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# On linear operators

Let and suppose that , are linear operators from into satisfying

(1)

1. Show that for all one has

2. Show that there exists such that .

Solution

1. Using the assumptions we have

2. Consider the linear operator acting over all matrices . It may have at most different eigenvalues. Assuming that for every we get that has infinitely many different eigenvalues in view of (i). This is a contradiction.