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# Joint spectrum shrinking maps on projections

Dec 2022

Let $\mathcal H$ be a finite dimensional complex Hilbert space with dimension$n \ge 3$ and $\mathcal P(\mathcal H)$ the set of projections on $\mathcal H$.Let $\varphi: \mathcal P(\mathcal H) \to \mathcal P(\mathcal H)$ be asurjective map. We show that $\varphi$ shrinks the joint spectrum of any twoprojections if and only if it is joint spectrum preserving for any twoprojections and thus is induced by a ring automorphism on $\mathbb C$ in aparticular way. In addition, for an arbitrary $k \ge 3$, $\varphi$ shrinks thejoint spectrum of any $k$ projections if and only if it is induced by a unitaryor an anti-unitary. Assume that $\phi$ is a surjective map on the Grassmannspace of rank one projections. We show that $\phi$ is joint spectrum preservingfor any $n$ rank one projections if and only if it can be extended to asurjective map on $\mathcal P(\mathcal{H})$ which is spectrum preserving forany two projections. Moreover, for any $k >n$, $\phi$ is joint spectrumshrinking for any $k$ rank one projections if and only if it is induced by aunitary or an anti-unitary.

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