An algebra of operators combining interior pseudodifferential operators with boundary operators (trace, Poisson, singular Green parts) that encodes classical elliptic boundary problems on manifolds with boundary and their parametrices.

Principle

Form a closed algebra under composition modulo smoothing operators by adjoining boundary operators to the pseudodifferential calculus so that boundary-value problems admit parametrices inside the algebra and Fredholm theory can be carried out microlocally.

Demonstration

The Dirichlet problem for a classical elliptic operator on a compact manifold with boundary can be represented by a Boutet de Monvel block-operator; constructing an inverse modulo smoothing in the algebra yields a parametrix and shows Fredholmness on Sobolev spaces.

Misapplication

Expecting the algebra to handle arbitrary nonlocal or highly singular boundary conditions, manifolds with corners, or operators lacking the transmission property will fail: the algebra assumes specific mapping types and symbol behaviour at the boundary.

Consequence

When applicable, one obtains explicit parametrices, precise mapping properties of boundary-value operators, index formulas, and a microlocal framework to study elliptic boundary problems and regularity up to the boundary.

Reversal

If one restricts to the interior pseudodifferential calculus only, boundary conditions are external and parametrices may not exist within that calculus; conversely, using Boutet de Monvel where transmission fails yields operators outside the algebraic closure.

Boundary

Designed for classical pseudodifferential operators on smooth manifolds with smooth boundary, with boundary operators of trace, Poisson and Green type and with symbols satisfying compatibility (transmission) conditions; excludes corners, nonclassical singular symbols, and many nonlocal boundary rules without extension.

Semantic Tension

There is tension between Boutet de Monvel algebra and calculi that focus on rescaled tangential behaviour (b-calculus) or on scattering asymptotics; additionally the algebra relies on the transmission property, which is not automatic for all pseudodifferential operators.

Synthesis

The Boutet de Monvel Algebra packages interior pseudodifferential operators together with boundary traces, Poisson and Green components into a compositional algebra that realizes classical elliptic boundary problems, producing parametrices, Fredholm statements and microlocal control up to the boundary.