Ellipticity in pseudodifferential algebras of Toeplitz type

Let L^\star be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T^\star be a subalgebra of operators of the form P_1AP_0, where P_0,P_1\in L^\star are projections, i.e., P_j^2=P_j. The elements of L^\star act as linear continuous operators in a scale of abstract Sobolev spaces, those of T^\star in the corresponding subspaces determined by the projections. We study how the ellipticity in L^\star descends to T^\star, focusing on parametrix construction, equivalence with the Fredholm property, characterisation in terms of invertibility of principal symbols, and spectral invariance. Applications concern SG-pseudodifferential operators, pseudodifferential operators on manifolds with conical singularities, and Boutet de Monvel's algebra for boundary value problems. In particular, we derive invertibilty of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvel's algebra. We indicate how the concept generalizes to parameter-dependent operators.