Decomposing numerical ranges along with spectral sets

Abstract

This note is to indicate the new sphere of applicability of the method developed by Mlak (W. Mlak, Partitions of spectral sets, Ann. Polon. Math. 25 (1971), pp. 281–288) as well as by the author. Restoring those ideas is summoned by current developments concerning K-spectral sets on numerical ranges.

Keywords:

• K-spectral set
• numerical range
• function algebra
• set of antisymmetry
• Gleason part; F. and M. Riesz theorem
• elementary spectral measure
• similarity
• orthogonal decomposition
• contraction; 𝒞_ρ operators

AMS Subject Classifications:

• primary 47A12
• 47A25
• secondary 47A60
• 46J10

Acknowledgements

This work was partially supported by the MNiSzW grant N201 026 32/1350. The author would like to acknowledge an assistance of the EU Sixth Framework Programme for the Transfer of Knowledge ‘Operator theory methods for differential equations’ (TODEQ) # MTKD-CT-2005-030042.

Notes

Notes

1. Pretty often it is known under the name of the maximal ideal space of the algebra.

2. From the monographs on function algebras, we recommend 9 as the most accessible one, also for missing here terminology; the source 19 has also to be mentioned.

3. A full description of this relation, in the general case of a function algebra, can be found, for instance in 19, Theorem 3.12], see 4 and 2 for the master results.

4. These results have passed unnoticed because, presumably, the people recognized in the area have not found it deserving of any attention; even in so authoritative monographs like 16 there is no mention of it though similarity is one of the leading topics therein; winner takes all!

5. Please notice a little deviation from the standard terminology.

6. Except that in 17, cf. Theorem 9, we ought to mention also (some of) other which come up: 6 as well as 7, 1 and 3.