On the Kato square root problem

TL;DR

This paper constructs new examples of unbounded operators in Hilbert spaces where the domain of the square root differs from that of its adjoint, and establishes criteria for when these domains are equal.

Contribution

It introduces novel examples of sectorial operators with domain discrepancies and provides new criteria for the equality of these domains.

Findings

Constructed new abstract examples of sectorial operators with domain differences.

Established criteria for the equality of domains of square roots of operators and their adjoints.

Abstract

In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators $B$ for which ${\rm dom\,}B^{\frac{1}{2}}\ne{\rm dom\,}B^{*{\frac{1}{2}}}$. New criterions for the equality are established.