[Paper Review] A Contribution to the Fong-Tsui Conjecture Related to Self-adjoint Operators

Mohammed Hichem Mortad|arXiv (Cornell University)|Aug 21, 2012

Spectral Theory in Mathematical Physics3 references3 citations

TL;DR

This paper investigates the Fong-Tsui conjecture, which posits that a bounded operator T with |T| ≤ |Re T| must be self-adjoint. The author proves the conjecture holds under specific conditions—such as when T commutes with the unitary part of Re T's polar decomposition or when T belongs to the Θ-class (including normal and isometric operators)—and provides a counterexample for unbounded operators, showing the conjecture does not extend to that setting.

ABSTRACT

We are interested in an open question raised by Fong-Tsui (dating back to the beginning of the eighties of last century) as to whether a bounded operator whose absolute value is less than the absolute value of its real part is self-adjoint. The analogue in the unbounded operators setting is also treated.

Motivation & Objective

• To investigate the validity of the Fong-Tsui conjecture: if |T| ≤ |Re T|, then T is self-adjoint.
• To determine whether the conjecture holds for bounded operators beyond finite-dimensional or compact cases.
• To explore the extension of the conjecture to unbounded operators, given domain complications in such settings.
• To identify sufficient conditions under which |T| ≤ |Re T| implies self-adjointness in bounded operator theory.
• To resolve the conjecture by providing new classes of operators for which it holds, and to test its limits via counterexamples.

Proposed method

• Uses the polar decomposition of Re T = U|Re T|, where U is unitary, and assumes UT = TU* to derive structural properties of T.
• Applies Fuglede's theorem to deduce that U* commutes with T, enabling the transformation of the norm inequality into a form suitable for Theorem 2.
• Employs Theorem 2 (if |T| ≤ Re T, then T is positive) after transforming the inequality via unitary conjugation.
• Leverages the Θ-class condition: T+T* commutes with T*T, which implies Re T commutes with |T|, leading to |T|² ≤ (Re T)².
• Applies Theorem 1 (if |T|² ≤ (Re T)², then T is self-adjoint) to conclude self-adjointness under Θ-class assumptions.
• Constructs an unbounded counterexample by defining T = S − S on a proper dense domain D(S) ⊂ H, showing T is symmetric but not self-adjoint, with |T| = |Re T|.

Experimental results

Research questions

• RQ1 Does the Fong-Tsui conjecture hold for all bounded operators satisfying |T| ≤ |Re T|?
• RQ2 Under what additional conditions does |T| ≤ |Re T| imply that T is self-adjoint?
• RQ3 Can the conjecture be extended to unbounded operators?
• RQ4 What structural properties of T (e.g., commutativity with unitary part of Re T) ensure the implication holds?
• RQ5 Are there unbounded operators for which |T| ≤ |Re T| but T is not self-adjoint?

Key findings

• The conjecture holds if T commutes with the unitary part U of Re T’s polar decomposition (i.e., UT = TU*), leading to T being self-adjoint.
• The conjecture is true for all operators in the Θ-class, including normal and isometric operators, under the condition |T| ≤ |Re T|.
• For Θ-class operators, |T| ≤ |Re T| implies |T|² ≤ (Re T)², which by Theorem 1 guarantees self-adjointness.
• The conjecture fails for unbounded operators: a counterexample is constructed where T = S − S on a proper dense domain, with |T| = |Re T| but T not self-adjoint.
• The counterexample shows that T is symmetric but not closed, and D(T*) = H, so T is not self-adjoint despite satisfying |T| = |Re T|.
• The result highlights the critical role of domain restrictions in unbounded operator theory, where norm inequalities alone do not imply self-adjointness.