FAMILIES OF ANNIHILATING SKEW-SELFADJOINT OPERATORS AND THEIR CONNECTION TO HILBERT COMPLEXES

. In this short note we show that Hilbert complexes are strongly related to what we shall call annihilating sets of skew-selfadjoint operators. This provides for a new perspective on the classical topic of Hilbert complexes viewed as families of commuting normal operators.


Introduction
The classical differential geometry topic of "chain complexes" has entered functional analysis as the topic of so-called "Hilbert complexes".The purpose of this note is to inspect Hilbert complexes from another functional analytical perspective linked to a four decades old construction of the skew-selfadjoint extended Maxwell operator see [13,14].For some pre-history and the scope of this construction see [15].We merely mention here that the extended Maxwell system provides not only a deeper structural insight into the system of Maxwell's equations but also shows a deep connection to the Dirac equation.Indeed, the extended Maxwell operator has proven to be useful in important applications such as boundary integral equations in electrodynamics at low frequencies, see e.g.[5] and [16,17,18,19].
As it will turn out Hilbert complexes are intimately related to, indeed generalized by, an abstract concept, which we shall refer to as annihilating sets of skew-selfadjoint operators, which in turn is based on observations made in connection with the extended Maxwell system.This is the subject of the main part in Section 2. Our final section, Section 3 serves to illustrate the abstract setting by a number of more or less classical applications.For the rest of this section, let S be an annihilating set of skew-selfadjoint operators.
(1) S is a set of commuting operators.
(2) It holds T S = 0 on dom (S) for S = T .
(3) The observation that S = f S (Q) for some suitable complex valued functions f S : R → C in the sense of a function calculus associated with a single selfadjoint operator Q provides for other examples, which are not necessarily tridiagonal.
As a consequence we have a straight-forward application of the projection theorem the following generalized (orthogonal) Helmholtz decomposition.

2.2.
A Special Case: Tridiagonal Operator Matrices.We consider operator matrices of the form closed and densely defined on a Cartesian product and we have the following main result.
Proof.The result follows by a straightforward calculation.See Appendix A for more details.
Remark 6. Often a Hilbert complex (a) := (a 1 , . . ., a N ) is written as It is noteworthy that the Hilbert complex (a) is equivalently turned into the property or ran (S k ) ⊆ ker (S j ) for all j, k = 1, . . ., N, where the sequential character of Hilbert complexes seems to have disappeared.Theorem 4 suggests to consider annihilating sets of skew-selfadjoint operators as an appropriate generalization of Hilbert complexes.
Remark 7. Note that, if preferred, the set S may be considered as (1) a set of homomorphisms by restriction of the elements to their respective domains, i.e., S hom := S 1 , . . ., S N , where S k := S k ι dom(S k ) : dom(S k ) → H are now bounded linear operators, (2) a set of bounded isomorphisms by restriction of the elements to their respective domains and orthogonal complements of their kernels (and projections onto the ranges), i.e., where In the latter remark ι X denotes the embedding of the subspace X into H.If X is closed in H the orthonormal projector onto X is given by π Remark 8.For consistency we set a 0 := 0 and a N +1 := 0. Note that and that by the complex property In particular, the product of the cohomology groups . By replacing the skew-selfadjoint operators with selfadjoint operators the presented theory works literally as well.The only modifications are This is in a sense a matter of taste.We prefer, however, the skew-selfadjoint setting, since it has the advantage of being closer to various applications, such as Maxwell's and Dirac's equation (written in real form).We note, in particular, that skew-selfadjointness is at the heart of energy conservation.

Applications
In this final section we give several examples of annihilating sets of skew-selfadjoint operators, i.e., of Hilbert complexes, cf.Theorem 4. All operators will be considered as closures of unbounded linear operators densely defined on smooth and compactly supported test fields.For example, g rad, sym Curl T , and div Div S -where the tiny circle on top of an operator indicates the full Dirichlet boundary condition associated to the respective differential operator -are the closures of where Ω ⊂ R 3 is an open set, C∞ (Ω) denotes the space of smooth and compactly supported fields in Ω, and S and T indicate symmetric and deviatoric tensor fields, respectively.The corresponding adjoints − div, Curl S , and Grad grad are then given by 3.1.The Classical de Rham Complexes.

De Rham Complex of Vector Fields.
Let Ω be an open set in R 3 with boundary Γ := ∂Ω.
The most prominent example is the classical de Rahm complex of vector fields involving the classical operators of vector calculus grad, curl, and div with full Dirichlet or Neumann boundary conditions: Inhomogeneous and anisotropic coefficients and mixed boundary conditions can also be considered: Here the boundary Γ is decomposed into two parts Γ 0 and Γ 1 where the Dirichlet and Neumann boundary condition is imposed, respectively.Note that the de Rham off-diagonals are skew-adjoint to each other.
Again, the de Rham off-diagonals are skew-adjoint to each other.Note that S Dir and S Neu are unitarily congruent via transposition, permutation, sign change and Hodge- * -isomorphism.

Other Complexes in Three
Dimensions.There are plenty of extensions and restrictions of the de Rham complex.A nice overview and list of complexes is given in [1], from which we extract the following discussion.Their construction is based on the BGG-resolution using copies of the de Rham complex.For this section let Ω be an open set in R 3 .
• first Hessian complex: • second Hessian complex (formal dual of the first Hessian complex): • conformal Hessian complex (formally self-dual): In particular, we have for the Dirichlet de Rham complex Remark 14. Recalling Theorem 10 it has been shown in [2,3,7] that in case of the de Rham complexes the embeddings are compact, provided that (Ω, Γ 0 ) is a bounded weak Lipschitz pair, see [21,20,12,22] and [6,4] for the first results about the respective compact embeddings.For corresponding results in case of elasticity and biharmonic complexes and bounded strong Lipschitz pairs (Ω, Γ 0 ) see [8,9] and [10,11].
Remark 15.There is no doubt that all the latter complexes may be generalised to inhomogeneous and anisotropic coefficients and to mixed boundary conditions, cf.(3.1).Moreover, the techniques of [7,8,9] (for the de Rham, Kröner, and Hessian complexes) can be extended to show that all the embeddings dom(S

2 .
Annihilating Sets of Skew-Selfadjoint Operators and Hilbert Complexes 2.1.Finite Sets of Annihilating Skew-Selfadjoint Operators.We start with particular finite sets of commuting skew-selfadjoint operators S on a Hilbert space H, i.e., S : dom(S) ⊂ H → H, S * = −S, which we shall refer to as an (pair-wise) annihilating set of skew-selfadjoint operators.Definition 1.A finite set S of skew-selfadjoint operators satisfying ran (S) ⊆ ker (T ) , S = T, S, T ∈ S , is called an annihilating set of skew-selfadjoint operators.

3. 1 . 2 .
De Rham Complex of Differential Forms.Let Ω be an N -dinensional Riemannian manifold, e.g., an open set in R N .Another prominent example is the classical de Rahm complex of differential forms involving the exterior derivative d and its formal skew-adjoint the co-derivative δ = − d * , d = − δ * with full Dirichlet or Neumann boundary conditions: ran(S k ) are now bounded and bijective, (3) a set of topological isomorphisms S iso if all ranges ran(S k ) are closed.Note that in this case we have ran(S k ) = ker (S k ) ⊥H and that ran(S k ) is closed if and only if ran(a k ) is closed.
equals the kernel of S. Definition 9. Recall Remark 6.A Hilbert complex (a) is called (1) closed if all ranges ran(a k ) are closed.(2) compact if all embeddings dom(a k ) ∩ dom(a * k−1 ) ֒→ H k are compact.
(2)compact if and only if the embedding dom(S) ֒→ H is compact.Proof.Use Remark 8 and orthogonality.