Block numerical range and estimable total decompositions of normal operators

Abstract

This paper deals with a conjecture posed by Abbas Salemi in 2011 (Banach J. Math. Anal.), claiming that for the spectrum of every bounded linear operator on a separable Hilbert space there exists an estimable total decomposition. We partially solve this problem, in the sense that, for the normal operator, there exists an estimable decomposition, under approximately unitary equivalence.

Keywords:

• Block numerical range
• estimable total decomposition
• spectrum
• normal operator

AMS Subject Classifications:

• Primary: 47A12
• Secondary: 49M27

Acknowledgements

The authors are grateful to the referees for valuable comments on this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by the National Nature Science Foundation of China [grant number 11371185], [grant number 11761029], [grant number 11362011].