Joint spectrum shrinking maps on projections
Wenhua Qian, Dan an Xiao, Tanghong Tao, Wenm ng Wu, Xin Yi · Operators and Matrices · 2024
Let H be a finite dimensional complex Hilbert space with dimension n 3 and P(H ) the set of projections on H . Let : P(H ) P(H ) be a surjective map. We show that shrinks the joint spectrum of any two projections if and only if it is induced by a semilinear automorphism on H . In addition, shrinks the joint spectrum of I,P,Q for any two projections P,Q P(H ) if and only if it is induced by a unitary or an anti-unitary. Assume that is a surjective map on the Grassmann space of rank one projections. We show that is joint spectrum shrinking for any n rank one projections if and only if it is induced by a semilinear automorphism on H . Moreover, for any k > n , is joint spectrum shrinking for any k rank one projections if and only if it is induced by a unitary or an anti-unitary.