Chalmers–Metcalf operator and uniqueness of minimal projections

We know that not all minimal projections in are unique (see [B. Shekhtman, L. Skrzypek, On the non-uniqueness of minimal projections in Lp spaces]). The aim of this paper is examine the connection of the Chalmers–Metcalf operator (introduced in [B.L. Chalmers, F.T. Metcalf, A characterization and equations for minimal projections and extensions, J. Oper. Theory 32 (1994) 31–46]) to the uniqueness of minimal projections. The main theorem of this paper is Theorem 2.2. It relates uniqueness of minimal projections to the invertibility of the Chalmers–Metcalf operator. It is worth mentioning that to a given minimal projection (even unique) we may find many different Chalmers–Metcalf operators, some of them invertible, some not—see Example 2.6. The main application is in Section 3, where we have proven that minimal projections onto symmetric subspaces in smooth Banach spaces are unique (Theorem 3.2). This leads (in Section 4) to the solution of the problem of uniqueness of classical Rademacher projections in Lp[0,1] for 1