Eigenvalues of seminormal operators, examples
Richard Carey, Joel David Pincus · Pacific Journal of Mathematics · 1974
If T* is completely hyponormal and [T, T*] has one dimensional range, a necessary and sufficient condition for a point z to belong to the point spectrum of T is known.Using this criterion two examples are constructed.In the first example the point spectrum of T is empty, in the second example the spectrum of T is nowhere dense but almost every point of it is an eigenvalue.The construction of both examples uses results about trigonometric series and the so-called principal function map T ?l<εJ JO JO Thus, \ Xoipe™ + z)ρdρ = 0 a.a.θ , Jo and thus άpe™ + z)dρ = 0 a.a.θ .Also setting φ = 0 -π, we have Γ d ^ Cz^'^"^ + φd/o -0