Spectral radius inequalities for Hilbert space operators

We prove several spectral radius inequalities for sums, products, and commutators of Hilbert space operators. Pinching inequalities for the spectral radius are also obtained.

     

Norm equalities and inequalities for operator matrices

Several norm equalities and inequalities for operator matrices are proved in this paper. These results, which depend on the structure of circulant and skew circulant operator matrices, include pinching type inequalities for weakly unitarily invariant norms.     

The numerical radius and bounds for zeros of a polynomial

Let be a monic polynomial. We obtain two bounds for zeros of via the Perron root and the numerical radius of the companion matrix of the polynomial.

     

Bounds and a majorization for the real parts of the zeros of polynomials

We apply some eigenvalue inequalities to the real parts of the Frobenius companion matrices of monic polynomials to establish new bounds and a majorization for the real parts of the zeros of these polynomials.

     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Refinements of the Heinz operator inequalities

In this article, we study the Heinz inequalities for two positive operators. We refine the ordering relations among the Heinz means with different parameters and obtain some improvements of the Heinz operator inequalities.     

Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

Quadratic operator inequalities and linear-fractional relations

We study properties of solution sets of inequalities of the form

$X^* AX + B^* X + X^* B + C \leqslant 0,$

, where A, B, and C are bounded Hilbert space operators and A and C are self-adjoint. The following properties are considered: closedness and inferior points in Standard operator topologies, convexity, and nonemptiness.

     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

无界分块算子矩阵的谱分布

本文首先给出次对角元有界的2×2阶无界算子矩阵的Gershgorin定理,然后利用主对角元算子的谱和数值域刻画整个算子矩阵的谱分布.特别地,当次对角元算子互为共轭(反共轭)算子时,结合二次数值域和Gershgorin定理对谱分布给出更精细的描述.     

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.     

The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.

     

Improved Young and Heinz inequalities for matrices

We give refinements of the classical Young inequality for positive real numbers and we use these refinements to establish improved Young and Heinz inequalities for matrices.     

Numerical radius for the asymptotic stability of delay differential equations

In this note, we make use of the numerical radius to investigate the asymptotic stability of linear systems of delay differential equations. We present examples to illustrate our assumption is weaker than the classical ones.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

一类缺项算子矩阵正补的谱刻画

对 于 具 有 离 散 谱 的 正 算 子 Ａ, Ｂ, 我 们 给 出 了 一 个 闭 集 成 为 ２ × ２ 缺 项 算 子 矩 阵 Ａ ？？ Ｂ 的某个正 补的谱的一些 判别条件