Biorthogonal Bases & Metric Structures

• Biorthogonal bases are paired sequences satisfying ⟨fₙ, gₘ⟩ = δₙₘ, establishing stable reconstructions and norm equivalence in various functional settings.
• They underpin spectral theory and operator analysis by ensuring bounded invertibility and robust metric structures in Hilbert, Krein, and quasi-Banach spaces.
• Practical applications include signal processing, quantum mechanics, and numerical methods, where recursive construction and metric operators enhance stability in computations.

A biorthogonal basis is a pair of sequences $\{f_n\}$ and $\{g_n\}$ in a (typically infinite-dimensional) vector space equipped with an inner product or more general sesquilinear form, such that $\langle f_n, g_m \rangle = \delta_{nm}$ for all indices $n, m$. These systems and their associated metric structures underpin much of modern functional analysis in both Hilbert, indefinite inner product, and quasi-Banach settings, with deep connections to spectral theory, operator theory, and applied harmonic analysis.

1. Foundational Definitions: Biorthogonality, Riesz Bases, and Norm Structures

Biorthogonal Sequences:

Two sequences $\{f_n\}$, $\{g_n\}$ in a Hilbert space $H$ are biorthogonal if

$$\langle f_n, g_m \rangle = \delta_{nm} \quad (n, m \text{ in index set}).$$

Completeness and minimality are crucial: if only one sequence is total, their expansions reconstruct any $f$ in $H$ or in a dense subspace.

Riesz Basis:

A sequence $\{f_n\}$ is a Riesz basis if it is the image of an orthonormal basis under a bounded, boundedly invertible operator $T$, i.e., $f_n = T e_n$. Equivalently, $\{f_n\}$ admits constants $A, B>0$ such that for all finite scalar sequences $\{c_n\}$,

$$A \sum |c_n|^2 \le \left\| \sum c_n f_n \right\|^2 \le B \sum |c_n|^2.$$

The biorthogonal dual $\{g_n\}$ can be constructed as $g_n = (T^{-1})^* e_n$, and satisfies $\langle f_n, g_m \rangle = \delta_{nm}$ (Zikkos, 2022, Stoeva, 2019).

Norm Equivalences:

A Riesz basis $\{f_n\}$, $\{g_n\}$, via the reconstruction

$$x = \sum_n \langle x, g_n \rangle f_n = \sum_n \langle x, f_n \rangle g_n,$$

induces an equivalent Hilbert norm: $$A \|x\|^2 \le \sum_n |\langle x, g_n \rangle|^2 \le B \|x\|^2.$$

2. Characterization Theorems and Metric Implications

Classical results and recent refinements establish crisp characterizations of Riesz bases in terms of biorthogonal systems and associated metrics:

• Equivalences: A sequence is a Riesz basis if and only if it is a complete Bessel sequence and admits a complete biorthogonal Bessel sequence (Stoeva, 2019). A substantial refinement shows that completeness is required for only one sequence: as soon as $\{f_n\}$ and $\{g_n\}$ are biorthogonal Bessel sequences and either is complete, both are automatically complete and each is a Riesz basis (Zikkos, 2022, Stoeva, 2019).
• Metric Consequences: These equivalences ensure that the corresponding synthesis (analysis) operators are bounded and invertible, and the biorthogonal pair induces two-sided norm equivalence—establishing a control of the geometry of the space by the spectral data of the biorthogonal bases.

Characterization	Requirements	Norm Equivalence
Riesz basis (both)	Each is Bessel, complete	$$A\|x\|^2 \le \sum |\langle x, g_n \rangle|^2 \le B\|x\|^2$$
Riesz basis (one-side)	Both Bessel, one complete	As above

The sharpness of these theorems underpins the modern theory of stability and reconstruction in analysis and signal processing (Zikkos, 2022, Stoeva, 2019).

3. Biorthogonality Beyond Hilbert Spaces: Krein Spaces, Quasi-Banach, and Generalizations

Krein Spaces:

In an indefinite inner product space $(\mathcal{K}, [\cdot,\cdot])$, biorthogonal sequences are defined with respect to the indefinite inner product, and Riesz bases must respect the fundamental decomposition $\mathcal{K} = \mathcal{K}^+ \dotplus \mathcal{K}^-$. A sequence $\{f_n\}$ is a Riesz basis for $(\mathcal{K}, [\cdot, \cdot])$ iff it is the image of the canonical basis under a bounded, invertible operator on each definite subspace. The metric structure is encoded in the Gram matrices $\left([f_n, f_j]\right)$ on $\mathcal{K}^+$ and $\mathcal{K}^-$, which must be boundedly invertible for a Riesz basis (Jahan et al., 2024).

Quasi-Banach and $D$-quasi Bases:

In quasi-Banach spaces or for unbounded transformations, one encounters generalized Riesz systems—biorthogonal pairs whose constructing operator may be unbounded but closed and densely defined (Bagarello et al., 2018, Inoue, 2016). In this context:

• Biorthogonal systems may only satisfy "weak" resolutions of the identity on suitable dense subspaces (the $D$-quasi basis property).
• Positive (possibly unbounded) self-adjoint metric operators $\Theta$ or $G$ can be constructed via the biorthogonal expansions, under which one sequence becomes orthonormal in the new inner product (Bagarello et al., 2018, Bagarello et al., 2017, Inoue, 2016).
• These structures are central to the study of non-self-adjoint Hamiltonians, where reality of the spectrum can be linked to the existence of a positive metric intertwining $H$ and $H^\dagger$ (Bagarello et al., 2017, Bagarello et al., 2013).

4. Operator-Theoretic Constructions and Harmonic Analysis Applications

Biisometric Operators:

A pair of operators $(V,W)$ with $V^*W = I$ generates biorthogonal sequences by iterative action on their joint kernels, yielding shift-like, biorthogonal pairs. If these sequences are total, they constitute biorthogonal (possibly Riesz) bases, and stable series expansions and norm equivalences are achieved as in the Hilbert case. Concrete applications include Laguerre-type shift operators with explicit biorthogonal expansions in $L^2$ (Kubrusly et al., 2019).

Spectral Theory on Directed Graphs:

Non-self-adjoint graph operators (e.g., the adjacency or random-walk Laplacian of a directed network) admit left/right eigenvector biorthogonal systems, which are essential for the spectral analysis, sampling, and filtering on directed graphs. The Biorthogonal Graph Fourier Transform (BGFT) and its metric structure—via weighted norms and the Gram matrix $V^*V$—quantify the stability and conditioning of spectral decompositions. Non-normality (departure from orthogonality) amplifies noise, quantified precisely by eigenbasis condition numbers and Gram-matrix metrics (Gokavarapu, 25 Dec 2025, Gokavarapu, 13 Dec 2025).

Domain	Biorthogonal Data	Metric Structure	Stability Principle
Krein spaces	$[f_n, g_m] = \pm \delta_{nm}$	Block Gram matrices $G_\pm$	Invertibility of $G_\pm$
Directed graphs	$u_k^* v_\ell = \delta_{k\ell}$	Gram $G = V^*V$, $\kappa(V)$	Conditioning in BGFT
Quantum mechanics	$\langle \phi_n, \psi_m \rangle = \delta_{nm}$	Metric operator $\Theta$	Pseudo-hermiticity

5. Greedy and Almost-Greedy Biorthogonal Bases in Geometry of Banach and Quasi-Banach Spaces

In quasi-Banach settings, biorthogonal systems (Markushevich bases) underpin the theory of greedy, quasi-greedy, and almost-greedy bases. The key metric invariants—democracy functions, unconditionality constants, and associated norm bounds—are characterized via the geometry induced by the biorthogonal expansions. Greedy-type bases admit renormings with sharp or isometric constants, although additional subtleties arise relative to the Hilbert space theory. These structures generalize classical results in approximation theory and underpin bases in Lorentz, Garling, and other intricate sequence spaces (Albiac et al., 2019).

6. Recursive and Stable Construction of Biorthogonal Duals in Finite-Dimensional Settings

Finite-dimensional biorthogonal bases can be efficiently constructed via recursive algorithms that leverage well-conditioned orthonormal polynomial bases (Legendre, Laguerre, Chebyshev, etc.), thereby sidestepping the instability of direct Gram-matrix inversion. The duals to a given monomial basis are constructed by forward recursion (basis enlargement) or rank-one downdates (basis reduction), maintaining stability and explicit metric control throughout. These methods have application in polynomial regression, projection, and numerical analysis, where the condition number of the Gram matrix typically grows prohibitively with degree (Rebollo-Neira et al., 2024).

Approach	Key Step	Stability Feature
Gram-matrix inversion	$G c = m$	Direct but ill-conditioned
Recursive construction	Use orthonormal polynomials	Numerically stable, bypasses inversion

7. Metric Operators, Quasi-Hermiticity, and Physical Interpretations

Metric operators $G$ or $\Theta$ derived from biorthogonal bases play a fundamental role in providing alternative (possibly equivalent) Hilbert or Krein-like structures, crucial in spectral theory and non-self-adjoint operator analysis. Quasi-Hermiticity, implemented via a positive metric operator, restores self-adjointness in a transformed inner product and can guarantee reality of spectra. In quantum mechanics, this supports the physical interpretability of non-Hermitian Hamiltonians, provided the required metric structures are sufficiently well-behaved (e.g., boundedness or dense definition) (Bagarello et al., 2017, Bagarello et al., 2018, Bagarello et al., 2013).

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References:

• Characterization of Riesz bases and metric equivalence via biorthogonal Riesz-Fischer and Bessel sequences (Zikkos, 2022, Stoeva, 2019)
• Biorthogonality and Gram-metric structure in Krein spaces (Jahan et al., 2024)
• Operator constructions, shift-invariant biorthogonal bases (Kubrusly et al., 2019)
• $D$-quasi bases and metric operators in non-self-adjoint quantum mechanics (Bagarello et al., 2017, Bagarello et al., 2018, Inoue, 2016, Bagarello et al., 2013)
• Greedy-type bases and metric invariants in quasi-Banach spaces (Albiac et al., 2019)
• Recursive algorithms for biorthogonal duals and polynomial regression (Rebollo-Neira et al., 2024)
• Biorthogonal bases, metric conditioning, and harmonic analysis on directed networks (Gokavarapu, 25 Dec 2025, Gokavarapu, 13 Dec 2025)