### A characterization of operator order via grand Furuta inequality.

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We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${\left(T\u207f\right)}_{n=1,2,...}$ by the continuous semigroup ${\left({e}^{-t(I-T)}\right)}_{t\ge 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $supn\parallel T\u207f-{T}^{n+1}\parallel :n=1,2,...<\infty $. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

It is shown that if A is a bounded linear operator on a complex Hilbert space, then $w\left(A\right)\le 1/2\left(\right|\left|A\right||+|\left|A\xb2\right|{|}^{1/2})$, where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.

A spectral radius inequality is given. An application of this inequality to prove a numerical radius inequality that involves the generalized Aluthge transform is also provided. Our results improve earlier results by Kittaneh and Yamazaki.

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define $\lambda {\u2099}^{N}=sup\lambda \left(V\right):dim\left(V\right)=n,V\subset {l}_{\infty}^{\left(N\right)}$, λₙ = supλ(V): dim(V) = n. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ₂ = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented

It is proved that, for some reverse doubling weight functions, the related operator which appears in the Fefferman Stein's inequality can be taken smaller than those operators for which such an inequality is known to be true.

Let $L\left(H\right)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L\left(H\right)$, we define the elementary operator ${\Delta}_{A}:L\left(H\right)\u27f6L\left(H\right)$ by ${\Delta}_{A}\left(X\right)=AXA-X$. In this paper we study the class of operators $A\in L\left(H\right)$ which have the following property: $ATA=T$ implies $A{T}^{*}A={T}^{*}$ for all trace class operators $T\in {C}_{1}\left(H\right)$. Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of ${\Delta}_{A}$ is closed under taking...

This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras....