Paranormal operator on a Hilbert space

<p>In this thesis an extensive study is made of the set <i>P</i> of all paranormal operators in B(<i>H</i>), the set of all bounded endomorphisms on the complex Hilbert space <i>H</i>. T ϵ B(<i>H</i>) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. <i>P</i> contains the set <i>N</i> of normal operators and <i>P</i> contains the set of hyponormal operators. However, <i>P</i> is contained in <i>L</i>, the set of all T ϵ B(<i>H</i>) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, <i>N</i>≤<i>P</i>≤<i>L</i>.</p> <p>If the uniform operator (norm) topology is placed on B(<i>H</i>), then the relative topological properties of <i>N</i>, <i>P</i>, <i>L</i> can be discussed. In Section IV, it is shown that: 1) <i>N P </i> and <i>L</i> are arc-wise connected and closed, 2) <i>N, P,</i> and <i>L</i> are nowhere dense subsets of B(<i>H</i>) when dim <i>H</i> ≥ 2, 3) <i>N</i> = <i>P</i> when dim<i>H</i> ˂ ∞ , 4) <i>N</i> is a nowhere dense subset of <i>P</i> when dim<i>H</i> ˂ ∞ , 5) <i>P</i> is not a nowhere dense subset of <i>L</i> when dim<i>H</i> ˂ ∞ , and 6) it is not known if <i>P</i> is a nowhere dense subset of <i>L</i> when dim<i>H</i> ˂ ∞. </p> <p>The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ <i>P</i> to lie on a C²-smooth rectifiable Jordan curve G_o, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ G_o can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ <i>P</i> with σ(T) ≤ G_o, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ <i>P</i> with σ(T) ≤ G_o, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ G_o is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods. </p>