# On subspace convex-cyclic operators

## TL;DR

This paper introduces the concept of subspace convex-cyclicity for operators on infinite-dimensional Hilbert spaces, providing conditions for such operators and exploring their relation to invariant subspace problems.

## Contribution

It defines subspace convex-cyclicity, establishes sufficient conditions for transitive operators to be subspace convex-cyclic, and presents a counterexample illustrating the concept's nuances.

## Key findings

- Provided a sufficient condition for subspace convex-cyclicity
- Related subspace convex-cyclicity to a specialized Kitai criterion
- Constructed a counterexample of a non-transitive subspace convex-cyclic operator

## Abstract

Let $\mathcal{H}$ be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator $T$ and its important relation with invariant subspace problem on $\mathcal{H}$: operator $T$ is said to be is subspace convex-cyclic for a subspace $\mathcal{M}$, if there exists a vector whose orbit under $T$ intersects the subspace $\mathcal{M}$ in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator $T$ to be subspace convex-cyclic. We also give a special type of Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. We conclude showing a counterexample of a subspace convex-cyclic operator which is not subspace convex-cyclic transitive.