Linear maps relating different unitary similarity orbits or different generalized numerical ranges Original Research Article

Let image be the complex linear space Mn of n × n complex matrices or the real linear space Hn of n × n hermitian matrices. For C set membership, variant image, its unitary similarity orbit is the set image and its circular unitary similarity orbit is the set image where image is the scalar field image or image according as image = Mn or image = Hn. Related to image(C) and image(C) are the C-numerical range and the C-numerical radius of A set membership, variant image defined by image and imagerC(A)=max{z:zset membership, variantWC(A)}, respectively. Let C, D set membership, variant Hn, we study the linear operators T on image satisfying one of the following properties: (I) WD(T(A)) = WC(A) for all A set membership, variant image, (II) rD(T(A)) = rC(A) for all A set membership, variant image, (III) T(image(D)) = U(C), (IV) T(image(D)) = image(C). In particular, we determine the conditions on C and D for the existence of a linear operator T on image satisfying any one of the conditions (I)–(IV), and characterize such an operator if it exists.