Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

: Let T : H → H be a bounded linear operator on a separable Hilbert space H . In this paper, we construct an isomorphism F xx ∗ : L 2 ( σ ( | T − a | ) , µ | T − a | , ξ ) → L 2 ( σ ( | ( T − a ) ∗ | ) , µ | ( T − a ) ∗ | , F H xx ∗ ξ ) such that ( F xx ∗ ) 2 = identity and F H xx ∗ is a unitary operator on H associated with F xx ∗ . With this construction, we obtain a noncommutative functional calculus for the operator T and F xx ∗ = identity is the special case for normal operators, such that S = R | ( S − a ) | , ξ ( M z φ ( z ) + a ) R − 1 | S − a | , ξ is the noncommutative functional calculus of a normal operator S , where a ∈ ρ ( T ) , R | T − a | , ξ : L 2 ( σ ( | T − a | ) , µ | T − a | , ξ ) → H is an isomorphism and M z φ ( z ) + a is a multiplication operator on L 2 ( σ ( | S − a | ) , µ | S − a | , ξ ) . Moreover, by F xx ∗ we give a sufﬁcient condition to the invariant subspace problem and we present the Lebesgue class B Leb ( H ) ⊂ B ( H ) such that T is Li-Yorke chaotic if and only if T ∗− 1 is for a Lebesgue operator T .


Invariant Subspace
The invariant subspace problem has been stated by Beurling and von Neumann [1]. It can be formulated as follows.

Problem 1. Does every bounded linear operator on a given linear space have a non-trivial invariant subspace?
In 1966, Bernstein et al. [2] showed that if T is a bounded linear operator on a complex Hilbert space H and p is a nonzero polynomial such that p(T) is compact, then T has non-trivial invariant subspace. Especially, when p(t) = t, which is, T itself is compact, the result was proved independently by von Neumann and N. Aronszajn, and in [3], this result was extended to compact operators on a Banach space.
Let T be a bounded linear operator on a Banach space. In 1973, Lomonosov [4] proved that if T is not a scalar multiple of the identity and commutes with a nonzero compact operator, then T has a non-trivial hyperinvariant subspace, which is, any bounded linear operator commuting with T has a non-trivial invariant subspace (other results see [5][6][7]).
In 1976, Enflo [8] was the first to construct an operator on a Banach space having no non-trivial invariant subspace and Nordgren et al. [9] proved that every operator has an invariant subspace if and only if every pair of idempotents has a common invariant subspace.
In 1983, Atzmon [10] constructed a nuclear Fréchet space F and a bounded linear operator, which has no non-trivial invariant subspace. Especially, in 1984, C. J. Read made an example, such that there is a bounded linear operator without non-trivial invariant subspace on 1 [11].
In 2011, Argyros et al. [12] constructed the first example of a Banach space for which every bounded linear operator on the space has the form λ + K where λ is a real scalar and K is a compact operator, such that every bounded linear operator on the space has a non-trivial invariant subspace.
In 2019, Tcaciuc [19] proved that, for any bounded operator T acting on an infinite-dimensional Banach space, there exists an operator F of rank at most one such that T + F has an invariant subspace of infinite dimension and codimension.
For finite-dimensional vector spaces or nonseparable Hilbert spaces, the result is trivial. However, for infinite-dimensional separable Hilbert spaces, the problem is, after a long period of time, not yet completely solved.

Linear Dynamics
With the development of operator theory and dynamics progress, there are many papers about C*-algebras and dynamics. Additionally, "the fundamental theorem of C*-algebras [20]" is Gelfand-Naimark theorem [21]. Subsequently, in [22], Fujimoto said that this theorem eventually opened the gate to the subject of C*-algebras. Hence, there are various attempts to generalize this theorem [23][24][25][26][27].
For the research on Problem 1 and with the development of chaos, Operator Dynamics or Linear Dynamics has aroused extensive attention as an important branch of functional analysis, which was probably born in 1982 with the Toronto Ph. D. thesis of C. Kitai [28]. More details of this subject can be found in [29][30][31][32][33].
If X is a metric space and T is a continuous self-map on X, then the pair (X, T) is called a topological dynamic systems, which is induced by the iteration Moreover, if T is a continuous invertible self-map on X, then (X, T) is called an invertible dynamic and if the metric space X and the continuous self-map T are both linear, then the topological dynamic systems (X, T) is called a linear dynamic.
For invertible dynamics, the relationship of Li-Yorke chaos between (X, f ) and (X, f −1 ) was raised by Stockman as an open question [34]. Additionally, in [35,36] and [37], the authors give counterexamples for this question in noncompact spaces and compact spaces, respectively. For an invertible bounded linear operator T ∈ B(H), the chaotic relationship between (H, T) and (H, T * −1 ) is also interesting.
Next, we give the following definition Definition 1 (Li-Yorke chaos). Let T ∈ B(H). If there exists x ∈ H, such that satisfies: then the operator T is said to be Li-Yorke chaotic, and x is called a Li-Yorke chaotic point of T.
An example of an operator T that is Li-Yorke chaotic but T * −1 is not can be found in [38]. However, presently there is no general method to do this research. In fact, the C*-algebra A(T) generated by T cannot be used for that.

Motivation and Main Results
For an n-tuple T of not necessarily commuting operators, Colombo et al. [39] put to use the notion of slice monogenic functions [40] to define a new functional calculus, which is consistent with the Riesz-Dunford calculus in the case of a single operator and that allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T (more results, see [41][42][43][44]).
In 2010, for bounded operators defined on quaternionic Banach spaces, Colombo et al. [45] developed a noncommutative functional calculus that is based on the new notion of slice-regularity and that is based on the key tools of a new resolvent operator and a new eigenvalue problem, also, they extended this calculus to the unbounded case [46] (more results, see [47]).
In 2018, Monguzzi et al. [48] characterized the closed invariant subspaces for the (*−) multiplier operator of the quaternionic space of slice L 2 functions, obtained the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and provided a characterization of quaternionic outer functions in terms of cyclicity.
In this paper, we give a noncommutative functional calculus for T ∈ B(H). Additionally, by this construction, we give some applications, such as its applications on the invariant subspace problem and chaos. The precise meaning of the multiplication operator M zψ Let H be a separable Hilbert space over C, B(H) be the set of all bounded linear operator on H. For any given T ∈ B(H), we obtain a C*-algebra A(|T − a|) associated with the polar decomposition T − a = U|T − a|, where a ∈ ρ(T) and ξ is a A(|T − a|)-cyclic vector, such that H = A(|T − a|)ξ. In this paper, we construct an isomorphism F xx * , such that the following diagram is valid.
where F H xx * is the corresponding unitary operator associated with the isomorphism F xx * and Fix(F H xx * ) = ∅, (F H xx * ) 2 = identity and (F xx * ) 2 = identity. With this construction, we get a noncommutative functional calculus for the operator T such that Especially, F xx * = identity, which is the special case for normal operators, will become clear in Corollary 3, and, in this special case, we get that the noncommutative functional calculus of a normal operator S is just only S = R |(S−a)|,ξ (M zφ(z) + a)R −1 |S−a|,ξ , which is compatible with the classical normal operator functional calculus of [49]. Where ψ(z) ∈ L ∞ (σ(|T − a|), µ |T−a|,ξ ) and φ(z) ∈ L ∞ (σ(|S − a|), µ |S−a|,ξ ).
Moreover, from F xx * , we deduce a sufficient condition to Problem 1 on infinite-dimensional separable Hilbert spaces and present the Lebesgue class B Leb (H) ⊂ B(H), such that, if T is a Lebesgue operator, then T is Li-Yorke chaotic if and only if T * −1 is.
In fact, we get that

Decomposition and Isomorphic Representation
In this paper,f (·) means the conjugate of the complex function f (·). Let X be a compact subset of C, C(X) be the set of all continuous function on X, and P (x) be the set of all polynomial on X. For any given T ∈ B(H), let σ(T) be its spectrum.
Let A(|T|) be the complex C * -algebra generated by |T| and 1. Obviously, if T is invertible, then U is a unitary operator. Lemma 1. Let X ⊆ C be a compact subset not containing zero. If P (x) is dense in C(X), then P ( 1 x ) is also dense in C(X).
Proof. By the properties of complex polynomials, we get that P ( 1 x ) is a subalgebra of C(X), which is closed under the standard algebraic operations. In addition, we have: . We get the conclusion from the Stone-Weierstrass theorem [49] (p. 145).
For X ⊆ R + , there is x = y ⇐⇒ x 2 = y 2 . With Lemma 1, we get the following result.

Lemma 3.
Let T be an invertible bounded linear operator on H. Then there exists a sequence of nonzero A(|T|)-invariant subspaces H 1 , H 2 , · · · , H i , · · · , such that: (2) For every H i , there is a A(|T|)-cyclic vector ξ i such that Proof. The decomposition of (1) is obvious [51] (p. 54), Therefore, From Lemma 1, we get that Let ξ ∈ H be a A(|T|)-cyclic vector, such that A(|T|)ξ is dense in H . Because the spectrum is closed and σ(|T|) = ∅ , on C(σ(|T|)), we can define the nonzero linear functional It is easy to get that ρ |T|,ξ is a positive linear functional. By [51] (p. 54), and the Riesz-Markov theorem, on C(σ(|T|)), we get that there is a uniquely finite positive Borel measure µ |T|,ξ , such that

Theorem 1.
Let T be an invertible bounded linear operator on H, A(|T n |) be the complex C * -algebra generated by |T n | and 1 and let ξ n be a A(|T n |)-cyclic vector, such that A(|T n |)ξ n = H , where n ∈ N. Subsequently: (1) there is a uniquely positive linear functional (2) there is a uniquely isomorphic representation R |T n |,ξ n : L 2 (σ(|T n |), µ |T n |,ξ n ) → H associated with the uniquely finite positive Borel measure µ |T n |,ξ n , which is complete.
We get that, on C(σ(|T n |)), there is a uniquely finite positive Borel measure µ |T n |,ξ n , such that Moreover, we can complete this Borel measure µ |T n |,ξ n on σ(|T n |). For this completion, we keep the notation µ |T n |,ξ n . We know that this Borel measure is unique [52] .
Obviously, C(σ(|T n |)) and A(|T n |)ξ n are dense subspaces of L 2 (σ(|T n |), µ |T n |,ξ n ) and H , respectively. Additionally, its closed extension Therefore, we get that R |T n |,ξ n is the uniquely isomorphic representation of H associated with the uniquely finite positive Borel measure µ |T n |,ξ n , which is complete.
Subsequently, ξ is a A(|T| −1 )-cyclic vector, such that H = A(|T| −1 )ξ and we get the following equation

Noncommutative Functional Calculus
We know that the spectral theory and functional calculus of normal operators [49] is very important in the study of operator theory and C * -algebras [50]. Inspired by the Hua Loo-kang theorem on the automorphisms of a sfield [53], in this section, we give a useful construction from L 2 (σ(|T −1 |), µ |T −1 |,η ) to L 2 (σ(|T| −1 ), µ |T| −1 ,ξ ) and with this construction, we give a noncommutative functional calculus for any given T ∈ B(H) . However, there is valueless information just only from Lemma 4. Let T be an invertible bounded linear operator on H. Subsequently, we get Let T be an invertible bounded linear operator on H, ξ be a A(|T|)-cyclic vector, such that H = A(|T|)ξ . On P (z), with z ∈ σ(|T|), we define the mapping Following Lemma 3 and Theorem 1, P (z) and P ( 1 z ) are dense subspaces of L 2 (σ(|T|), µ |T|,ξ ) and L 2 (σ(|T| −1 ), µ |T| −1 ,ξ ) , respectively. Its closed extension is linear and for this closed extension we keep the notation F z −1 .
Subsequently, we obtain that By a simple computation, we get that Hence, it follows that By an application of the Banach inversion theorem [49] (p. 91), we get that F z −1 is an invertible bounded linear operator from Next, we define the operator |m| .
Moreover, we obtain the following diagram.

Definition 2.
For invertible T ∈ B(H), let the symbol xx * stand for T −1 T * −1 . Subsequently, we get that there is a linear algebraic isomorphism from P (xx * ) to P (x * x), such that Let ξ be a A(|T| −1 )-cyclic vector, such that A(|T| −1 )ξ = H and U 0 ∈ B(H) be a unitary operator such Then its closed extension is For this closed extension, we keep the notation F xx * .
In fact, when T is invertible, we can choose a special unitary operator, which shows that the operators |T| −1 and |T −1 | are unitary equivalent. This is explained in the following theorem. Theorem 2. Let T be an invertible bounded linear operator on H and U 0 ∈ B(H) be a unitary operator, such that U 0 P (|T| −1 ) = P (|T −1 |)U 0 . Afterwards, there is a unitary operator F H xx * , such that Moreover, F H xx * is the corresponding unitary operator associated with the almost everywhere nonzero function |φ |T| (z)|, such that where |φ |T| (z)| ∈ L 1 (σ(|T|), µ |T|,ξ ) and ξ is a A(|T|)-cyclic vector, such that A(|T|)ξ = H .

With Theorem 1 and Definition 2, we have the operator
,ξ , such that the following diagram is valid.
Therefore, we said that the linear operator F H xx * is associated with F xx * . Subsequently, we see that F H xx * is a unitary operator and by Lemma 3, we obtain Afterwards, F H xx * is the corresponding unitary operator associated with F xx * , which is, associated with the almost everywhere nonzero function |φ |T| ( 1 z )|, such that where |φ |T| (z)| ∈ L 1 (σ(|T|), µ |T|,ξ ) .

Definition 3. For any given T ∈ B(H), we say that R
In the final part of this section, we give some properties of normal operator through the noncommutative functional calculus. Proof. For TT * = T * T and a ∈ ρ(T) , we get that Therefore, we see that That is, With the proof of Theorem 3, we get T − a ∈ A (|T − a|), which is, T ∈ A (|T − a|) .

A Sufficient Condition
In this section, we study Problem 1 on infinite-dimensional separable Hilbert spaces. With the fact that the exist of non-trivial invariant subspace is unchanged by the similarity of bounded linear operators on Banach spaces [1], which is, for R ∈ B(B 1 ) and S ∈ B(B 2 ), if T : B 1 → B 2 is an invertible bounded linear operator and S = TRT −1 , then R has non-trivial invariant subspace if and only if S has, where B 1 and B 2 are Banach spaces. Therefore, for any given T ∈ B(H), using the construction of F xx * , we give a sufficient condition to Problem 1 on infinite-dimensional separable Hilbert spaces.
For convenience, we define Proof. It is enough to prove the result for infinite-dimensional separable complex Hilbert space H.

Obviously, if A ⊂ H is a non-trivial invariant subspace of T if and only if
A is a non-trivial invariant subspace of T − a, where a ∈ C. Let a ∈ ρ(T) and let ξ be a A(|T − a|)-cyclic vector such that H = A(|T − a|)ξ . Subsequently, following Corollary 2, we get that there exists ψ(z) ∈ L ∞ (σ(|T − a|), µ |T−a|,ξ ), such that By the construction of F H xx * in Theorem 2, we get (F H xx * ) 2 = identity and Fix(F H xx * ) = ∅.
by the proof of Corollary 3, then T is unitary equivalent to M zψ(z) + a. Because M zψ(z) + a is a normal operator, it possesses a non-trivial invariant subspace and, hence, the same is true for T. For details, see, e.g., [49]. ( With the proof of Theorem 3, we get that That is,

Lebesgue Operator
In this section, we study chaos of an invertible bounded linear operator on an infinite-dimensional separable Hilbert space. For the example of integral calculus in mathematical analysis, we know that the convergence or the divergence of the weighted integral calculus of x and x −1 should be independent of each other; however, sometimes it happens that this indeed depends on a special choice of the weight function.
In the view of integral calculus, we define the Lebesgue class and prove that if T is a Lebesgue operator, then T is Li-Yorke chaotic if and only if T * −1 is. With the idea of the noncommutative functional calculus R |(T−a) * |,F H xx * ξ F xx * M zψ(z) R −1 |T−a|,ξ , we give an example of a Lebesgue operator that is not a normal operator.
Let dx be the Lebesgue measure on L 2 (R + ). By Theorem 1, there exists a Borel measure dµ |T n |,ξ n , which is complete, such that L 2 (σ(|T n |), dµ |T n |,ξ n ) is a Hilbert space. If there exists N > 0, such that, for all n ≥ N, the measure dµ |T n |,ξ n is absolutely continuity with respect to dx, then using the Radon-Nikodym theorem [49] (p. 380), there exists f n ∈ L 1 (R + ), such that dµ |T n |,ξ n = f n (x)dx, where n ∈ N, n ≥ N and H = A(|T n |)ξ n . (1) There exists N ∈ N, such that, for all n ≥ N      dµ |T n |,ξ n = f n (x)dx, f n ∈ L 1 (R + ) (2) There exists N ∈ N, such that for all n ≥ N and for any given nonzero x ∈ H, there exists a nonzero function g n (t) ∈ L 2 (σ(|T n |), dµ |T n |,ξ n ) and a nonzero vector y ∈ H, such that y = g n (|T n | −1 )ξ n whenever x = g n (|T n |)ξ n .
Subsequently, the operator T is said to be a Lebesgue operator, and the family of all Lebesgue operators on H is denoted by B Leb (H). If x 0 is a Li-Yorke chaotic point of T, then by Definition 4, we see that, for n ∈ N large enough, there exist g n (x) ∈ L 2 (σ(|T n |), dµ |T n |,ξ n ), f n (x) ∈ L 1 (R + ) and y 0 ∈ H, such that x 0 = g n (|T n |)ξ n , y 0 = g n (|T n | −1 )ξ n , and dµ |T n |,ξ n = f n (x)dx.
Following [54], for T ∈ B(H), x ∈ H and n ∈ N, we introduce the distributional function In addition, we denote and introduce the following definition.
Corollary 5. Let T be a Lebesgue operator on the separable Hilbert space H over C. Then T is I-distributionally chaotic (or I I-distributionally chaotic or I I I-distributionally chaotic) if and only if T * −1 is I-distributionally chaotic (or I I-distributionally chaotic or I I I-distributionally chaotic).

Theorem 6.
There exists an invertible bounded linear operator T on the separable Hilbert space H over C, such that T is Lebesgue operator that is not a normal operator.
is a separable Hilbert space over C. Any separable Hilbert space over R can be expanded to a separable Hilbert space over C. Without loss of generality, let L 2 ([a, b]) be the separable Hilbert space over R. We prove the conclusion by six parts: Subsequently, Φ is an invertible measure preserving transformation on the Borel algebra ξ(M). With [55] (p. 63), U Φ = 1 and U Φ is a unitary operator associated with Φ, where U Φ is the operation of composition (2) Define M x h = xh on L 2 ([a, b]). Subsequently, M x is an invertible positive operator.
(3) For f (x) = | ln x| x , x > 0, we define dµ = f (x)dx. Afterwards, f (x) is continuous and we get that T is not a normal operator and σ(|T|) = [a, b]. We get that f n (t) is continuous and almost everywhere positive. Hence, f n (t)dt is a finite positive Borel measure that is complete.
For any E ⊆ R + , we define I E = 1 when x ∈ E else I E = 0. Subsequently, I E is the identity function on E. With a simple computing, we get that f n (t −1 ) = 1 n I [a n ,b n ] f (t We see that x 2 f n (x) = f n (x −1 ) . From σ(|T n |) = [a n , b n ] and b n a n t 2 f (t Easily, we get that I [a,b] is a A(|M n x |)-cyclic vector of the multiplication M n x = M x n and I [a n ,b n ] is a A(|T n |)-cyclic vector of |T n |. By Definition 4, we get that T is Lebesgus operator, but is not a normal operator.

Corollary 6.
There exists an invertible bounded linear operator T on the separable Hilbert space H over C, such that T is a Lebesgue operator that is a positive operator. In fact, both these families contain non-trivial members.

Conclusions
By the idea of the isomorphism construction F xx * of this paper, we could study the operator using the integral calculus on R. This way maybe neither change the properties of chaos nor the difficulty of computing, but with this we should find some operator class and study its properties, as we give the Lebesgue class in this section. Hence, if some properties of operators on H only depending the norm that is compatible with the inner product, then these properties only depend on the corresponding properties of elements in {M ψ n (z) + a n : ψ n (z) ∈ L ∞ (σ(|T n − a n |), µ |T n −a n |,ξ n ), a n ∈ ρ(T n ), n ∈ N, H = A(|T n − a n |)ξ n }, just keeping the noncommutative functional calculus in mind. suggestions. Also, I shall extend my thanks to all those who have offered their help to me. Lastly, I sincerely appreciate the support and cultivation of Fudan University, Jilin University and Xidian University.

Conflicts of Interest:
The author declare no conflict of interest.