Schmidt decomposable products of projections

• Autores: Andruchow, Esteban; Corach, Gustavo
• Año de publicación: 2017
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Integral Equations and Operator Theory. Dic. 2017; 89(4): 557-580
• Materia: Projections; Products of projections; Differences of projections
• Institución: Universidad Nacional de General Sarmiento

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
Fil: Corach, Gustavo. Universidad Nacional de General Sarmiento; Instituto de Ciencias; Argentina.
Fil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina.
We characterize operators T = P Q (P, Q orthogonal projections in a Hilbert space H) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases {ψn} of R(P) and {ξn} of R(Q) such that ξn, ψm = 0 if n = m. Also it is shown that this is equivalent to 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 = P Q has a singular value decomposition, then the generic parts of P and Q are joined by a minimal geodesic with diagonalizable exponent.