written as projective limits of corresponding scales of e E e E e E e E

Boundary value problems 311 be the set of all ay, η ∈ C ∞ U × q , ✁

E, e E

such that there are elements a µ− j y, η ∈ S µ− j U × q \ 0; E, e E , j ∈ ✁ , with ay, η − N X j =0 χ η a µ− j y, η ∈ S µ− N +1 U × q for all N ∈ ✁ with χ being any excision function in η. Set σ ∂ ay, η := a µ y, η for the homogeneous principal symbol of ay, η of order µ. In the case U =  × ,  ⊆ q open, the variables in U will also be denoted by y, y ′ . Similarly to 2 we set L µ cl ; E, e E = n Op a : ay, y ′ , η ∈ S µ cl  ×  × q ; E, e E o , 17 where Op refers to the action in the y-variables on , while the values of amplitude functions are operators in ✁

E, e E

. For A ∈ L µ cl ; E, e E we set σ ∂ Ay, η = a µ y, y ′ , η| y ′ =y , called the homogeneous principal symbol of A of order µ. Every A ∈ L µ ; E, e E induces continuous operators A : s comp , E −→ s−µ loc , e E for each s ∈ . More details of this kind on the pseudo-differential calculus with operator-valued symbols may be found in [26], [30]. In particular, all elements of the theory have a reasonable generalization to Fr´echet spaces E and e

E, written as projective limits of corresponding scales of

Hilbert spaces, where the strong continuous actions are defined by extensions or restrictions to the Hilbert spaces of the respective scales [30], Section 1.3.1. This will tacitly be used below. Let us now pass to an analogue of the global pseudo-differential calculus of Section 2.3 with operator-valued symbols. Let S µ;δ q × q ; E, e E for µ, δ ∈ denote the space of all ay, η ∈ C ∞ q × q , ✁

E, e E

that satisfy the symbol estimates ˜κ −1 η n D α y D β η ay, η o κη ✂

E,e E

≤ chηi µ−|β| hyi δ−|α| for all α, β ∈ ✁ q , y, η ∈ 2q , with constants c = cα, β 0. This space is Fr´echet, and again, like for standard symbols, we have generalizations of the structures from the local spaces to the global ones. Further details are given in [30], [5], see also [31]. We now define operator-valued symbols that are classical both in η and y, where the group actions on E, e E are taken as the identities for all λ ∈ + when y is treated as a covariable. Similarly to the scalar case we set S µ η = ay, η ∈ C ∞ q × q \ 0 , ✁

E, e E

: ay, λη = λ µ ˜κ λ ay, ηκ −1 λ for all λ 0, y, η ∈ q × q \ 0 , S δ y = ay, η ∈ C ∞ q \ 0 × q , ✁

E, e E

: aλy, η = λ δ ay, η for all λ 0, y, η ∈ q \ 0 × q , and S µ;δ η; y = ay, η ∈ C ∞ q \ 0 × q \ 0 , ✁

E, e E

: aλy, τ η = λ δ τ µ ˜κ τ ay, ηκ −1 τ for all λ 0, τ 0, y, η ∈ q \ 0 × q \ 0 . 312 D. Kapanadze – B.-W. Schulze Moreover, let S µ;δ η; cl y defined to be the subspace of all ay, η ∈ S µ η such that ay, η| |η|=1 ∈ C ∞ S q−1 , S δ cl y q ; E, e E