The numerical range of elementary operators

For n-tuplesA=(A ₁,...,A _n ) andB=(B ₁,...,B _n ) of operators on a Hilbert spaceH, letR _A,B denote the operator onL(H) defined by . In this paper we prove that

Norms and CB norms of Jordan elementary operators

We establish lower bounds for norms and CB-norms of elementary operators on . Our main result concerns the operator T_a,bx=axb+bxa and we show ‖T_a,b‖?‖a‖‖b‖, proving a conjecture of M. Mathieu. We also establish some other results and formulae for ‖T_a,b‖_cb and ‖T_a,b‖ for special cases.     

A remark on numerical range of semi-hyponormal operators

Let U be the unilateral shift on ?². For any complex numbers α and β, put T = αU + βU* and S = T ². Then we show that the operator S is convexoid.     

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.     

Norm equality for a basic elementary operator

Let be the algebra of bounded linear operators on a Hilbert space H. For , define the elementary operator M_A,B by M_A,B(X)=AXB (). We give necessary and sufficient conditions for any pair of operators A and B to satisfy the equation ‖I+M_A,B‖=1+‖A‖‖B‖, where I is the identity operator on H.     

Spectral properties and numerical range of off-diagonal infinite-dimensional Hamiltonian operators

The spectral properties of the off-diagonal infinite-dimensional Hamiltonian operator are studied in this article. Characterizations of the spectrum, the point spectrum and the residual spectrum are given. A spectral inclusion property of numerical range is proved.     

保不交算子值域的一些性质

研究了保不交算子值域的性质,建立了保不交算子值域为Riesz子空间的一个刻画;又讨论了主理想和主带在保不交算子作用后的象的性质,一些相关结果也得以讨论.     

Norms of elementary operators

Let and , , be bounded linear operators acting on a separable Hilbert space . In this note, we prove that Moreover, we prove that there exists an operator with such that if and only if there exists a unitary such that

     

The block numerical range of matrix polynomials

Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials.     

Orthogonality of the range and the kernel of some elementary operators

We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings , , where is the algebra of all bounded Hilbert space operators, and , , , are normal operators, such that , and . Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given.

     

Lucas' theorem and numerical range

The Lucas' theorem is generalized to the numerical ranges of some 3?×?3 companion matrices. We determine monic polynomials of degree 3 which assert the generalization. Examples are provided to show the generalization is restricted which gives a negative answer to the question raised by Zemanek [J. Zemanek, The derivative and linear algebra, Kolo Mlodych Matematykow, Ogolnopolskie Warsztaty, dla Mlodych Matematykow, Krakow, 2003, pp. 207–212.].     

Maps preserving numerical radius or cross norms of products of self-adjoint operators

Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.     

The polynomial numerical hull of a matrix and algorithms for computing the numerical range

Let A be an n × n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A.     

On the numerical range in tropical algebra

ABSTRACT

We introduce the concept of numerical range in tropical linear algebra, explore its properties and show in particular that it contains all the tropical eigenvalues. An explicit formula is obtained for the tropical numerical range and an application in the asymptotic analysis of condition number of symmetric matrices is proposed.