Schmidt Decomposable Products of Projections

Schmidt Decomposable Products of Projections We characterize operators $$T=PQ$$ T = P Q (P, Q orthogonal projections in a Hilbert space $${\mathcal {H}}$$ H ) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $$\{\psi _n\}$$ { ψ n } of R(P) and $$\{\xi _n\}$$ { ξ n } of R(Q) such that $$\langle \xi _n,\psi _m\rangle =0$$ ⟨ ξ n , ψ m ⟩ = 0 if $$n\ne m$$ n ≠ m . Also it is shown that this is equivalent to $$A=P-Q$$ A = P - Q being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener–Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if $$T=PQ$$ T = P Q has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

Schmidt Decomposable Products of Projections

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2402-x
Publisher site
See Article on Publisher Site

Abstract

We characterize operators $$T=PQ$$ T = P Q (P, Q orthogonal projections in a Hilbert space $${\mathcal {H}}$$ H ) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $$\{\psi _n\}$$ { ψ n } of R(P) and $$\{\xi _n\}$$ { ξ n } of R(Q) such that $$\langle \xi _n,\psi _m\rangle =0$$ ⟨ ξ n , ψ m ⟩ = 0 if $$n\ne m$$ n ≠ m . Also it is shown that this is equivalent to $$A=P-Q$$ A = P - Q being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener–Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if $$T=PQ$$ T = P Q has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent.

Journal

Integral Equations and Operator TheorySpringer Journals

Published: Oct 11, 2017

References

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