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 x x * : L 2 ( σ ( | T − a | ) , μ | T − a | , ξ ) → L 2 ( σ ( | ( T − a ) * | ) , μ | ( T − a ) * | , F x x * H ξ ) such that ( F x x * ) 2 = i d e n t i t y and F x x * H is a unitary operator on H associated with F x x * . With this construction, we obtain a noncommutative functional calculus for the operator T and F x x * = i d e n t i t y is the special case for normal operators, such that S = R | ( S − a ) | , ξ ( M z ϕ ( z ) + a ) R | S − a | , ξ − 1 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 x x * we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class B L e b ( H ) ⊂ B ( H ) such that T is Li-Yorke chaotic if and only if T * − 1 is for a Lebesgue operator T .