Subnormal operators

Abstract

Some of the natural questions about normal operators have the same answers for finite-dimensional spaces as for infinite-dimensional ones, and the techniques used to prove the answers are the same. Some questions, on the other hand, are properly infinite-dimensional, in the sense that for finite-dimensional spaces they are either meaningless or trivial; questions about shifts, or, more generally, questions about subnormal operators are likely to belong to this category (see Problem 154). Between these two extremes there are the questions for which the answers are invariant under change of dimension, but the techniques are not. Sometimes, to be sure, either the question or the answer must be reformulated in order to bring the finite and the infinite into harmony. As for the technique, experience shows that an infinite-dimensional proof can usually be adapted to the finite-dimensional case; to say that the techniques are different means that the natural finite-dimensional techniques are not generalizable to infinite-dimensional spaces. It should be added, however, that sometimes the finite and the infinite proofs are intrinsically different, so that neither can be adapted to yield the result of the other; a case in point is the statement that any two bases have the same cardinal number.