Ellipticity in pseudodifferential algebras of Toeplitz type

Let L⋆ be a filtered algebra of abstract pseudodifferential operators equipped with a notion of ellipticity, and T⋆ be a subalgebra of operators of the form P1AP0, where P0,P1∈L⋆ are projections, i.e., . The elements of L⋆ act as linear continuous operators in a scale of abstract Sobolev spaces, those of T⋆ in the corresponding subspaces determined by the projections. We study how the ellipticity in L⋆ descends to T⋆, 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, operators on manifolds with conical singularities, and Boutet de Monvelʼs algebra for boundary value problems. In particular, we derive invertibility of the Stokes operator with Dirichlet boundary conditions in a subalgebra of Boutet de Monvelʼs algebra. We indicate how the concept generalizes to parameter-dependent operators.