Every transcendental operator has a non-trivial invariant subspace

TL;DR

This paper advances the invariant subspace problem by classifying contraction operators into three types and proving that operators in the second class possess non-trivial invariant subspaces.

Contribution

It provides a proof that contraction operators without non-trivial algebraic elements have non-trivial invariant subspaces, addressing a key case in the invariant subspace problem.

Findings

Operators in Case 3 have non-trivial invariant subspaces

Operators in Case 2 are shown to have non-trivial invariant subspaces

Remaining case (Case 1) is left as an open question

Abstract

In this paper, to solve the invariant subspace problem, contraction operators are classified into three classes ; (Case 1) completely non-unitary contractions with a non-trivial algebraic element, (Case 2) completely non-unitary contractions without a non-trivial algebraic element, or (Case 3) contractions which are not completely non-unitary. We know that every operator of (Case 3) has a non-trivial invariant subspace. In this paper, we answer to the invariant subspace problem for the operators of (Case 2). Since (Case 1) is simpler than (Case 2), we leave as a question.