Geometry of unitary orbits of pinching operators

CHIUMIENTO EDUARDO; M. DI IORIO Y LUCERO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2012

0022-247X

 Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let {  p_i }_1  ^w  (1leq w leq infty)  be   a family of mutually orthogonal projections on H.  The pinching operator associated with the former family of projections is given by   P: I ---> I,   P(x)=sum_{i=1}^{w} p_i x p_i. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I.  We study geometric properties of the orbitUI(P)={ L_u P L_u^*  :  u in  UI  }, where L_u is the left  representation of UI on the algebra B(I) of bounded operators acting on I. The results include necessary and sufficient conditions for UI(P) to be a submanifold of B(I). Special features arise in the case of  the ideal K of compact operators. In general, UK(P)   turns out to be a non complemented submanifold of B(K). We find a necessary and sufficient condition for UK(P) to  have complemented tangent spaces in B(K). We also show that UI(P) is a  covering space of another   orbit of pinching operators.