Every transcendental operator has a non-trivial invariant subspace

Every bounded linear operator T on a separable Hilbert space H of dimension ≥ 2 over C has a non-trivial invariant subspace.

Since the problem is solved for all finite dimensional complex vector spaces of dimension at least 2, in this note, H denotes a separable Hilbert space whose dimension is infinite. It is enough to think for a contraction T , i.e., T ≤ 1 on H. Thus, in this note, T denotes a contraction.

If T is a contraction, then (Case 1) T is a completely non-unitary contraction with a non-trivial algebraic element, or (Case 2) T is a transcendental operator ; that is, T is a completely non-unitary contraction without a non-trivial algebraic element, or (Case 3) T is not completely non-unitary.

In this note, we discuss the invariant subspace problem for operators of (Case 2). By using fundamental properties (Proposition 2.2, Proposition 2.3, and Corollary 2.4) of transcendental operators, we answer to the invariant subspace problem for the operators of (Case 2) in Theorem 2.5;

Every transcendental operator defined on a separable Hilbert space H has a non-trivial invariant subspace.

Thus, we answered to the invariant subspace problem for the (Case 2) in Theorem 2.5 and, clearly, we know that every operator of (Case 3) has a nontrivial invariant subspace. Thus, to answer to the invariant subspace problem, it suffices to answer for (Case 1).

We do not consider project (Case 1) in this note, and leave as a question;

Question. Let T ∈ L(H) be a completely non-unitary contraction such that T has a non-trivial algebraic element. Then, does the operator T have a non-trivial invariant subspace?

The author would like to appreciate the advice of Professor Ronald G. Douglas.

In this note, C, M and L(H) denote the set of complex numbers, the (norm) closure of a set M , and the set of bounded linear operators from H to H where H is a separable Hilbert space whose dimension is not finite, respectively.

For a set A = {a i : i ∈ I} ⊂ H, A denotes the closed subspace of H generated by {a i : i ∈ I}.

If T ∈ L(H) and M is an invariant subspace for T , then T |M is used to denote the restriction of T to M , and σ(T ) denotes the spectrum of T .

1.1. A Functional Calculus. Let H ∞ be the Banach space of all (complexvalued) bounded analytic functions on the open unit disk D with supremum norm [4]. A contraction T in L(H) is said to be completely non-unitary provided its restriction to any non-zero reducing subspace is never unitary.

B. Sz.-Nagy and C. Foias introduced an important functional calculus for completely non-unitary contractions.

Proposition 1.1. Let T ∈ L(H) be a completely non-unitary contraction. Then there is a unique algebra representation Φ T from H ∞ into L(H) such that :

We simply denote by u(T ) the operator Φ T (u). B. Sz.-Nagy and C. Foias [4] defined the class C 0 relative to the open unit disk D consisting of completely non-unitary contractions T on H such that the kernel of Φ T is not trivial.

is a weak * -closed ideal of H ∞ , and hence there is an inner function generating ker Φ T . The minimal function m T of an operator T of class C 0 is the generator of ker Φ T ; that is, ker Φ T = m T H ∞ . Also, m T is uniquely determined up to a constant scalar factor of absolute value one [1].

1.2. Algebraic Elements. In this section, we provide the notion of algebraic elements for a completely non-unitary contraction T in L(H). Definition 1.2. [3] Let T ∈ L(H) be a completely non-unitary contraction. An element h of H is said to be algebraic with respect to T provided that θ(T )h = 0 for some θ ∈ H ∞ \ {0}. If h = 0, then h is said to be a non-trivial algebraic element with respect to T .

If h is not algebraic with respect to T , then h is said to be transcendental with respect to T .

If T is a contraction, then (Case 1) T is a completely non-unitary contraction with a non-trivial algebraic element, or (Case 2) T is a completely non-unitary contraction without a non-trivial algebraic element; that is, every non-zero element in H is transcendental with respect to T , or (Case 3) T is not completely non-unitary.

). To answer to the invariant subspace problem for (Case 2), we provide the following definition; Definition 2.1. If T is a completely non-unitary contraction without a nontrivial algebraic element; that is, every non-zero element in H is transcendental with respect to T , then T is said to be a transcendental operator.

Proof. Suppose that θ(T ) is not one-to-one for a function θ ∈ H ∞ \ {0}. Then, there is a non-zero element h in H such that θ(T )h = 0; that is, h is a non-trivial algebraic element with respect to T . This, however, is a contradiction, since T is a transcendental operator. Thus, θ(T ) is one-to-one for any θ ∈ H ∞ \ {0}.

Recall that an arbitrary subset M of H is said to be linearly inde