About the Legendre type operators

The article considers Legendre type operators acting in the corresponding weight separable Hilbert spaces. The choice of these spaces is due to the fact that these operators preserve all properties of the Legendre operator acting on L 2 ( − 1 , 1). In particular, 1) the Legendre type operators, operating in respective weight separable Hilbert spaces, remain discrete, 2) the spectrum does not change relative to the classical Legendre operator, 3) the corresponding eigenfunctions are compositions of Legendre polynomials and some functions.


Introduction
Retrieval of operators which have known spectral properties in advance is one of the most important problems for contemporary mathematics. If we construct two operators, P + (ν) and P − (ν), in a certain separable Hilbert space (hereinafter referred to as SHS), and one of them will recurrently find certain sequence of functions from SHS, while the other will find the same sequence but in reverse order, though recurrently as well, when we multiply these two operators, we may get explicitly operative operator in SHS having the following peculiarities: the constructed functions turn out to be the eigenfunctions for this operator.
The article also describes the construction of some of the weighted separable Hilbert spaces L ω 2 (a, b), ω = ω(x), x ∈ [a, b], denoted by H ω,a,b . These spaces H ω,a,b are homeomorphic to the space H. The homeomorphism will be proved by the properties of the operatorÂ hereinafter. We will also study the action of the operatorÂ in H ω,a,b .

Construction of the Legendre type operator
Let us assume that in a certain separable Hilbert space H = L 2 (−1, 1), the following differential operators are operative: where the functions f 1 = f 1 (x) and f 2 = f 2 (x) meet the following requirements: 1) f 2 is doubly continuously differentiable function within For any x ∈ [−1, 1] the following equation will be true: Operator I is an identity operator. The definition range for P + (ν) (−1 = ν ∈ R 1 ) and P − (ν) (0 = ν ∈ R 1 ) operators includes all the functions which are absolutely continuous together with their first-order derivatives within the range of [−1, 1], while second-order derivatives within [−1, 1] must be square summable. It is easy to verify that the definition ranges for P − (ν + 1)P + (ν) (−1 = ν ∈ R 1 ) and P + (ν −1)P − (ν) (0 = ν ∈ R 1 ) operators include all the functions which are absolutely continuous together with their first-order derivatives within the range of [−1, 1], herewith, within the same range, second-order derivatives must be square summable. In this case, it is obvious that D(P − (ν + 1)P + (ν)) = D(P + (ν − 1)P − (ν)) and D(P − (ν + 1)P + (ν)) = D(P + (ν − 1)P − (ν)) = H, where the line above them means norm closure in H. Definition 1. Operator A which is operative in H is such that D( A) = D(P − (ν + 1)P + (ν)) = D(P + (ν − 1)P − (ν)) and for D ν ( A) := D( A)∩T ν ∀ν ∈ C the following operator equations are simultaneously true: will be called the Legendre type operator. Here O is the annihilator in H, Let us prove that this definition is correct. At first, let us consider the operator multiplications: Given that P − (ν + 1)P + (ν) = I and P + (ν − 1)P − (ν) = I for D ν ( A), we will multiply expression (3) by (ν + 1) 2 : and expression (4) we will multiply by ν 2 : From (5) and (6), for any ν ∈ C the following equation will be true: Based on (7), from (5) and (6) we have as follows: Taking into account (8), from (5) and (6) we will get: Right-hand members of the last two operator equations are equal. It is easy to extract the Legendre operator from them: thus, the correctness of the above definition is obvious.
3 Several properties of the operator A

Equivalence
Statement 1. For any complex number ν, the operator equation A = ν(ν + 1)I (9) and the equations set shall be equivalent within the set of all the functions from The above Statement may be proved obviously using (2).

Self-adjointness of operatorÂ
Let's look at the identity: and define as follows: y [1] .
In this case, y [1] and y [2] are quasiderivatives which correspond to the differential expression −(f 1 y) + f 2 y. In this can we can introduce the following operator By := y [2] whose properties have been well studied in [2], which, in its turn, will allow us to look at the operator A from 'classic' point of view. To do so we will need some data from [2, Chapter 5], [3]. Let us remind them: a) Quasiderivative functions y = y(x) (= y [0] (x)), corresponding to the expression below l(y) := (−1) n (p 0 y (n) ) (n) + (−1) n−1 (p 1 y (n−1) ) (n−1) + ... + p n y, are the functions y [1] , y [2] , . . . , y [2n] which are defined be the following formulas: from this it follows that l(y) = y [2n] ; b) l(y) is a self-adjoint differential expression, if its coefficients are real-valued functions differentiable sufficient number of times; c) If (a, b) is the range where the differential expression l(y) is considered, and functions 1 p 0 (x) , p 1 (x), . . . , p n (x) are measurable within (a, b) and summable in its every isolated finate subinterval [α, β], then l(y) is a regular differential expression; d) If the above item c) will have at least one of the conditions violated, with the conditions being the measurability of functions 1 p 0 (x) , p 1 (x), . . . , p n (x) within the range of (a, b) and summability within its every finate subinterval [α, β], then l(y) is singular differential expression.
In our case, if the function 1 is summable within (-1;1), then the operator B := f 2 A+I := y [2] will be called regular (according to Naimark, compare to item c) of the above paragraph); if the summability condition is violated for the function 1 f 1 (x) , it will be a singular operator (compare to item d) of the above paragraph). This property can be proved using item b) for the selfadjoint differential expression l(y).

Eigenvalues and eigenfunctions of operator A
Throrem 1. The following multitudes are sets of eigenvalues and corresponding eigenfunctions of operator A which is operative in H; P n (·) is Legendre's polynomial.
P n+1 (f 2 ) = P + (n)P n (f 2 ), are the formulas for recurrent calculation of eigenfunctions P n (f 2 ) of the operator A.

Examples of Legendre type operator acting in H
Let us consider several particular examples for the Legendre type operators which are operative in SHS H := L 2 (−1, 1). Since the type and properties of operators are significantly dependent on the type and properties of functions f 1 and f 2 which may be of an unlimited choice (as long as the requirements 1)-3) are met for them), the number of Legendre type operators is quite vast. Eigenvalues and eigenfunctions for these operators may be found using formulas (11) and (12), accordingly.

About homeomorphism of spaces
We will prove the following lemma. Lemma 1. For every interval [a, b] (a < b) of finite length and every continuously differentiable function f 2 (x) with a derivative of fixed sign on this interval, there exist α and β such that every monotone, continuously differentiable function f 2 (x) = α f 2 (x) + β on [a, b] will have the following values: f (a) = −1, f (b) = 1.
Proof. Let f 2 (a) = c, f 2 (b) = d, then c = d and we have to solve a linear system where α and β: α . This concludes the proof of the Lemma. Theorem 2. Let [a, b] be a random interval of finite length (a < b) and f 2 (x) is any continuously differentiable function with a derivative of fixed sign on this interval. Let's take the function f (x) ∈ H. Then g(t) = f (f 2 (t)) ∈ H ω,a,b : Now let us consider the function g(t) ∈ H ω,a,b . We can see from the properties of f 2 that t = f −1 2 (y) is a monotone, continuously differentiable function on [−1, 1].
It is obvious that f (y) = g(f −1 2 (y)) ∈ H: We can see that the spaces H and H ω,a,b are isomorphic. Considering b a P n (f 2 (x))P m (f 2 (x))f 2 (x)dx = 1 −1 P n (u)P m (u)du = δ nm we get that ψ n (t) = P n (f 2 (t)) P n (f 2 (t)) L ω 2 , n ∈ N ∪ {0}, is an orthonormal basis in H ω,a,b ; δ nm it is the symbol of Kronecker. It is then easy to verify usingthe Lebesgue integral properties that G(f ) : H f (x) → f (f 2 (t)) ∈ H ω,a,b