General Calculus of Pseudodifferential Operators

Abstract

Let H, H _j denote complex Hilbert spaces which are always assumed to be separable, B _ij = Hom(H _i , H _j ) denotes the space of bounded linear operators mapping H _i into H _j . The norm of A ∈ Hom(H _i , H _j ) is denoted by ‖A : H _i → Hj‖. The group of operators in H, with bounded inverses, is denoted by Aut H and the unit of the algebra B _ii = End H _i is denoted by EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa % aaleaacaWGibWaaSbaaWqaaiaadMgaaeqaaaWcbeaaaaa!38E1!]]</EquationSource><EquationSource Format="TEX"><![CDATA[ $$ {I_{{H_i}}} $$ or I _(i) (or I for short, if H _i is known from the context). If H _i = ℂ^m, we write I _m instead of I _(i).