[Paper Review] Products of Unbounded Normal Operators

Mohammed Hichem Mortad|arXiv (Cornell University)|Feb 28, 2012

Holomorphic and Operator Theory12 references3 citations

TL;DR

This paper investigates the normality of products of unbounded normal operators, extending Kaplansky's bounded operator results to the unbounded setting. It establishes that for a bounded unitary operator $ A $ and an unbounded normal operator $ B $, $ BA $ is normal if and only if $ AB $ is normal, underpinned by the Fuglede-Putnam theorem and properties of closed, subnormal, and hyponormal operators.

ABSTRACT

The present paper partly constitutes an "unbounded" follow-up of a paper by I. Kaplansky dealing with bounded products of normal operators. Results on the normality of unbounded products are also included.

Motivation & Objective

• To extend Kaplansky’s bounded operator results on normal products to the unbounded setting.
• To investigate conditions under which the product $ AB $ or $ BA $ of an unbounded normal operator $ B $ and a bounded operator $ A $ is normal.
• To establish analogues of classical normality criteria for unbounded operators using the Fuglede-Putnam theorem and closedness properties.
• To examine the role of commutativity, hyponormality, and subnormality in ensuring normality of unbounded operator products.
• To resolve the failure of converse implications in unbounded settings, as shown by counterexamples.

Proposed method

• Utilizes the Fuglede-Putnam theorem for unbounded normal operators to relate $ A^{*}AB \subset BA^{*}A $ to normality of $ BA $.
• Applies the polar decomposition $ A = UR $, where $ U $ is unitary and $ R $ is positive, to reduce the problem to unitary transformations.
• Employs the maximally symmetric (MS) property of self-adjoint operators to conclude equality of operators from inclusion under closedness.
• Uses the closedness of $ AB $, $ BA $, and $ B $ to ensure domain and graph consistency in operator inclusions.
• Applies Theorem 4 from Stochel (2003) on closed hyponormal/subnormal operators to deduce normality under kernel conditions.
• Constructs a counterexample showing that $ A^{*}AB \subset BA^{*}A $ does not imply normality of $ BA $ when $ B $ is unbounded.

Experimental results

Research questions

• RQ1 Under what conditions is the product $ BA $ normal when $ B $ is an unbounded normal operator and $ A $ is a bounded operator?
• RQ2 Does the implication $ A^{*}AB \subset BA^{*}A \Rightarrow BA \text{ normal} $ hold in the unbounded setting, as in the bounded case?
• RQ3 Can the normality of $ AB $ be characterized in terms of $ A^{*}AB \subset BA^{*}A $ when $ A $ is unitary and $ B $ is unbounded normal?
• RQ4 What role does the unitarity of $ A $ play in ensuring symmetry between the normality of $ AB $ and $ BA $?
• RQ5 How do hyponormality and subnormality of $ BA $ interact with the condition $ A^{*}AB \subset BA^{*}A $ to imply normality?

Key findings

• The converse of the implication $ BA \text{ normal} \Rightarrow A^{*}AB \subset BA^{*}A $ does not hold in the unbounded case, as demonstrated by a counterexample.
• For a bounded unitary operator $ A $ and an unbounded normal operator $ B $, $ BA $ is normal if and only if $ AB $ is normal.
• When $ A $ is unitary and $ BA $ is hyponormal or subnormal, the condition $ A^{*}AB \subset BA^{*}A $ implies that $ BA $ is normal.
• The proof relies on the polar decomposition $ A = UR $, unitary invariance, and the MS-property of self-adjoint operators to establish equality of operators from inclusion.
• The Fuglede-Putnam theorem is extended in spirit to unbounded settings, enabling the transfer of normality properties across operator products.
• The closedness of $ AB $, $ BA $, and $ B $ ensures that domain inclusions can be manipulated rigorously to derive normality conditions.