## 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| ≤ γ^{n}} ⊂ W(T^{n}) whenever n ≥ N

(ii) T^{n} 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 T^{n} by T^{n}D, 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(T^{n})), 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 T^{n} ∉ {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).

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | Mathematics; Engineering Science | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Physics, Mathematics and Astronomy | ||||||

Major Option: | Mathematics | ||||||

Minor Option: | Engineering | ||||||

Thesis Availability: | Public (worldwide access) | ||||||

Research Advisor(s): | - De Prima, Charles R.
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Thesis Committee: | - De Prima, Charles R. (chair)
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Defense Date: | 12 May 1975 | ||||||

Funders: |
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Record Number: | CaltechTHESIS:08242021-184834279 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:08242021-184834279 | ||||||

DOI: | 10.7907/344c-kv97 | ||||||

Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 14337 | ||||||

Collection: | CaltechTHESIS | ||||||

Deposited By: | Benjamin Perez | ||||||

Deposited On: | 25 Aug 2021 00:08 | ||||||

Last Modified: | 25 Aug 2021 00:08 |

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