Diagonalizable Locally Bounded Operators

• Diagonalizable locally bounded operators are operators on infinite-dimensional spaces that decompose as pointwise scalar multiplication across a measurable field of Hilbert spaces.
• They play a critical role in spectral theory by facilitating explicit computations of spectral radii and functional calculus through direct integral decompositions.
• Their structure underpins abelian von Neumann algebras, linking harmonic analysis, quantum mechanics, and noncommutative geometry in practical operator algebra applications.

A diagonalizable locally bounded operator is a locally bounded operator—typically acting on an infinite-dimensional space such as a Banach space, Hilbert space, or a direct integral of locally Hilbert spaces—that admits a representation through a basis or a decomposition in which its action is diagonal in a precise technical sense. This concept connects spectral theory, operator algebras, and the geometry of Banach and Hilbert spaces, and finds rigorous elaboration in the contexts of unbounded operators, direct integrals, and noncommutative operator algebras. The diagonalizability property intertwines with local boundedness, spectral decompositions, commutant structures, and the realization of abelian von Neumann subalgebras.

1. Precise Definitions and Structural Features

A locally bounded operator T, acting on a space with appropriate topological or measure-theoretic structure (e.g., the direct integral of locally Hilbert spaces), is diagonalizable if, after possibly passing to a dense domain or a direct integral decomposition, there exists an orthonormal basis or decomposition in which T acts as multiplication by a scalar function on each “fiber.” Explicitly, for a direct integral

$$\mathcal{D} = \int^{\oplus}_{\text{loc}(X)} \mathcal{D}_p \, d\mu(p)$$

of locally Hilbert spaces over a locally measure space (X, μ), a locally bounded operator T is called diagonalizable if there is a measurable function $f : X \to \mathbb{C}$ such that

$$(Tu)(p) = f(p) \cdot u(p) \quad \text{for } \mu\text{-almost every } p \in X,\; u \in \mathcal{D}$$

(Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024). In this setting, “diagonalizable” refers to pointwise scalar multiplication, generalizing classical diagonalization for bounded operators on Hilbert spaces.

Diagonalizable locally bounded operators form an abelian (locally) von Neumann algebra, and, crucially, are characterized as the commutant of the algebra of decomposable locally bounded operators. Decomposable locally bounded operators are operators of the form

$$(Tu)(p) = T_p u(p), \quad \text{where } T_p\in B(\mathcal{D}_p)$$

with $\{T_p\}$ a measurable field of bounded operators.

Table 1: Operator Classes in Direct Integral Setting

Operator Class	Form	Algebraic Structure
Diagonalizable locally bounded	$(Tu)(p) = f(p)u(p)$	Abelian (locally) von Neumann alg.
Decomposable locally bounded	$(Tu)(p) = T_p u(p)$	(Locally) von Neumann alg.

Every diagonalizable operator is decomposable, but not conversely. The commutant relationship is:

$$\mathrm{DIAG} = \mathrm{DEC}' \qquad \mathrm{DEC} = \mathrm{DIAG}'$$

(Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).

2. Spectral and Functional Analytic Context

Diagonalizable locally bounded operators are intrinsically linked to spectral theory. In various analytic or topological frameworks (e.g., Banach spaces, Hilbert spaces, direct integrals) their defining property is that each vector decomposes into spectral components on which the operator acts by multiplication. This enables explicit computation of spectral radii, functional calculus, and provides strong control over local spectra.

For unbounded or locally diagonalizable operators arising as differential operators (e.g., constant coefficient differential operators, Jacobi, Hermite, and Laguerre operators), a diagonalizing (Fourier-type) transform maps the operator’s domain isomorphically onto a sequence space or function field where the action is diagonal and the local spectrum of T at $x$ is explicitly determined by the eigenvalues present in the transform support:

$$\lim_{n\to\infty} \|T^n x\|^{1/n} = \sup_{\alpha \in \mathrm{supp}(F(x))} |\varepsilon(\alpha)|$$

(Andersen et al., 2010). This “local spectral radius formula” extends the classical spectral radius theorem to diagonalizable, possibly unbounded, locally bounded operators.

3. Algebraic and Operator Algebra Framework

The algebra of diagonalizable locally bounded operators is a key example of an abelian locally von Neumann algebra. Structurally, for a direct integral of locally Hilbert spaces, every diagonalizable locally bounded operator corresponds to multiplication by a locally essentially bounded measurable function (see (Kulkarni et al., 2 Sep 2024), Section 3). Under suitable projective and inductive limit conditions, this provides a one-to-one correspondence between abelian locally von Neumann algebras and the algebra of all diagonalizable locally bounded operators arising from direct integral decompositions.

In terms of double commutant characterizations: if $\mathcal{M}_{\mathrm{DEC}}$ is the (locally) von Neumann algebra of decomposable operators, and $\mathcal{M}_{\mathrm{DIAG}}$ that of diagonalizable ones, then

$$\mathcal{M}_{\mathrm{DIAG}} = \mathcal{M}_{\mathrm{DEC}}' \quad \text{and} \quad \mathcal{M}_{\mathrm{DEC}} = \mathcal{M}_{\mathrm{DIAG}}'$$

(Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).

4. Examples and Model Constructions

Several canonical examples elucidate these concepts:

• Constant coefficient differential operators on the torus: via the Fourier transform, the operator is diagonal on the sequence space indexed by frequencies, with eigenvalues determined by the polynomial evaluated at the frequencies (Andersen et al., 2010).
• Direct sum examples: For the direct sum of countably many locally Hilbert spaces, a diagonalizable locally bounded operator acts as multiplication by a sequence $\{f(n)\}$:

$$T = \begin{bmatrix} f(1) & 0 & 0 & \cdots \ 0 & f(2) & 0 & \cdots \ 0 & 0 & f(3) & \cdots \ \vdots & \vdots & \vdots & \ddots \end{bmatrix}$$

(Kulkarni et al., 5 Aug 2025).

• Direct integral models: In representation theory, spectral theory, and the theory of operator algebras, diagonalizable locally bounded operators are modeled by multiplication operators on the fibers in the direct integral decomposition (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025).

5. Role in Noncommutative Analysis and Commutant Structures

The relationship between diagonalizable and decomposable locally bounded operators mirrors classic results from von Neumann algebra theory and direct integral decompositions. In both the classical and locally Hilbert space contexts, diagonalizable operators parameterize the spectrum of more general decomposable operators (serving as “coordinate functions”), and their mutual commutant relationship underpins the structure of operator algebras on these spaces (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025).

For locally bounded operator algebras (in the sense of inductive limits or projective systems), these commutant relationships provide tools for classifying abelian subalgebras, understanding spectral decompositions, and constructing functional calculi.

6. Applications and Broader Perspectives

Diagonalizable locally bounded operators form the backbone of spectral theory in several fields, including:

• Harmonic analysis: Direct integral decompositions in representation theory and Fourier analysis (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025).
• Quantum mechanics and local quantum field theory: For systems where underlying spaces are best modeled by inductive limits or variable local dimension, the abelian structure and spectral decomposition provided by diagonalizable locally bounded operators facilitate the construction of physically meaningful observables and the paper of spectral invariants.
• Noncommutative geometry and operator algebraic dynamics: Through their appearance as commutants and their role in double commutant theorems, they help elucidate structure and duality in operator algebras arising from group actions and representations.

A plausible implication is that, in general, the machinery developed for diagonalizable locally bounded operators in the context of direct integrals and locally Hilbert spaces can be transplanted—with appropriate modifications—to the paper of operator algebras on more general topological and measure-theoretic structures, offering new avenues for the analysis of non-separable or non-classically decomposable modules.

7. Generalizations, Limitations, and Further Directions

The explicit identification of the algebra of all diagonalizable locally bounded operators with the space of (locally) essentially bounded measurable functions establishes a complete duality between operator-theoretic and function-theoretic perspectives in the direct integral of locally Hilbert spaces (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025). However, limitations arise when considering extensions to nonabelian or more general classes of locally bounded operators, or when working outside the context of strictly inductive/projective limit structures.

Further generalizations, such as in the context of type I and II von Neumann algebras, reveal that the diagonalizable (or scalar-type) part of a locally bounded operator can, in general, be unbounded even if the operator itself is bounded (Nayak et al., 5 May 2025). This suggests that in noncommutative settings, the pursuit of a robust “Jordan–Chevalley” decomposition necessitates careful extension of the framework to include unbounded affiliated operators.

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In summary, diagonalizable locally bounded operators represent the abelian, spectral-theoretic core within the broad family of locally bounded operators in functional analysis and operator algebra. Their definition, structure, and algebraic properties provide a foundational tool for spectral decompositions, harmonic analysis, and the characterization of operator algebras over locally convex and locally Hilbert spaces. This theory generalizes, unifies, and extends spectral and diagonalization techniques to settings accommodating locally variable topology, measure, and algebraic structure.