{'fraction_non_alphanumeric': 0.07230769230769231, 'fraction_numerical': 0.019130434782608695, 'mean_word_length': 4.049991555480493, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The De Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in multidimension and its applications. Part 1

Spectral properties od Delsarte transmutation operators are studied, their differential geometrical and topological structure in multidimension is analyzed, the relationships with De Rham-Hodge-Skrypnik theory of generalized differential complexes is stated.


Spectral operators and generalized eigenfunctions expansions
We shall use below the following definition motivated by considerations from [8], Chapter 5.
Definition 1.1. An operator L ∈ L(H) will be called spectral if for all Borel subsets ∆ ⊂ σ(L) of the spectrum σ(L) ⊂ C and for all pairs (u, v) ∈ H + × H + there are defined the following expressions: where ρ σ is some finite Borel measure on the spectrum σ(L), E is some self-adjoint projection operator measure on the spectrum σ(L), such that E(∆)E(∆ ′ ) = E(∆ ∩ ∆ ′ ) for any Borel subsets ∆, ∆ ′ ⊂ σ(L), and P(λ) : H + → H − , λ ∈ σ(L), is the corresponding family of nucleous integral operators from H + into H − .
As a consequence of the expression (1.2) one can write down that formally in the weak topology of H for any Borel subset ∆ ⊂ σ(L).
Similarly to (1.2) and (1.3) can write down the corresponding expressions for the adjoint spectral operator L * ∈ L(H) whose domain D(L * ) ⊂ H is assumed to be also dense in H : , where E * is the corresponding projection spectral measure on Borel subsets ∆ ∈ σ(L * ), P * (λ) : H → H, λ ∈ σ(L * ), is the corresponding family of nucleous integral operators in H and ρ * σ is some finite Borel measure on the spectrum σ(L * ). We will assume, moreover, that the following conditions (1.5) P(µ)(L − µI)v = 0, P * (λ)(L * −λI)u = 0 hold for all u ∈ D(L * ), v ∈ D(L), whereλ ∈ σ(L * ), µ ∈ σ(L). In particular, one assumes also that σ(L * ) =σ(L). 1.2. Proceed now to a description of the corresponding to operators L and L * generalized eigenfunctions via the approach devised in [8]. We shall speak that an operator L ∈ L(H) with a dense domain D(L) allows a rigging continuation, if one can find another dense in H + topological subspace D + (L * ) ⊂ D(L * ), such that the adjoint operator L * ∈ L(H) maps it continuously into H + . It is evident that in the case when ψ λ ∈ D(L), λ ∈ σ(L), then Lψ λ = λψ λ as usually. The definition (1.6) is related [8] and is assumed also that the unity operator I : H − → H − ⊂ D − is extended naturally as the imbedding operator from H − into D − . Then equalities (2.2) can be equivalently written down [8] as follows: for any λ ∈ σ(L) ∩σ(L * ). Take now a kernelK λ ∈ H − ⊗ H − and suppose that the following operator equality (2.5) (L ext ⊗ I)K λ = (I ⊗ L * ext )K λ holds. Since the equation (2.4) can be written down in the form (2.6) (L ext ⊗ I)Ẑ λ = (I ⊗ L * ext )Ẑ λ for any λ ∈ σ(L) ∩σ(L * ), the following characteristic theorem [8] holds. Theorem 2.3. (see [8], chapter 8, p.621) Let a kernelK ∈ H − ⊗ H − satisfy the condition (2.5). Then due to (2.6) there exists such a finite Borel measure defined on Borel subsets ∆ ⊂ σ(L) ∩σ(L * ), that the following weak spectral representation holds. Moreover, due to (2.4) one can write down the following representation Proof. ⊳ It is easy to see due to (2.6) that the kernel (2.7) satisfies the equation (2.5). On one hand-side, consider the second expression of (1.2) related with our operator L ∈ L(H) and observe that it can be represented exactly in the form (2.1): for any u, v ∈ H + due to the Schwartz theorem 2.1. On another hand-side due to the condition (2.5) by means of the kernel K ∈ H − ⊗ H − one can define as in [8] a new Hilbert space H K ⊃ H + with the scalar product for any u, v ∈ H + , where, by definition, |K|:= K * K . As the norm (2.10) ||u|| K = (|K|, u ⊗ u) H×H ≤ ||K|| − · ||u ⊗ u|| + = ||K|| − ||u|| 2 + for any u ∈ H + , then one can deduce from (2.10) that really H K ⊃ H + . Thus one has a new Hilbert-Schmidt rigged chain with the basic Hilbert space taken now to be H K : where embeddings H ++ → H K is also quasi-nucleous [8] as the composition of the quasi-nucleous imbedding H ++ → H + and the continuous imbedding H + → H K due to (2.10). Since now the operator L ∈ L(H) can be considered as an operator L ∈ L(H K ), there exists a representation similar to (2.8) but just for u, v ∈ H ++ . Thereby for the expression (2.9) one derives from (2.4) the searched for all (u, v) ∈ H + × H + representation : equivalent, obviously, to (2.7). The integral (2.7) is defined well since the norm || Z λ || − − < ∞ and the measure ρ σ is finite due to the construction ⊲ Definition 2.4. A kernelK ∈ H − × H − satisfying the conditions (2.5) will be called self-similar congruent with respect to a given operator L ∈ L(H).
The construction done above for a self-similar congruent kernelsK ∈ H − ⊗ H − in the form (2.7) subject to a given operator L(H) appears to be very inspiring if the condition self-similarity to make changed by a simple similarity. This topic will be discussed below.
3. Congruent kernel operators, related Delsarte transmutation mappings and their structure 3.1. Consider in a Hilbert space H a pair of densely defined linear differential operators L andL ∈ L(H). The following definition will be useful. (L ext ⊗ 1)K s = (1 ⊗ L * ext )K s for a given pair of the correspondingly extended linear operators L,L ∈ L(H). Then the kernelsK s ∈ H − ⊗ H − , s = ±, will be called congruent to this pair (L,L) of operators in H.
Since not any pair of operators L,L ∈ L(H) can be congruent, the natural problem arises if they exist: how to describe the set of corresponding kernelsK s ∈ H − ⊗ H − , s = ±, congruent to a given pair (L,L) of operators in H. The first question being important for further is that of existence of kernelsK s ∈ H − ⊗ H − , s = ±, congruent to this pair. The question has an evident answer for the case wheñ L = L and the congruence is then self-similar. The interesting case whenL = L appears to be very nontrivial and can be treated more or less successfully if there exist such bounded and invertible operators Ω s ∈ H, s = ±, that the transmutation conditions Definition 3.2. (Delsarte, Lions; [1,2]) Let a pair of densely defined differential closeable operators L,L ∈ L(H) in a Hilbert space H is endowed with a pair of closed subspaces H 0 ,H 0 ⊂ H − subject to a rigged Hilbert spaces chain (1.1). Then invertible operators Ω s ∈ Aut(H) ∩ B(H), s = ±, are called Delsarte transmutations if the following conditions hold: i) the operator Ω s and its inverse Ω −1 s , s = ±, are continuous in H, that is Ω s ∈ Aut(H) ∩ B(H), s = ±; ii) the images Im Ω s | H0 =H 0 , s = ±; iii) the relationships (3.2) are satisfied.
Suppose now that an operator pair (L,L) ⊂ L(H) is differential of the same order n(L) ∈ Z + , that is the following representations is some open connected region in R m , the coefficients a α ,ã α ∈ S(Q; End C N ) for all α ∈ Z m + , |α| = 0, n(L) and N ∈ Z + . The differential expressions s , s = ±, which, by definition, must persist to be also differential. An additional natural constraint involved on operators Ω s ∈ Aut(H) ∩ B(H), s = ±, is the independence [5,12] of differential expressions for operators (3.4) on indices s = ±. The problem of constructing such Delsarte transmutation operators Ω s ∈ Aut(H) ∩ B(H), s = ±, appeared to be very complicated and in the same time dramatic as it one could observe from special results obtained in [5,17] for two-dimensional Dirac and three-dimensional Laplace type operators.
3.2. Before proceeding to setting up our approach to treating the problem mentioned above, let us consider some formal generalizations of the results described before in Chapter 2. Take an elementary kernel Z λ ∈ H − ⊗ H − satisfying the conditions generalizing (2.4): , being, evidently, well suitable for treating the equation (3.1). Then one sees that an elementary kernel Z λ ∈ H − ⊗ H − for any λ ∈ σ( L) ∩ σ( L * ) solves the equation (3.1), that is (3.6) ( L ext ⊗ 1) Z λ = (1 ⊗ L * ext ) Z λ . Thereby one can expect that for kernelsK s ∈ H − ⊗ H − , s = ±, there exist the similar to (2.9) spectral representations (3.7)K s = σ( L)∩σ(L * ) Z λ dρ σ,s (λ), s = ±, with finite spectral measures ρ σ,s , s = ±, localized upon the Borel subsets of the common spectrum σ(L) ∩σ(L * ). Based of the spectral representation like (2.8) applied separately to operatorsL ∈ L(H) and L * ∈ L(H) one states similarly as before the following theorem.  Now we will be interested in the inverse problem of constructing kernelsK s ∈ H − ⊗ H − , s = ±, like (3.7) a priori satisfying the congruence conditions (3.1) subject to the same pair (L,L) of differential operators in H and related via the Delsarte transmutation condition (3.2). In some sense we shall state that only for such Delsarte related operator pairs (L,L) in H one can construct a dual pair {K s ∈ H − ⊗ H − : s = ±} of the corresponding congruent kernels satisfying the conditions like (3.1), that is 3. Suppose now that there exists another pair of Delsarte transmutation operators Ω s and Ω ⊛ s ∈ Aut(H) ∩B(H), s = ±, satisfying condition ii) of Definition 3.2 subject to the corresponding two pairs of differential operators (L,L) and (L * ,L * ) ⊂ L(H). This means, in particular, that there exists an additional pair of closed subspace Im for the Delsarte transmutation operator Ω ⊛ s ∈ Aut(H) ∩ B(H), s = ±, satisfying the obvious conditions (3.10)L * · Ω ⊛ s = Ω ⊛ s · L * s = ±, involving the adjoint operatorsL * , L * ∈ L(H) defined before and given by the following from (3.3) usual differential expressions: Construct now the following [18,19] Delsarte transmutation operators of Volterra type (3.12) Ω ± := 1 + K ± (Ω), corresponding to some two different kernelsK + andK − ∈ H − ⊗ H − , of integral Volterrian operators K + (Ω) and K − (Ω) related with them in the following way: x,± ) are some characteristic functions of two mdimensional smooth hypersurfaces S x,− ) = ∂Q. In the case when Q := R m , it is assumed naturally that ∂R m = ⊘. Making use of the Delsarte operators (3.12) and relationship like (3.2) one can construct the following differential operator expressions: Since the left-hand sides of (3.14) are, by definition, purely differential expressions, one follows right away that the local kernel relationships like (3.7) hold: (3.15) (L ext,± ⊗ 1)K ± = (1 ⊗ L * ext )K ± . The expressions (3.14) define, in general, two different differential expressionsL ± ∈ L(H) depending correspondingly both on the kernelsK ± ∈ H − ⊗ H − and on the chosen hypersurfaces S (m) x,± ∈ K(Q). As will be stated later, the following important theorem holds.    3.17) [Ω −1 + Ω − , L] = 0 or, equivalently, kernel relationship Remark 3.5. It is a place to notice here that special degenerate cases of theorem 3.4 where before proved in works [5,17] for two-dimensional Dirac and threedimensional Laplace type differential operators. The constructions and tools devised there appeared to be instructive and motivative for the approach developed here by us in the general case.   Ω = Ω −1 + Ω − by means of two Delsarte operators Ω + and Ω − ∈ Aut(H) ∩ B(H) in the form (3.12). Then there exists the unique operator kernelΦ ∈ H − ⊗ H − corresponding naturally to the compact operator Φ(Ω) ∈ B ∞ (H) and satisfying the following selfsimilar congruence commutation condition: From the equality (3.20) and Theorem 2.2 one gets easily the following corollary.
Corollary 3.7. There exists such a finite Borel measure ρ σ defined on the Borel subsets of σ(L) ∩ σ(L * ), that the following weak equality holds.
Really, since equalities (3.16) hold, one gets easily that Suppose also that, first, for another Fredholm operator [18] integral operators K ⊛ ± (Ω) related naturally with some kernelŝ K ± ∈ H − ⊗ H − , and, second, the factorization condition is satisfied, then the following theorem holds. x,± ⊂ K(Q) satisfy all of the conditions from Theorem 3.2. Then the Delsarte transformed operatorsL * ± ∈ L(H) are differential and equal, that is Proof. ⊳ A proof of this theorem is stated by reasonings similar to those done before when analyzing the congruence condition for a given pair (L,L) ⊂ L(H) of differential operators and their adjoint ones in H. ⊲ By means of the Delsarte transmutation from the differential operators L and L * ∈ L(H) we have obtained above two differential operators which must be compatible and, thereby, related as The condition (3.29) due to (3.28) gives rise to the following additional commutation expressions for kernels Ω ⊛ ± and Ω * ± ∈ Aut(H) ∩ B(H) : (3.30) [L * , Ω * ± Ω ⊛ ± ] = 0, being equivalent, obviously, to such a commutation relationship: (3.31) [L, Ω ⊛, * ± Ω ± ] = 0. As a result of representations (3.31) one can formulate the following corollary. Corollary 3.9. There exist finite Borel measures ρ σ,± localized upon the common spectrum σ(L) ∩σ(L * ), such that the following weak kernel representations hold, whereΩ ⊛, * ± andΩ ± ∈ H − ⊗ H − are the corresponding kernels of integral Volterrian operators Ω ⊛, * ± and Ω ± ∈ Aut(H) ∩ B(H).
3.5. The integral operators of Volterra type (3.12) constructed above by means of kernels in the form (3.7) are, as well known [13,7,5,11,10], very important for studying many problems of spectral analysis and related integrable nonlinear dynamical systems [12,11,6,17,25] on functional manifolds. In particular, they serve as factorizing operators for a class of Fredholm operators entering the fundamental Gelfand -Levitan -Marchenko operator equations [13,11,12] whose solutions are exactly kernels of Delsarte transmutation operators of Volterra type, related with the corresponding congruent kernels subject to given pairs of closeable differential operators in a Hilbert space H. Thereby it is natural to try to learn more of their structure properties subject to their representations both in the form (3.7), (3.12), and in the dual form within the general Gokhberg -Krein theory [18,12,19] of Volterra type operators.
To proceed further with we need to introduce some additional notions and definitions from [18,10] important for what will follow below. Define a set P of projectors P 2 = P : H → H which is called a projector chain if for any pair P 1 , P 2 ∈ P, P 1 = P 2 , one has either P 1 < P 2 or P 2 < P 1 , and P 1 P 2 = min(P 1 , P 2 ). The ordering P 1 < P 2 above means, as usually, that P 1 H ⊂ P 2 H, P 1 H = P 2 H. If P 1 H ⊂ P 2 H, then one writes down that P 1 ≤ P 2 . The closure P of a chain P means, by definition, that set of all operators being weak limits of sequences from P. The inclusion relationship P 1 ⊂ P 2 of any two sets of projector chains possesses obviously the transitivity property allowing to consider the set of all projector chains as a partly ordered set. A chain P is called maximal if it can not be extended. It is evident that a maximal chain is closed and contains zero 0 ∈ P and unity 1 ∈ P operators. A pair of projectors (P − , P + ) ⊂ P is called a break of the chain P if P − < P + and for all P ∈ P either P < P − or P + < P. A closed chain is called continuous if for any pair of projectors P 1 , P 2 ⊂ P there exist a projector P ∈ P, such that P 1 < P < P 2 . A maximal chain P will be called complete if it is continuous. A strongly ascending with to inclusion projector valued function P : Q ∋ ∆ → P is called a parametrization of a chain P, if the chain P = Im(P) such a parametrization of the self-adjoint chain P is called smooth, if for any u ∈ H the positive value measure ∆ → (u, P(∆)u) is absolutely continuous. It is well known [18,19,12] that very complete projector chain allows a smooth parametrization. In what will follow a projector chain P will be self-adjoint, complete and endowed with a fixed smooth parametrization with respect to an operator valued function F : P → B(H) the expressions like P F(P)dP and P dPF(P) will be used for the corresponding [18] Riemann-Stiltjes integrals subject to the corresponding projector chain. Take now a linear compact operator K ∈B ∞ (H) acting in a separable Hilvert space H endowed with a projector chain P. A chain P is also called proper subject to an operator K ∈ B ∞ (H) if PKP = KP for any projector P ∈ P, meaning obviously that subspace PH is invariant with respect to the operator K = B ∞ (H) for any P ∈ P. As before the note by σ(K(Ω)). The spectrum of any operator K ∈ L(H).
As it can be shown [18], a Volterrian operator K ∈ B ∞ (H) possesses the maximal proper projector chain P such, that for any its break (P − , P − ) the following relationship holds. Since integral operators (3.12) constructed before are of Volterra type and congruent to a pair (L, L) of closeable differential operators in H, we will be now interested in their properties with respect both to the definition given above and to the corresponding proper maximal projector chains P(Ω). 3.6. Suppose now that we are given a Fredholm operator Ω ∈ B(H)∩Aut(H) self-congruent to a closeable differential operator L ∈ L(H). As we are also given with an elementary kernel (2.6) in the spectral form (2.7), our present task will be a description of elementary kernelsẐ λ , λ ∈ σ(L) ∩σ(L * ), by means of some smooth and complete parametrization suitable for them. For treating this problem we will make use of very interesting recent results obtained in [19] and devoted to the factorization problem of Fredholm operators. As a partial case this work contains some aspects of our factorization problem for Delsarte transmutation operators Ω ∈ Aut(H) ∩ B(H) in the form (3.12).
Let us formulate now some preliminary results from [18,19] suitable for the problem under regard. As before, we will the note by B ( Denote also by P + (P − ) the corresponding projectors of the linear space , and call them after [18] by transformators of a triangle shear. The transformators P + and P − are known [18] to be continuous operators in ideals B p (H), p ∈ [1, ∞]. From definitions above one gets that Remark 3.11. It is clear and important that transformators P + and P − strongly depend on a fixed projector chain P.
Put now, by definition, Consider also the following two operator sets: which are characterized by the following (see [18,19]) theorem.
iii) for any Φ ∈ W f it is necessary and sufficient that at least one of integrals is convergent in the uniform operator topology, and, moreover, if the one integral of (3.24) is convergent then the another one is convergent too; iiii) the factorization representation for Φ ∈ W f is satisfied.
The theorem above is still abstract since it doesn't take into account the crucial relationship (3.22) relating the operators representation (3.42) with a given differential operator L ∈ L(H). Thus, it is necessary to satisfy the condition (3.22). If this condition is due to (3.1) and (3.16) satisfied, the following crucial equalities in H and the corresponding congruence relationships Here byK ± ∈ H − ⊗ H − we denoted the corresponding kernels of Volterra operators K ± (Ω) ∈ B ± ∞ (H).Since the factorization (3.42) is unique, the corresponding kernels must a priori satisfy the conditions (3.43) and (3.44). Thereby the self-similar congruence condition must be solved with respect to a kernelΦ ∈ H − ⊗ H − corresponding to the integral operator Φ ∈ B ∞ (H), and next, must be found the corresponding unique factorization (3.42), satisfying a priori condition (3.43) and (3.44).

3.7
To realize this scheme define preliminarily a unique positive Borel finite measure on the Borel subsets ∆ ⊂ Q of the open set Q ⊂ R m , satisfying for any projector P x ∈ P x of a chain P x , marked by a running point x ∈ Q, the following condition for any x ∈ Q a measurable with respect to some Borel measure µ Px on Borel subsets of Q operator-valued mapping of Hilbert-Schmidt type. The representation (3.45) follows due the reasoning similar to that in [8], based on the standard Radon-Nikodym theorem [8,32]. This means in particular, that in the weak sense for any Borel set ∆ ∈ Q and a running point x ∈ Q. Making use now of the weak representation (3.46) the integral expression like I f,g (x) = Px f (P x )dP x g(P x ), x ∈ Q, for any continuous mappings f, g : P x → B(H) can be, obviously, represented as Thereby for the Volterrian operators (3.41) one can get the following expressions: where, due to the representation (3.21) and Theorem 2.2. we put for any running points x, y and x ′ ∈ Q the following convolution of two kernels: for λ ∈ σ(L) ∩σ(L * ) and someψ λ ∈ H − . Taking now into account the representation (3.7) at s = " + ", from (3.49) one gets easily that the elementary congruent kernel where, by definition, S + Ω − and stated their relationships with the approach suggested in [18,19]. In particular, we demonstrated the existence of some Borel measures µ Px,± localized upon hypersurfaces S (m) ±,x ∈ K(Q) and related naturally with the corresponding integral operators K ± (Ω), whose kernelsK ± (Ω) ∈ H − ⊗H − are congruent to a pair of given differential operators (L,L) ⊂ L(H), satisfying the relationships (3.12). In what will follow below we shall study some differentialgeometric properties of the Lagrange identity naturally associated with two Delsarte related differential operators L andL in H and describe by means of some specially constructed integral operator kernels the corresponding Delsarte transmutation operators exactly in the same spectral form as it was studied in Chapter 3 above.
Let a multi-dimensional linear differential operator L : H → H of order n(L) ∈ Z + be of the form and defined on a dense domain D(L) ⊂ H, where, as usually, α ∈ Z m + is a multiindex, x ∈ R m , and for brevity one assumes that coefficients a α ∈ S(R m ; EndC N ), α ∈ Z m + . Consider the following easily derivable generalized Lagrangian identity for the differential expression (4.1) : where (ϕ, ψ) ∈ H * × H, mappings Z i : H * × H → C, i = 1, m, are semilinear due to the construction and L * : H * → H * is the corresponding formally conjugated to (4.1) differential expression, that is Having multiplied the identity (4.2) by the usual oriented Lebesgue measure dx = ∧ j= − − → 1,m dx j , we get that Consider now all such pairs (ϕ(λ), ψ(µ)) ∈ H * 0 × H 0 ⊂ H − × H − , λ, µ ∈ Σ, where as before (4.5) H + ⊂ H ⊂ H − is the usual Gelfand triple of Hilbert spaces [8,9] related with our Hilbert-Schmidt rigged Hilbert space H, Σ ∈ C p , p ∈ Z + , is some fixed measurable space of parameters endowed with a finite Borel measure ρ, that the differential form (4.4) is exact, that is there exists a set of (m − 2)−differential forms Ω (m−2) [ϕ(λ), ψ(µ)] ∈ Λ m−2 (R m ; C), λ, µ ∈ Σ, on R m satisfying the condition A way to realize this condition is to take some closed subspaces H * 0 and H 0 ⊂ H − as solutions to the corresponding linear differential equations under some boundary conditions: The triple (4.5) allows naturally to determine properly a set of generalized eigenfunctions for extended operators L , L * : H − → H − , if Γ ⊂ R m is taken as some (n-1)-dimensional piece-wise smooth hypersurface embedded into the configuration space R m . There can exist, evidently, situations [13,7,5] when boundary conditions are not necessary.
Based now on this Lemma and exact expressions of operators (4.10), similarly to calculations done in [23], one shows right away that operatorsL andL * , depending correspondingly only both on the homological cycles σ ∈ C m−2 (M ; C) from a simplicial chain complex K(M ), marked by points x, x 0 ∈ R m , and on two sets of functions (ϕ(λ), ψ(µ)) ∈ H * 0 × H 0 and (φ(λ),ψ(µ)) ∈H * 0 ×H 0 , λ, µ ∈ Σ, are purely differential thereby finishing the proof.◮ The differential-geometric construction suggested above can be nontrivially generalized for the case of m ∈ Z + commuting to each other differential operators in a Hilbert space H giving rise to a new look at theory of Delsarte transmutation operators based on differential-geometric and topological de Rham-Hodge techniques. These aspects will be discussed in detail in the next two chapters below.
5. The general differential-geometric and topological structure of Delsarte transmutation operators: the De Rham-Hodge-Skrypnik theory 5.1. Below we shall explain the corresponding differential-geometric and topological nature of these spectral related results obtained above and generalize them to a set L of commuting differential operators Delsarte related with another commuting setL of differential operators in H. These results are deeply based on the De Rham-Hodge-Skrypnik theory [28,29,27,3,30] of special differential complexes giving rise to effective analytical expressions for the corresponding Delsarte transmutation Volterra type operators in a given Hilbert space H. As a by-product one obtains the integral operator structure of Delsarte transmutation operators for polynomial pencils of differential operators in H having many applications both in spectral theory of such multidimensional operator pencils [21,22,24,33] and in soliton theory [17,12,6,25,11] of multidimensional integrable dynamical systems on functional manifolds, being very important for diverse applications in modern mathematical physics.
Let M :=R m denote as before a suitably compactified metric space of dimension m = dimM ∈ Z + (without boundary) and define some finite set L of smooth commuting to each other linear differential operators It is easy to see that the operation ( Proof. It follows easily from the equality d L d L = 0 holding due to the commutation of operators (5.1) .⊲ 5.2. Below we will follow the ideas developed before in [3,29]. A differential form β ∈ Λ(M ; H) will be called d L -closed if d L β = 0, and a form γ ∈ Λ(M ; H) will be called d L -homological to zero if there exists on M such a form ω ∈ Λ(M ; H) that γ = d L ω.
Consider now the standard algebraic Hodge star-operation Subject to the scalar product (5.5) we can naturally construct the corresponding Hilbert space well suitable for our further consideration. Notice also here that the Hodge star ⋆-operation satisfies the following easily checkable property: for any β, γ ∈ H k Λ (M ), k = 0, m, Such a form is called harmonic. One can also verify that a harmonic form β ∈ H Λ (M ) satisfies simultaneously the following two adjoint conditions: (5.9) d ′ L β = 0, d L β = 0, easily stemming from (5.7) and (5.9).
It is not hard to check that the following differential operation in H Λ (M ) (5.10) d * L := ⋆d ′ L (⋆) −1 defines also a usual [26,27,28] external anti-differential operation in H Λ (M ). The corresponding dual to (5.3) co-chain complex is evidently exact too, as the property d * L d * L = 0 holds due to the definition (5.7). Another variant of the statement similar to that above was formulated in [3] and reads as the following generalized de Rham-Hodge-Skrypnik theorem. A proof of this theorem is based on some special sequence [3] of differential Lagrange type identities. Define the following closed subspace

Denote further by
,− (M ) is, as above, a suitable Hilbert-Schmidt rigged [8,9] zero-order cohomology group Hilbert space from the chain given by (5.14), σ(L) and σ(L * ) are, correspondingly, mutual spectra of the sets of commuting operators L and L * . Thereby the dimension dim H * 0 = card Σ is assumed to be known. The next lemma stated by Skrypnik I.V. [3] being fundamental for the proof holds.

Conclusion
The results obtained above and developing the De Rham-Hodge-Skrypnik theory [28,29,27,3,30] of special differential complexes gave rise to effective analytical expressions for the corresponding Delsarte transmutation Volterra type operators in a given Hilbert space H. In particular, it was shown also that they can be effectively applied to studying the integral operator structure of Delsarte transmutation operators for polynomial pencils of differential operators in H having many applications both in spectral theory of such multidimensional operator pencils and in soliton theory [17,12,6,22,25] of multidimensional integrable dynamical systems on functional manifolds, being very important for diverse applications in modern mathematical physics. If one considers a differential operator L : H → H and assumes that its spectrum σ(L) consists of the discrete σ d (L) and continuos σ c (L) parts, by means of the general form of the Delsarte transmutation operators obtained in Chapters 4 and 5 one can construct a new more complicated differential operatorL := Ω ± LΩ −1 ± in H, such that its continuous spectrum σ c (L) = σ c (L) but σ d (L) = σ d (L). Thereby these Delsarte transformed operators can be effectively used for both studying generalized spectral properties of differential operators and operator pencils [8,5,11,13,9,32,31] and constructing a wide class of nontrivial differential operators with a prescribed spectrum as it was done [11,6,33] in one dimension.
As it was shown before in [5,17] for the two-dimensional Dirac and threedimensional perturbed Laplace operators, the kernels of the corresponding Delsarte transmutation operator satisfy some special of Fredholm type linear integral equations called the Gelfand-Levitan-Marchenko ones, which are of very importance for solving the corresponding inverse spectral problem and having many applications in modern mathematical physics. Such equations can be easily constructed for our multidimensional case too, thereby making it possible to pose the corresponding inverse spectral problem for describing a wide class of multidimensional operators with a priori given spectral characteristics. Also, similar to [17,25,6,11], one can use such results for studying so called completely integrable nonlinear evolution equations, especially for constructing by means of special Darboux type transformations [14,20,24] their exact solutions like solitons and many others. Such an activity is now in progress and the corresponding results will be published later.