Denseness for norm attaining operator-valued functions
Abstract In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss' famous result on the denseness of norm attaining operators. Specifically, we show given any A∈ L ( H ) there is a sequence of rank-1 operators K n such that A+K n is norm attaining for each n and K n converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operator-valued functions.
On the numerical radius of a quaternionic normal operator
We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternionic compact normal operators.. Finally, we show that every quaternionic compact operator is norm attaining and prove the Lindenstrauss theorem on norm attaining operators, namely, the set of all norm attaining quaternionic operators is norm dense in the space of all bounded quaternionic operators defined between two quaternionic Hilbert spaces.
GEOMETRY OF THE UNIT SPHERE OF A C -ALGEBRA AND ITS DUAL
In this study we exploit as a main tool a polar decomposition for linear functionals on operator algebras, introduced in 1958 by Sakai, to determine various types of extremal behavior in the unit spheres of C*-algebras and their duals. We discuss exposed points and complex extreme points as well as extreme points. Throughout this paper M, F, and A will be generic symbols for a von Neumann algebra, its predual, and a C*-algebra which may not have a unit, respectively. By a C*-algebra is meant a Banach *-algebra in which ||#*a;|| = \\x\\2 holds for all x. A von Neumann algebra is a C*-algebra of operators on a Hubert space which is closed in the weak operator topology and contains the identity operator. Each von Neumann algebra M is equivalent (as a Banach space) to the dual of the Banach space F of ultra-weakly continuous ( = normal) linear functionals on M. The space F is called the predual of M (and is unique). References for the preceding facts, as well as any others to follow concerning C* -algebras and von Neumann algebras, are the two monographs of Dixmier [3], [4], and the lecture notes of Sakai [13]. We will denote by w, s, uw, us the weak operator, strong operator, ultra-weak and ultra-strong topologies of M respectively, and n refers to the norm topology of a Banach