Interpolation Conditions for Linear Operators

• Interpolation Conditions for Linear Operators are criteria based on Gram matrices and spectral bounds that guarantee the existence of an operator matching specified input–output relations.
• They leverage convex optimization, polar decomposition, and semidefinite programming to simplify complex interpolation problems into symmetric and isometric components.
• These frameworks underpin key applications in performance estimation, model reduction, and function space theory, offering actionable insights for operator approximation.

Interpolation conditions for linear operators are analytical characterizations—typically in terms of scalar product data, Gram matrices, or operator-theoretic constraints—which are necessary and sufficient to ensure the existence of a linear operator (or a class thereof) interpolating given finite sets of input–output relations, with or without additional spectral structure. These conditions underpin tractable formulations for operator approximation, performance estimation, and control, integrating areas such as convex optimization, operator theory, and interpolation of function spaces.

1. Gram Matrix-Based Interpolation and Necessity/Sufficiency

Interpolation conditions for real (or complex) finite-dimensional linear operators are most effectively expressed via convex constraints on Gram matrices of the specified vectors. For the class $L_{[\sigma_\text{min}, \sigma_\text{max}]}$ of $m \times n$ real matrices $M$ whose singular values $\sigma_i(M)$ lie in $[\sigma_\text{min}, \sigma_\text{max}]$, given two sets of input–output pairs $\{(x_i, y_i)\}_{i=1}^{N_1}$ ($x_i \in \mathbb{R}^n$, $y_i \in \mathbb{R}^m$) and $\{(u_j, v_j)\}_{j=1}^{N_2}$ ($u_j \in \mathbb{R}^m$, $v_j \in \mathbb{R}^n$), there exists an interpolant $M$ such that $y_i = M x_i$, $v_j = M^T u_j$ if and only if there exist auxiliary variables $Z \in \mathbb{R}^{n \times N_1}$, $W \in \mathbb{R}^{n \times N_2}$ fulfilling the constraints (Bousselmi et al., 20 Nov 2025) (Theorem 1):

• $Y^T Y = Z^T Z$
• $Y^T U = Z^T W$
• $U^T U \succeq W^T W$
• $[X; W]^T [Z; V] = [Z; V]^T [X; W]$
• $$([Z; V] - \sigma_\text{min}[X; W])^T (\sigma_\text{max}[X; W] - [Z; V]) \succeq 0$$

This five-line system gives a complete necessary and sufficient description for Gram-data-based interpolation in terms of operator spectral intervals.

A similar but more specialized necessary and sufficient characterization holds for symmetric linear operators $Q$ with spectrum in $[\lambda_\text{min}, \lambda_\text{max}]$:

• $X^T Y = Y^T X$ and
• $$(Y - \lambda_{\min} X)^T (\lambda_{\max} X - Y) \succeq 0$$ (Bousselmi et al., 2023, Bousselmi et al., 20 Nov 2025)

Any convex interpolation condition depending only on Gram data that is both necessary and sufficient must reduce to such spectral (singular value or eigenvalue) characterizations; no further convex scalar-product constraints are admissible (Bousselmi et al., 20 Nov 2025).

2. Polar Decomposition and Reduction to Symmetric/Isometric Interpolation

The proofs and construction of interpolation conditions exploit the operator polar decomposition: $M = R Q$ where $Q$ is positive semidefinite symmetric with spectrum in $[\sigma_{\min}, \sigma_{\max}]$, and $R$ is a partial isometry. This allows reduction of the general interpolation system to (i) a symmetric interpolation problem for $Q$ and (ii) an isometric interpolation problem for $R$ (Bousselmi et al., 20 Nov 2025).

The isometry interpolation block is fully characterized by

• $Y^T Y = Z^T Z$
• $Y^T U = Z^T W$
• $U^T U \succeq W^T W$

while the symmetric block involves the spectral constraints above.

These matrix inequalities can all be written as linear matrix inequalities and thus embedded into semidefinite programming frameworks.

3. Extensions: Spectral Unions, Gram Decompositions, and Limitations

The interpolation conditions generalize to classes of symmetric operators whose spectra are constrained to unions of intervals $\mathcal{S}_{\bigcup_k S_k}$. Here, the corresponding Gram matrix $G$ admits a decomposition $G = \sum_{k} G^k$, each $G^k$ satisfying the interval interpolation inequalities for $S_k$ (Bousselmi et al., 20 Nov 2025). This enables exact interpolation for more general spectral classes (e.g., block-diagonal or sectorial constraints).

A fundamental limitation is that, up to convex closure, the only information about classes of operators encoded in scalar products of data is the spectrum (or multiset of singular values) of the interpolant. Thus, imposing non-convex or non-spectral constraints (such as trace-norm, rank, or other nonlinear functionals) is inexpressible in a convex Gram matrix system (Bousselmi et al., 20 Nov 2025).

4. Applications: Performance Estimation, Model Reduction, and Operator Theory

Interpolation conditions are foundational in worst-case performance analysis of iterative optimization algorithms—Performance Estimation Problems (PEPs)—where such conditions are imposed as constraints in a semidefinite program for all linear operators appearing in the iteration graphs (Bousselmi et al., 2023, Bousselmi et al., 20 Nov 2025). For example, in analyzing the rate of the gradient method for $F(x) = g(Mx)$ with $M$ spectrally constrained and $g$ smooth and strongly convex, convex interpolation inequalities for $M$ and for $g$ allow semi-analytic or numerical derivation of exact convergence rates that are minimax-optimal.

Similarly, in $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems, interpolatory optimality conditions—expressed as multipoint tangential interpolation constraints—ensure reduced-order surrogates can be constructed to exactly match low-order moments (including quadratic ones) of the original system, via Petrov-Galerkin projections (Reiter et al., 5 May 2025).

In function space theory, interpolation of operators connects deeply with factorization theorems, Fredholm theory (including the necessary/sufficient index inequalities in real interpolation spaces) (Asekritova et al., 2014), and maximal regularity for evolution equations (Batty et al., 2014).

5. Complex Interpolation in Operator Theory

Complex interpolation frameworks yield operator-norm bounds via analytic families of the type $A(z) = T_2^{-z} S T_1^{-1+z}$ under appropriate spectral (sectorial, self-adjoint, bounded imaginary powers) assumptions. Main estimates interpolate operator or trace ideal norms between two endpoint spaces (Gesztesy et al., 2014):

$$\|T_2^{-x} S T_1^{-1+x}\| \leq N_1 N_2 e^{(\theta_1 + \theta_2) \sqrt{x(1-x)}} \| S T_1^{-1}\|^{1-x} \| S^* T_2^{-1} \|^x$$

for $x \in [0,1]$. These results depend on balancing domain conditions, polar decomposition, and the three-lines theorem, delivering optimal two-point interpolation bounds which are critical in spectral and index theory.

6. Function Space and Sequence Interpolation Conditions

In Banach and Hilbert space settings, classical interpolation conditions—such as the Carleson condition and Besselian/hilbertian basis properties—are both necessary and sufficient for the existence of bounded linear extension/interpolation operators for function evaluation sequences in spaces of holomorphic functions (e.g., $H^\infty(B_H)$, Hardy/Bergman spaces) (Miralles, 2015, Amar, 2019). For $H^\infty(B_H)$, the extended Carleson product condition

$$\prod_{k \neq j} PH(x_j, x_k) \geq \delta > 0, \;\; \forall j$$

ensures complete interpolability with explicit, norm-controlled linear extensions.

In $L^p$ and operator ideal contexts, the improved Calderón–Ryff interpolation theorem characterizes the lattice ideals invariant under conditional expectations as precisely those which are interpolation spaces for $(L^1,L^\infty)$ bounded operators (Mekler, 2018).

7. Interpolation and Linear Algebraic Operations

The evaluation of a function $F(A)$ of a linear operator (or matrix) $A$ can be recast as an interpolation problem once $A$ satisfies a minimal polynomial $P(A)=0$ (Khovanskii et al., 2022). By constructing the unique degree $<n$ polynomial $Q$ that matches $F$ and its derivatives at all roots of $P$ up to the algebraic multiplicities, one has $F(A)=Q(A)$. This avoids explicit Jordan form computation, and the construction is via Lagrange-Hermite interpolation polynomials.

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Summary Table: Major Theoretical Types of Interpolation Conditions for Linear Operators

Interpolation Type	Main Constraint Structure	Reference
General linear (spectral bounds)	Gram block LMIs encoding singular values interval	(Bousselmi et al., 20 Nov 2025, Bousselmi et al., 2023)
Symmetric (eigenvalue bounds)	$X^T Y = Y^T X$, $(Y - m X)^T(L X - Y) \succeq 0$	(Bousselmi et al., 2023)
Unions of intervals (spectra)	Gram decomposition into blocks for interval unions	(Bousselmi et al., 20 Nov 2025)
Complex interpolation	Analytic family bounds, three-lines theorem	(Gesztesy et al., 2014)
Fredholm/interpolation spaces	K-functional index inequalities	(Asekritova et al., 2014)
Function spaces / sequences	Carleson-type/Besselian/Hilbertian/LEP conditions	(Miralles, 2015, Amar, 2019)

These frameworks are indispensable in advanced areas of mathematical analysis, operator theory, and optimization, and they underpin both theoretical and computational advances in applied mathematics (Bousselmi et al., 20 Nov 2025, Bousselmi et al., 2023, Asekritova et al., 2014, Gesztesy et al., 2014, Reiter et al., 5 May 2025, Miralles, 2015, Amar, 2019).