Chernoff Like Counterexamples Related to Unbounded Operators

TL;DR

This paper presents specific examples of unbounded operators in functional analysis, highlighting unusual properties of their domains and adjoints, which contribute to the understanding of operator theory.

Contribution

It introduces novel examples of unbounded operators with unique domain properties, including an essentially self-adjoint operator with a trivial square domain.

Findings

Example of a closed unbounded operator with equal and trivial domains for its square and adjoint's square.

Construction of an essentially self-adjoint operator whose square's domain is trivial.

Abstract

In this paper, we give an example of a closed unbounded operator whose square's domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.