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.
Denseness of operators which attain their numerical radius
We show that a bounded linear operator on a dual Banach space X may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius provided the weak star and norm topologies coincide on the unit sphere of X.
Analytic operator functions
Let H denote a complex Hilbert space. Let L(H) be the complex Banach space of all bounded linear operators on H, and L(H)*, the conjugate space of L(H). By an operator function f on a domain D, i.e., an opn connected subset of the complex plane, we mean that f(z) E L(H) for every z E D. An operator function f on a domain D is said to be analytic if &f(z)) is analytic on D in the classical sense for every a, E L(H)*. A,(D) will indicate the set of all analytic operator functions on D into L(H). It is well-known that most of the results of complex function theory, such as Cauchy’s integral theorem, Taylor expansion, Cauchy estemates, the maximum modulus principle and so on, are applicable to A,(D). We will appeal to them frequently. For references, we refer the reader to (71. In [5], K. Fan proves the following