Dichotomy between operators acting on finite and infinite dimensional Hilbert spaces

TL;DR

This paper illustrates the stark differences in the behavior of operators on finite versus infinite dimensional Hilbert spaces, highlighting the increased complexity in the infinite case.

Contribution

It provides illustrative examples demonstrating the contrasting properties of operators in finite and infinite dimensional Hilbert spaces.

Findings

Operators behave drastically differently in finite and infinite dimensions.

Infinite dimensional operators exhibit more complex and less intuitive behavior.

The paper clarifies the challenges in understanding infinite dimensional operators.

Abstract

In this expository article, we give several examples showing how drastically different can be the behavior of operators acting on finite versus infinite dimensional Hilbert spaces. This essay is written as in such a friendly-reader to show that the situation in the infinite dimensional setting is trickier than the finite one.