The spectrum of seminormal operators

Area of Science:

• Functional Analysis
• Operator Theory
• Spectral Theory

Background:

• Bounded self-adjoint operators {U,V} on Hilbert space satisfying VU - UV = (1/pii)C, where C is trace class, are associated with a determining function.
• Previous work established obtaining the determining function via a Riemann-Hilbert problem and deriving spectral multiplicity theory from it.

Purpose of the Study:

• To investigate the relationship between the determining function and the spectrum of a seminormal operator.
• To characterize the spectrum of the seminormal operator T = U + iV under the condition that C is semidefinite.

Main Methods:

• Utilizing the concept of the determining function associated with pairs of bounded self-adjoint operators.
• Applying the Riemann-Hilbert problem framework to derive the determining function.
• Analyzing the spectral properties of the seminormal operator T = U + iV when the operator C is semidefinite.

Main Results:

• The determining function method provides a straightforward characterization of the spectrum for the seminormal operator T = U + iV.
• This characterization is achieved under the specific condition that the operator C is semidefinite.

Conclusions:

• The determining function method offers an effective tool for understanding the spectral properties of seminormal operators.
• The study simplifies spectral analysis for T = U + iV when C is semidefinite, linking it to the determining function.