Discussion Overview
The discussion revolves around the spectra of a bounded linear operator T and its adjoint T* in the context of Hilbert spaces. Participants explore the relationships between these spectra, particularly in finite and infinite dimensions, and consider specific cases such as normal and self-adjoint operators.
Discussion Character
- Debate/contested
- Technical explanation
Main Points Raised
- Some participants note that the spectra of T and T* are equal for finite rank operators.
- Others argue that the spectra are not necessarily related for arbitrary operators in infinite-dimensional Hilbert spaces.
- It is mentioned that if T is self-adjoint, then the spectra of T and T* are equal.
- A question is raised regarding whether the spectral radii of T and T* are the same for normal operators.
- One participant asserts that for normal elements in a C*-algebra, the spectral radius equals the norm, implying that T and T* would have the same spectral radius due to their equal norms.
Areas of Agreement / Disagreement
Participants express differing views on the relationship between the spectra of T and T* in infinite dimensions, with some asserting equality under specific conditions (self-adjoint and normal operators) while others maintain that no general relationship exists for arbitrary operators.
Contextual Notes
The discussion highlights limitations regarding the applicability of results to infinite-dimensional spaces and the specific conditions under which certain properties hold, such as normality and self-adjointness.