Spectra of the spherical Aluthge transform, the linear pencil, and a commuting pair of operators

ABSTRACT

In this paper, we review and investigate some properties of a linear pencil in the frame of the point spectrum and an isolated point in its spectrum. We next give relationships among the spectrum of the linear pencil, the Taylor spectrums of a commuting pair and its spherical Aluthge transform through projection maps and the spectral mapping theorem.

KEYWORDS:

• Linear pencil of operators
• commuting pairs of operators
• spectral projection
• the functional calculus
• spherical Aluthge transform
• Taylor spectrum

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 15A18
• 47A10
• 47A60
• Secondary15A22
• 47A13
• 47A56

Acknowledgments

The authors wish to thank Professor Raul E. Curto for his many valuable suggestions and comments. The authors are deeply grateful to the referee for a careful reading of the paper and for several suggestions and edits that helped to improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second named author was supported by the National Research Foundation of Korea (NRF) funded by the Korea government (MSIT) (No. 2019R1F1A1047817) and 2019 (19-MS-7) research fund of the Korea Military Academy. This material is based upon work supported by a grant from the University of Texas System and the Consejo Nacional de Ciencia y Tecnología de México (CONACYT)