Abstract
Let T:H→H be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ) such that (Fxx*)2=identity and Fxx*H is a unitary operator on H associated with Fxx*. With this construction, we obtain a noncommutative functional calculus for the operator T and Fxx*=identity is the special case for normal operators, such that S=R|(S−a)|,ξ(Mzϕ(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:L2(σ(|T−a|),μ|T−a|,ξ)→H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|S−a|),μ|S−a|,ξ). Moreover, by Fxx* we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class BLeb(H)⊂B(H) such that T is Li-Yorke chaotic if and only if T*−1 is for a Lebesgue operator T.
Highlights
By Fxx∗ 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
Is not a scalar multiple of the identity and commutes with a nonzero compact operator, T has a non-trivial hyperinvariant subspace, which is, any bounded linear operator commuting with T has a non-trivial invariant subspace
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
Summary
The invariant subspace problem has been stated by Beurling and von Neumann [1]. It can be formulated as follows. 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, T has non-trivial invariant subspace. In 1973, Lomonosov [4] proved that if T is not a scalar multiple of the identity and commutes with a nonzero compact operator, 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 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. For infinite-dimensional separable Hilbert spaces, the problem is, after a long period of time, not yet completely solved
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