[Paper Review] Spectral Radii of Bounded Operators on Topological Vector Spaces

Vladimir G. Troitsky|ArXiv.org|Apr 8, 2000

Advanced Banach Space Theory16 references20 citations

TL;DR

This paper develops a generalized spectral theory for bounded linear operators on topological vector spaces (TVS), extending the Gelfand formula for spectral radius and Neumann series convergence beyond Banach spaces. It establishes that spectral radii and spectra depend on operator boundedness classes (nb, nn, bb, etc.), with the spectral radius in each class defined via the Gelfand formula and Neumann series converging when |λ| exceeds the spectral radius.

ABSTRACT

In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.

Motivation & Objective

• To extend classical spectral theory—previously limited to Banach spaces—onto general topological vector spaces (TVS), where standard tools like norms and completeness are absent.
• To address the lack of developed spectral theory for bounded operators on TVS, especially for applications in the Invariant Subspace Problem.
• To define and analyze multiple spectra and spectral radii based on different boundedness classes (nb, nn, bb, continuous, linear) of operators on TVS.
• To establish conditions under which the Neumann series converges in TVS, generalizing the classical result from Banach spaces.
• To prove that spectral radius equals the geometric spectral radius for nb-bounded and compact operators in sequentially complete locally convex spaces.

Proposed method

• Introduces five distinct classes of bounded operators on TVS: nb-bounded, nn-bounded, bb-bounded, continuous, and linear, forming a nested hierarchy.
• Defines the spectrum of an operator in each class as the set of λ for which λI − T is not invertible within that class.
• Generalizes the Gelfand formula: spectral radius r(T) = limₙ→∞ ||Tⁿ||^(1/n) for each boundedness class, using operator seminorms adapted to the class.
• Uses mixed seminorms and convergence in topology (X,U) induced by zero neighborhoods to define convergence of series and resolvents.
• Applies the Neumann series ∑ₙ₌₀^∞ Tⁿ/λⁿ⁺¹ to define the resolvent operator R(λ;T) when |λ| > r(T) in a given class.
• Employs closed extension arguments and density of domains to relate resolvents on dense subdomains to full operators on complete spaces.

Experimental results

Research questions

• RQ1 Can the Gelfand formula for spectral radius be generalized from Banach spaces to bounded operators on general topological vector spaces?
• RQ2 How do multiple spectra and spectral radii arise in the context of different boundedness classes on TVS?
• RQ3 Under what conditions does the Neumann series converge in a topological vector space, and how does this relate to the spectral radius?
• RQ4 Is the spectral radius equal to the geometric spectral radius (supremum of |λ| for which λI − T is not invertible) in the TVS setting?
• RQ5 Do the spectral radii coincide for nb-bounded and compact operators in sequentially complete locally convex spaces?

Key findings

• The Gelfand formula for spectral radius is generalized to all five boundedness classes (nb, nn, bb, continuous, linear) on topological vector spaces.
• For a continuous operator on a sequentially complete locally convex space, if |λ| > r(T) in any boundedness class, then the Neumann series ∑ₙ₌₀^∞ Tⁿ/λⁿ⁺¹ converges in the topology of that class and λ is not in the spectrum.
• The spectral radius in each boundedness class is greater than or equal to the geometric spectral radius, with equality holding for nb-bounded and compact operators.
• The resolvent operator R(λ;T) can be extended from a dense domain D to the full space X when T is the smallest closed extension of its restriction to D.
• For an operator T on a dense domain D, if λ is in the nn-resolvent set of T|D, then λ is in the resolvent set of T, provided T is the smallest closed extension.
• The spectral radius of the resolvent R(λ;T) on D satisfies rₙb(R|D) ≤ r(R), linking spectral radii across different classes.