The higher order differential operators in direct sum spaces

Spectral Operators in a Direct Sum of Hilbert Spaces.

Let E C X be the finite-dimensional subspace spanned by xi, ..., xm. Now, for y E Y and variable x e E, y(x) is a continuous linear functional on E which we denote by y'. Thus, there is defined in a natural way a function f'(x) with domain (E n B) and range in E*, the conjugate space of E. The domain contains and surrounds densely every point of E n K(D), hence every point of K(E n D). It is ea...

Second Order Differential Operators on the Configuration Spaces

Γ = { γ ⊂ R ∣∣ |γΛ| < +∞ for any compact Λ ⊂ Rd}, where | · | means the cardinality of a set and γΛ = γ ∩ Λ. Let us define the σ-algebra B (Γ) as the minimal σ-algebra such that all mappings Γ γ −→ |γΛ| are B(Γ)-measurable for any Λ ∈ Bc(R), where Bc(R) is the family of all Borel subsets of Rd with compact closure. The space Γ can be naturally embedded into the space M(Rd) of all measures on Rd...

Local Hardy Spaces Associated with Inhomogeneous Higher Order Elliptic Operators

Let L be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this article, for all p ∈ (0, ∞) and L satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces hpL(R ) associated with L, which coincide with Goldberg’s local Hardy spaces h(R) for all p ∈ (0, ∞) when L ≡ −∆ (the Laplace operator). The authors al...

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

Spectrum of the Direct Sum of Operators

We study the connection between spectral properties of direct the sum of operators in the direct sum of Hilbert spaces and its coordinate operators.