Numerical Ranges of Powers of Operators

Citation

Shiu, Elias Sai Wan (1975) Numerical Ranges of Powers of Operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/344c-kv97. https://resolver.caltech.edu/CaltechTHESIS:08242021-184834279

Abstract

We study the relations between a Hilbert space operator and the numerical ranges of its powers in this thesis.

Let β(ℋ) denote the set of bounded linear operators on a complex Hilbert space. For T ∈ β(ℋ), let σ(T) and W(T) denote its spectrum and numerical range, respectively. The following are proved using von Neumann's theory of spectral sets:

(i) σ(T) ⊂ (γ,∞) with γ > 0 and if T is not self-adjoint, then there is an index N such that {z ∈ ℂ : |z| ≤ γⁿ} ⊂ W(Tⁿ) whenever n ≥ N

(ii) Tⁿ is accretive, n = 1, 2, ..., k, if and only if the closed sector {z ∈ ℂ : |Arg z| ≤ π/2k} ⋃ {0} is spectral for T. In this case ∥ImTx∥ ≤ tan(π/2k) ∥ReTx∥ for each x ∈ ℋ.

(i) remains valid if we replace Tⁿ by TⁿD, where D is a surjective operator commuting with T. Various situations in which the commutativity assumption is relaxed are examined.

A theorem for finite dimensional matrices by C. R. Johnson is generalized to the operator case: If ∉ Cl(W(Tⁿ)), n = 1, 2, 3, ..., then an odd power of T is normal. Furthermore, if T is a convexoid, then T itself is normal; in fact, T is the direct sum of at most three rotated positive operators. Using these results, we prove: Let T ∈ β(ℋ), ℋ infinite dimensional and separable. If Tⁿ ∉ {Y ∈ β(ℋ) : Y = AX - XA, A,X ∈ β(ℋ), A positive}, n = 1, 2, 3, ..., then there is an odd integer m and a compact operator K_o such that T^m + K_o is normal. Moreover, T is a normal plus a compact if and only if ∩ {Cl(W(T + K)) : K compact} is a closed polygon (possibly degenerate).

Thesis Files