Joint spectrum shrinking maps on projections
Wenhua Qian, Dandan Xiao, Tanghong Tao, Wenming Wu, Xin Yi

TL;DR
This paper characterizes maps on projections in finite-dimensional Hilbert spaces that shrink or preserve the joint spectrum, showing they are induced by unitary, anti-unitary, or ring automorphisms, depending on the context.
Contribution
It provides a complete characterization of joint spectrum shrinking maps on projections, linking them to spectrum-preserving maps and automorphisms in finite-dimensional Hilbert spaces.
Findings
Maps shrinking joint spectrum are characterized by spectrum-preserving properties.
Such maps are induced by unitary or anti-unitary operators.
Extension to Grassmann spaces relates spectrum preservation to automorphisms.
Abstract
Let be a finite dimensional complex Hilbert space with dimension and the set of projections on . Let be a surjective map. We show that shrinks the joint spectrum of any two projections if and only if it is joint spectrum preserving for any two projections and thus is induced by a ring automorphism on in a particular way. In addition, for an arbitrary , shrinks the joint spectrum of any projections 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 preserving for any rank one projections if and only if it can be extended to a surjective map on which is spectrum…
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Differential Equations and Dynamical Systems · Algebraic structures and combinatorial models
