Product of Operators and Numerical Range
نویسندگان
چکیده
We show that a bounded linear operator A ∈ B(H) is a multiple of a unitary operator if and only if AZ and ZA always have the same numerical radius or the same numerical range for all (rank one) Z ∈ B(H). More generally, for any bounded linear operators A,B ∈ B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z ∈ B(H) if and only if A = eB (resp., A = B) is a multiple of a unitary operator for some t ∈ [0, 2π). We extend the result to other types of generalized numerical ranges including the k-numerical range and the higher-rank numerical range.
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