A new class intermediate between hyponormal operators and normaloid operators.

Let σ n (H) denote the class of operators A acting on a complex Hilbert space H such that W 0 (A) = c o (λ ∈ σ (A) : λ = A ) , where W 0 (A) and σ (A) are the maximal numericl range, the spectrum of A, respectively, and co stands for the convex hull. In this paper, we show that the class σ n (H) contains hyponormal operators. We also show that if two operators A ∈ σ n (H) and B ∈ σ n (H ′) have the same closure of the numerical range or have the same spectrum, then they also have the same maximal numerical range. An affirmative answer to the question 2.16 in Fialkow [Elementary operators and applications. Proceedings of the International Workshop; Bleuberen; 1991. p. 55–113] is also given. [ABSTRACT FROM AUTHOR]