Riesz-Spectral Operator

• Riesz-spectral operator is a closed linear operator in a Hilbert space whose complete system of eigenvectors forms a Riesz basis, enabling unconditional expansions.
• It underpins spectral analysis by facilitating resolvent, Green function, and spectral projector constructions for boundary value problems in functional analysis and PDEs.
• Applications include the Dirac operator with summable potentials, where regularity conditions ensure eigenvalue stability and effective spectral decompositions.

A Riesz-spectral operator is a closed linear operator in a Hilbert space whose complete system of eigenvectors and associated vectors (root vectors) forms a Riesz basis (or, in certain cases, a Riesz basis with parentheses—basis of subspaces) for the space. This notion arises in spectral analysis, particularly for operators such as the Dirac operator with summable potential and regular boundary conditions, where spectral properties ensure the unconditional stability and completeness of expansions in root vectors. The characterization of Riesz-spectrality underpins the analysis of boundary value problems and facilitates resolvent and spectral projector constructions essential for functional analysis and partial differential equations.

1. Definition and Riesz Basis Notions

A sequence $\{\varphi_n\}$ in a Hilbert space $\mathbb H$ is a Riesz basis if it is the image of an orthonormal basis under a boundedly invertible operator $A:\mathbb H \to \mathbb H$. Equivalently, there exist $0 < m \leq M < \infty$ such that for any finite sequence of scalars $\{c_n\}$,

$$m \sum |c_n|^2 \leq \left\|\sum c_n \varphi_n\right\|^2 \leq M \sum |c_n|^2.$$

A Riesz basis of subspaces (basis with parentheses) is a family of finite-dimensional subspaces $\{\mathbb H_n\}_{n\in \mathbb Z}$ for which there exists a bounded invertible $A$ such that $\{A(\mathbb H_n)\}$ forms an orthogonal decomposition. Gelfand's theorem asserts this condition is equivalent to uniform boundedness of finite sums of spectral projectors:

$$\sup_{\text{finite } J} \left\| \sum_{n\in J} P_n \right\| < \infty.$$

A closed operator $T$ is called Riesz-spectral if its system of root vectors (eigen- and associated vectors) forms a Riesz basis (or Riesz basis with parentheses) for $\mathbb H$ (Savchuk et al., 2015).

2. The Dirac Operator with Summable Potential

The Dirac operator

$$\ell_P(\mathbf{y}) = B\mathbf{y}' + P(x)\mathbf{y},$$

in $\mathbb H = (L_2[0,\pi])^2$, acts with $$B = \begin{pmatrix} -i & 0 \ 0 & i \end{pmatrix}$$ and $$P(x) = \begin{pmatrix} p_1(x) & p_2(x) \ p_3(x) & p_4(x) \end{pmatrix}$$, where all $p_j \in L^1[0,\pi]$ (summable complex-valued functions). Boundary conditions are imposed as:

$$U(\mathbf{y}) = C \mathbf{y}(0) + D \mathbf{y}(\pi) = 0,$$

with $C, D$ being $2 \times 2$ blocks forming matrix $\mathcal{J}$ with linearly independent rows.

Birhoff regularity requires

$J_{14} J_{23} + J_{13} J_{24} \neq 0,$

where $J_{\alpha\beta}$ are $2 \times 2$ minors of $\mathcal{J}$ (columns $\alpha$, $\beta$). Strict regularity further requires

$(J_{12} + J_{34})^2 + 4 J_{14} J_{23} \neq 0.$

By gauge transformation, $p_1=p_4=0$ is assumed, moving analysis to $$P(x) = \begin{pmatrix} 0 & p_2(x) \ p_3(x) & 0 \end{pmatrix}$$.

3. Spectral Properties and Asymptotics

The unperturbed operator $\mathcal{L}_{0,U}$ with $P \equiv 0$ possesses purely discrete spectrum: two arithmetic progressions,

$$\lambda_n^0 = \begin{cases} x_0 + 2n, & n \text{ even} \ x_1 + 2n, & n \text{ odd} \end{cases}$$

$x_0, x_1$ are roots of $J_{23} z^2 - (J_{12} + J_{34}) z - J_{14} = 0$, arranged so that $$-1 < \operatorname{Re} x_0 < \operatorname{Re} x_1 + 1 \leq 1$$.

For perturbed operator $\mathcal{L}_{P,U}$, eigenvalues $\lambda_n$ satisfy

$$\lambda_n = \lambda_n^0 + o(1), \qquad |n| \to \infty,$$

where $\Delta(\lambda)$, the characteristic determinant, is asymptotically close to its unperturbed counterpart:

$\Delta(\lambda) = \Delta_0(\lambda) + o(1)$

in strips $|\operatorname{Im} \lambda| \leq a$, which implies the eigenvalues' stability under perturbation (Savchuk et al., 2015).

4. Green Function, Resolvent, and Spectral Projectors

The resolvent $(\mathcal{L}_{P, U} - \lambda I)^{-1}$ is encapsulated by the Green kernel,

$$(R(\lambda)f)(x) = \int_0^\pi G(t, x, \lambda) f(t) dt,$$

where for $t<x$,

$$G(t, x, \lambda) = E(x, \lambda) M(\lambda)^{-1} C E(t, \lambda)^{-1} B^{-1},$$

with $E(x, \lambda)$ the fundamental matrix solution. Uniform bounds hold for $|G(t, x, \lambda)|$ off the eigenvalues.

Spectral projectors $P_n$ onto root subspaces are constructed via contour integrals around eigenvalues:

$$P_n = -\frac{1}{2\pi i} \oint_{|\lambda - \lambda_n| = \delta} R(\lambda) d\lambda,$$

where $\lambda_n$ is a simple eigenvalue (strictly regular case) or a double eigenvalue (regular but non-strict case), and for the latter, projectors onto paired two-dimensional root subspaces are similarly defined.

5. Completeness, Minimality, and Bessel Inequalities

Completeness and minimality are established for the global system $\{y_n\}$ of root vectors and its biorthogonal system $\{z_n\}$. Orthogonality to all $y_n$ implies triviality via Liouville's theorem applied to the resolvent scalar function, and minimality arises from biorthogonal construction in each root subspace.

In the strictly regular case, eigenfunction asymptotics yield

$y_n(x) = y_n^0(x) + r_n(x), \qquad \|r_n\| \to 0,$

with

$$y_n(x) = e^{\pm i\lambda_n x} \tau_n(x), \quad |\tau_n(0)| \leq C, \quad |\tau_n'(x)| \leq C \left(|p_2(x)| + |p_3(x)|\right).$$

Bessel (Riesz) inequalities guarantee unconditional expansion:

$$\sum_{n\in\mathbb{Z}} \left| \int_0^\pi f(x) e^{i\lambda_n x} dx \right|^2 \leq C \|f\|^2, \quad f \in L^2[0,\pi].$$

If a family $\{\varphi_n\}$ is Bessel and $\tau_n(x)$ are absolutely continuous with suitable bounds, then $\{\varphi_n \cdot \tau_n\}$ is also Bessel.

6. Strictly Regular vs. Regular but Non-Strict Cases

If the boundary conditions are strictly regular, eigenvalues are asymptotically simple and both $\{y_n\}$ and $\{z_n\}$ are Bessel systems. Bari’s theorem applies: completeness, minimality, and Bessel property for both systems imply that they form a Riesz basis for $\mathbb{H}$.

When boundary conditions are regular but not strictly regular, all unperturbed eigenvalues are double. The perturbed spectrum remains clustered in pairs, and two-dimensional root subspaces $\mathbb{H}_n$ emerge, each with suitably defined projectors. The family $\{\mathbb{H}_0\} \cup \{\mathbb{H}_n\}_{|n|>N}$ forms a Riesz basis of subspaces ("basis with parentheses") in $\mathbb{H}$. Uniform boundedness of finite sums of projectors is established via interpolation-type arguments and perturbative decompositions.

7. Summary of Riesz-Spectral Operator Structure

Under Birhoff-regular boundary conditions $U(\mathbf{y}) = 0$, the Dirac operator $\mathcal{L}_{P, U}$ has purely discrete spectrum $\{\lambda_n\}$ with asymptotic relations $\lambda_n \to \lambda_n^0 + o(1)$. In strictly regular cases, normalized eigenvectors form a Riesz basis in $\mathbb{H}$; in regular cases without strictness, the two-dimensional root spaces form a Riesz basis with parentheses. Consequently, $\mathcal{L}_{P, U}$ is a Riesz-spectral operator in the sense that all spectral projectors onto individual or paired root subspaces are uniformly bounded, and the total system of root vectors fulfills the Riesz basis property (or its subspace extension) (Savchuk et al., 2015).