Symplectic Geometry of the Koopman Operator

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.


KOOPMAN OPERATOR
Let be a space with a finite countably additive measure μ and be an invertible measure-preserving transformation. Let be the Hilbert space of real functions on M that are square integrable with respect to μ. The scalar product of functions f and g is defined as usual as The Koopman operator maps the function to the function It is well known that this operator is orthogonal: (1) or, equivalently, . Clearly, is always its eigenvalue. Nonzero constant functions on M are the corresponding eigenvectors.
An important result for our study is a consequence of (1) stating that the operator admits a quadratic invariant . In other words, , . In the nonresonance case (when the relation , , implies m = 0, k = 0) the mapping T is ergodic (Weyl's theorem). The corresponding discrete dynamical system is often called the Kronecker-Weyl cascade.

Example 2.
, where is a unimodular matrix with integer elements. If none of the eigenvalues of A belong to the unit circle of the complex plane, then T is a fortiori mixing. A classical example is as follows: n = 2 and It should be kept in mind that, for the transformations from Example 2, ergodicity is equivalent to mixing.
In Example 1, the functions , , are the eigenfunctions for the corresponding Koopman operator: They make up an orthogonal basis in . This simple observation means that we need to consider the space of square integrable functions with complex values. In the real case, the Koopman operator has twodimensional invariant planes spanned by the vectors sin(m, x) and ( ). If the mapping T is a mixing (or even a weak mixing), then the spectrum of the Koopman operator is "continuous" (more precisely, the only eigenvalue is and this eigenvalue is simple). The basic facts concerning the spectral theory of the Koopman operator can be found, for example, in [1,2].

SYMPLECTIC STRUCTURE
Consider a more general situation. Let be a real Hilbert space (the case is not excluded) with inner product ( , ), and let be an orthogonal operator. Obviously, the mapping admits the quadratic invariant . Define the operator (2) In view of the orthogonality conditions (1), Therefore, is a skew-self-adjoint operator. In view of (1), it can also be represented in the form Operator (2) is associated with the bilinear skewsymmetric form (4) Recall that this form is called nondegenerate if for all implies that x = 0. If the 2-form (4) is nondegenerate, it defines a symplectic structure in (and it itself is usually called a symplectic structure).
It is well known that the spectrum of an orthogonal transformation lies on the unit circle. It follows from (2) and (3) operator U is symplectic. In the finite-dimensional case, this fact was noted in [3]. Specifically, is even.
If U is the Koopman operator from Section 1, then the 2-form [ , ] is degenerate: constant functions remain intact under the operator U. To rectify the situation, we need to consider the Hilbert space orthogonal to the one-dimensional subspace consisting of constant functions. In other words, consists of square integrable functions with zero mean. Clearly, maps to . Specifically, the following result is true. Proposition 2. Suppose that the mapping T 2 : M is ergodic. Then the 2-form [ , ] is nondegenerate on .
Indeed, λ = 1 is then a simple eigenvalue of the corresponding Koopman operator . Therefore, the operator U 2 in does not have nonzero eigenvectors with eigenvalue λ = 1. It remains to use Proposition 1.
Note that the square of an ergodic transformation is not necessarily ergodic. However, this is a fortiori the case in two examples given in Section 1.
Summarizing, in the general case (for cascades with an ergodic square) the Koopman operator is a symplectic operator. However, the invariant symplectic structure depends on this operator.
It is well known that a real orthogonal operator that is a symplectic transformation is a unitary operator. This means that, in the real space , it is possible to introduce a complex structure such that the action of U is equivalent to the action of a unitary operator in the complex space . This construction is especially simple for operators with a discrete spectrum. These issues are discussed in [4] for real linear systems of differential equations with a quadratic invariant in a Hilbert space in order to represent them in the form of Schrödinger equations.

QUADRATIC INVARIANTS
be two such forms ( and are bounded self-adjoint operators). Their Poisson bracket is the quadratic form (5) The operator (like F and ) is self-adjoint (symmetric and bounded). In particular, the quadratic form h is also continuous.
The bracket { , } is bilinear, skew-symmetric, and satisfies the Jacobi identity. In [5,6] the Poisson bracket was defined under more general assumptions.
We introduce the self-adjoint operators Theorem 2. The continuous quadratic forms (6) are invariants of the mapping ; moreover, for all . To prove the invariance of F n , we need to check the equality , or Indeed, since is orthogonal, we have Therefore, the left-hand side of (8) is To prove the involution of these invariants, we need to check the equality (9) Since the operators , and commute with each other, both sides of (9) are symmetric with respect to k and l. This proves equality (7).
The quadratic invariants F n may be dependent. A simple example is as follows: if U = I, then all of them are proportional to . Suppose that is finite and U is an orthogonal operator having no eigenvalues . Then is even. Additionally, as was noted in [3], if the operator has no multiple eigenvalues, then the quadratic forms are functionally independent. Since they are pairwise in involution, their nonempty joint levels   x F x c … F x c * for almost all are N/2-dimensional tori invariant under the action of the operator U. Furthermore, on these tori, we can introduce angular coordinates such that the action of is reduced to the Kronecker-Weyl mapping Since the operator is linear, the numbers {α j } do not change from one torus to another.
Note that a linear symplectic mapping is completely integrable without assuming that the spectrum is simple. This result can be derived from the theory of Williamson normal forms [7]. However, in the case of a simple spectrum, a complete set of involution invariants can be presented without preliminarily solving the algebraic eigenvalue and eigenvector problem for a symplectic operator.
In the infinite-dimensional case, the situation is more complicated. We consider two (in a sense, opposite) cases, namely, when the operator U has a simple discrete spectrum and when its spectrum is continuous.

DISCRETE SPECTRUM
It is well known that the spectrum of an orthogonal operator lies on the unit circle of the complex plane. The possibility of existence of two real eigenvalues is ruled out. Suppose that (10) where and are vectors from the real Hilbert space and . Equality (10) is equivalent to two real relations Therefore, the real plane spanned by the vectors , is invariant under the action of the orthogonal operator U. Moreover, it is easy to show that so we can assume that the vectors and are of unit length; they make up an orthonormal basis in the plane .   [6]). Thus, in , there is an orthonormal set of vectors (13) Additionally, the operator U admits infinitely many quadratic invariants (14) where is the orthoprojector onto .
On the other hand, the planes are also invariant under the action of the skew-self-adjoint operator Ω. In the invariant plane , Ω is represented by the skew symmetric matrix All these matrices are nonsingular, since . Otherwise, the operator would have the eigenvalues ±1.
Theorem 3. Assume that the orthonormal system of vectors (13) is complete. Then the following assertions hold: (a) Quadratic forms (14) make up a complete set of independent quadratic invariants of the operator that are pairwise in involution with respect to the symplectic structure in defined by the skew-self-adjoint operator .
Px Px x * α + β , α + β , x f x c n c and c * (c) On these tori, it is possible to introduce angular variables such that the action of the operator U on in these variables is given by the formulas (15) This statement suggests that the mapping is completely integrable (as in the finite-dimensional case, which was mentioned in Section 3). The completeness of the family of involution invariants (14) means that this family cannot be supplemented with another quadratic form that is independent of (14) and is in involution with them. Mapping (15) is an infinitedimensional variant of the Kronecker-Weyl mapping. Its properties are quite similar to the ergodic properties of continuous Kronecker-Weyl flows on infinitedimensional tori (they are discussed in [8,9], where further references can be found).

LEBESGUE SPECTRUM
The case of a continuous spectrum is more complicated. We restrict our consideration to cascades with a Lebesgue spectrum. Specifically, they include automorphisms of the torus from Example 2 (see Section 1). Such systems a fortiori have mixing. For simplicity, we consider a simple Lebesgue spectrum.
In this case, has a complete orthonormal basis , such that In a sense, the general case is reduced to this partial one (see, e.g., [2]).

Let and . Then
Thus, if is identified with l 2 (the space of sequences infinite on both sides with the condition ), then the action of U is reduced to a left shift of the elements by unity. Of course, the scalar square is conserved under this shift. Theorem 2 gives an infinite set of quadratic invariants: More precisely, invariants (6) are reduced to finite linear combinations of quadratic forms (17).
Furthermore, for . Therefore,   . Then (18) implies that . Therefore, the elements of the vector x with even (odd) indices are equal to each other. However, all of them then vanish; otherwise, the series diverges. Hence, x = 0. Therefore, the skew-self-adjoint operator Ω defines a symplectic structure in . Formula (18) implies that this structure is invariant under the transformation U. By Theorem 2, quadratic invariants (17) are in involution with respect to the Poisson bracket generated by this symplectic structure.
It is easy to show that the quadratic forms , ... are independent: their gradients (as vectors from ) are linearly independent (at least at one point of ). Furthermore, it can be shown that any continuous quadratic form on that is invariant under the action of U can be represented in the form where , , are constants. This suggests that the Koopman operator is completely integrable for systems with a Lebesgue spectrum. However, the structure of joint levels of quadratic invariants (17) remains an open question. Can the action of the Koopman operator on these infinite-dimensional manifolds be reduced to the Kronecker-Weyl mapping? FUNDING This work was supported by the Russian Science Foundation, project no. 21-71-30011.
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