Square roots of some classical operators

TL;DR

This paper characterizes the square roots of various classical operators, offering explicit formulas and comprehensive descriptions for operators like the unilateral shift, Volterra operator, and Hilbert matrix.

Contribution

It provides complete descriptions and explicit formulas for the square roots of several well-known classical operators, advancing understanding in operator theory.

Findings

Explicit formulas for square roots of classical operators

Complete descriptions of square roots for specific operators

Enhanced understanding of operator square roots in functional analysis

Abstract

In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Ces\`{a}ro operator.