PhD Thesis Defence

We begin by presenting a spectral characterization theorem that settles Chevreau's problem of characterizing the class of absolutely norming operators --- operators that attain their norm on every closed subspace. We next extend the concept of absolutely norming operators to several particular (symmetric) norms and characterize these sets. In particular, we single out three (family of) norms on $\mathcal B(\mathcal H, \mathcal K)$: the "Ky Fan $k$-norm(s)", "the weighted Ky Fan $\pi, k$-norm(s)", and the "$(p,k)$-singular norm(s)", and thereafter define and characterize the set of absolutely norming operators with respect to each of these three norms. 

We then restrict our attention to the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space $\mathcal H$ and use the theory of symmetrically normed ideals to extend the concept of norming and absolutely norming from the usual operator norm to arbitrary symmetric norms on $\mathcal B(\mathcal H)$. In addition, we exhibit the analysis of these concepts and present a constructive method to produce symmetric norm(s) on $\mathcal B(\mathcal H)$ with respect to each of which the identity operator does not attain its norm. 

 Finally, we introduce the notion of "universally symmetric norming operators" and "universally absolutely symmetric norming operators" and characterize these classes. These refer to the operators that are, respectively, norming and absolutely norming, with respect to every symmetric norm on $\mathcal B(\mathcal H)$. In effect, we show that an operator in $\mathcal B(\mathcal H)$ is universally symmetric norming if and only if it is universally absolutely symmetric norming, which in turn is possible if and only if it is compact. In particular, this result provides an alternative characterization theorem for compact operators on a separable Hilbert space.