An operator means a bounded linear operator on a complex Hilbert space. Using two polar decompositions of idempotent and 2-nilpotent operators, we shall study numerical radii of these two operators and finally we shall discuss two operator transformations, one of which is the generalized Aluthge transformation and another is an extension of the operator transformation by Patel–Tanahashi–Uchiyama. †This work is dedicated to Professor Masanori Fukamiya on his 95th birthday with respect and affection.
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Takayuki Furuta (2007) studied this question.