(A) the supremum of those L satisfying the following inequality:

     

On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra M_n(F) of all n × n matrices with entries in a field F (i.e. a map Φ:M_n(F)→M_n(F) satisfying

Φ(ABA)=Φ(A)Φ(B)Φ(A)     

Maps completely preserving spectral functions

Let X,Y be infinite dimensional complex Banach spaces and A,B be standard operator algebras on X and Y, respectively. In this paper, we show that surjective maps completely preserving certain spectral function Δ(·) from A to B are isomorphisms, where Δ(·) stands for any one of 13 spectral functions σ(·), σ_l(·), σ_r(·), σ_l(·)∩σ_r(·), ∂σ(·), ησ(·), σ_p(·), σ_c(·), σ_p(·)∩σ_c(·), σ_p(·)∪σ_c(·), σ_ap(·), σ_s(·), and σ_ap(·)∩σ_s(·).     

A complement of the Ando-Hiai inequality

In this paper, we present a complement of a generalized Ando-Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando-Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541-545]. Let A and B be positive operators on a Hilbert space H such that 0<m₁?A?M₁ and 0<m₂?B?M₂ for some scalars m_i?M_i (i=1,2), and let α∈[0,1]. Put for i=1,2. Then for each 0<r?1 and s?1

     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R² in terms of the minimum number h₁(A,B) of parallel lines covering each of A and B. We show that, if h₁(A,B)?s and |A|?|B|?2s²−3s+2, then