A particular infinite matrix and related norms

Abstract

We investigate norms of vectors and operators obtained from the infinite scalar matrix where thereby improving and complementing results of [5]. For any choice of and , U gives rise to a bounded linear operator which enjoys compactness properties close to nuclearity. Whereas we cannot characterize for which (p, q) the operator is nuclear, we will show that frequently this is the case, and that for certain Banach operator ideals, the corresponding ideal norms for and/or related operators coincide with the usual operator norm. On the other hand for example, is easily seen to be a Hilbert-Schmidt operator, but neither its norm nor its Hilbert-Schmidt norm are explicitly known. However, at least for the Hilbert-Schmidt norm, reasonably efficient approximation schemes are available.