Yosida Distance in Linear Operator Perturbation

• Yosida distance is a resolvent-based metric that measures perturbations between possibly unbounded linear operators in Banach spaces.
• It provides a unified framework for perturbation theory in evolution and delay equations, ensuring stability through exponential dichotomy and invariant manifold persistence.
• The metric is applied in areas such as PDEs, control systems, and numerical approximations, offering robust insights into operator convergence and stability.

The Yosida distance is a quantitative framework for measuring the "size" of perturbations between (possibly unbounded) linear operators in Banach spaces, with foundational applications in the perturbation theory of evolution equations, delay equations, and the analysis of invariant manifolds. Developed to address limitations of classical operator-norm or resolvent-based approaches, the Yosida distance enables robust authorship of continuity and stability results under conditions where domains of involved operators may have no inclusion relationship. It is particularly effective for treating unbounded or sectorial operators, especially those generating $C_0$-semigroups, and it forms a central tool in contemporary infinite-dimensional dynamical systems theory (Bui et al., 2023, Bui et al., 2023).

1. Formal Definition and Yosida Approximations

Let $X$ be a Banach space, and let $U, V$ be linear (possibly unbounded) operators on $X$ with nonempty resolvent sets containing a half-line $[\omega, \infty) \subset \rho(U) \cap \rho(V)$. The Yosida approximation of $U$ for $\mu > \omega$ is given by

$$U_\mu := \mu^2 R(\mu, U) - \mu I = \mu U R(\mu, U),$$

where $R(\mu, U) = (\mu I - U)^{-1}$ denotes the resolvent operator.

The Yosida distance between $U$ and $V$ is then defined as

$$d_Y(U, V) := \limsup_{\mu \to +\infty} \| U_\mu - V_\mu \|.$$

This construction quantifies the asymptotic difference between the resolvent-based bounded approximations of the original operators as the spectral parameter $\mu$ grows large. Notably, $\lim_{\mu\to\infty} U_\mu x = Ux$ for each $x \in D(U)$ (Bui et al., 2023, Bui et al., 2023).

2. Mathematical Properties and Metric Structure

The Yosida distance exhibits the following fundamental properties:

• Symmetry: $d_Y(U, V) = d_Y(V, U)$ for any eligible $U, V$.
• Nonnegativity: $d_Y(U, V) \geq 0$; $d_Y(U, U) = 0$.
• Pseudometric Character: $d_Y$ satisfies the triangle inequality up to possible domain issues, and becomes a metric in certain subclasses (e.g., bounded operators, or unbounded generators with a common domain).
• Domain Independence: $d_Y$ does not require any inclusion between $D(U)$ and $D(V)$, generalizing previous perturbation metrics.
• Reduction to Operator Norm: If $U, V \in \mathcal L(X)$ (both bounded), $d_Y(U, V) = \|U - V\|$ (Bui et al., 2023, Bui et al., 2023).
• Separation on Generators: For $U, V$ generating contraction semigroups with $D(U)=D(V)$, $d_Y(U, V) = 0$ implies $U = V$.

Comparison to resolvent distance highlights that Yosida distance operates in the high-frequency regime ($\mu \to \infty$), focusing on the behavior of bounded approximants rather than uniform control across $\mu$.

3. Perturbation Theory and Evolution Equations

In the context of linear partial functional differential equations and delay evolution equations of the form

$u'(t) = Au(t) + Bu_t,$

where $A, B$ are linear operators on $X$ and $C([-r, 0], X) \to X$ respectively, the Yosida distance provides a tool for quantifying perturbations in both $A$ and $B$ (Bui et al., 2023). The main theorem (Theorem 4.7) establishes that if $d_Y(A_0, A_1)$ and $d_Y(B_0, B_1)$ are sufficiently small for a pair of systems, then the exponential dichotomy of the original system persists under the perturbation.

An explicit semigroup difference control is provided: if $A$, $B$ generate $C_0$-semigroups $T(t)$, $S(t)$ satisfying $\|T(t)\| \vee \|S(t)\| \leq M e^{\omega t}$, then

$$\|T(t) - S(t)\| \leq t M^2 e^{4\omega t} d_Y(A, B),$$

ensuring uniform closeness of the semigroups on compact time intervals (Bui et al., 2023).

In nonlinear dynamics, $d_Y$ enables new continuity notions for proto-derivatives of nonlinear operators $F$ in $u'(t) = F(u(t))$; persistence of invariant (stable/unstable) manifolds is achieved under small Yosida perturbations, regardless of varying operator domains (Bui et al., 2023).

4. Explicit Computations and Example Scenarios

Analysis of specific equations illustrates the effectiveness and flexibility of Yosida distance:

• Reaction–Diffusion with Delay: For

$$A_0 u = u'' - a u, \quad D(A_0) = H^2 \cap H_0^1, \quad B_0 \phi = b \phi(-r),$$

perturbation in diffusivity ($\varepsilon_1$) or potential ($\varepsilon_3$) yields

$$d_Y(A_0, A_1) \leq \varepsilon_1 + \|\varepsilon_3\|_\infty.$$

Volterra perturbation in $B_1$ gives $d_Y(B_0, B_1) \leq \text{Var}(\eta)$, controlling robustness of the dichotomy (Bui et al., 2023).

• Delay Equation Generator: For

$$(A_a \varphi)(s) = \varphi'(s), \; s \in [-1,0),\quad (A_a \varphi)(0) = a \varphi(-1),$$

one finds $d_Y(A_a, A_b) \leq 2|a-b|$, even though $D(A_a) \neq D(A_b)$ unless $a = b$ (Bui et al., 2023).

• Scalar Delay Equation: For $x'(t) = b x(t-1)$ on $C([-1,0], \mathbb{R})$, the generator domains may not overlap, but computed $d_Y(G_0, G_1) = |b_0 - b_1|$ shows sensitivity to parameter changes (Bui et al., 2023).

5. Impact on Exponential Dichotomy and Invariant Manifold Theory

Yosida distance ensures stability of exponential dichotomy under small perturbations measured in $d_Y$, in both ordinary and partial differential settings (Theorem 3.2 and Theorem 4.7 in (Bui et al., 2023)). In the analytic semigroup case, persistence in the framework $X^\alpha$ is obtained under analogous smallness conditions.

In infinite-dimensional dynamical systems, this framework enables construction and continuity of stable and unstable invariant manifolds near equilibria where generators may vary in domain or be unbounded. If the proto-derivative $DA(x)$ is continuous in the Yosida metric, then global and local invariant manifolds persist and enjoy Lipschitz regularity depending on the Yosida distance (Bui et al., 2023).

6. Broader Significance and Applications

The Yosida distance is effective for:

• Evolution equations with unbounded and non-domain-included perturbations;
• Neutral delay equations and PDEs with memory terms;
• Invariant manifold theory for infinite-dimensional systems;
• Robustness analysis in control and stabilization of PDEs with unbounded feedback laws;
• Quantitative assessment of numerical approximations and graph-convergence of discretized operators (Bui et al., 2023, Bui et al., 2023).

Its key contribution is to unify linear and nonlinear perturbation theory under a domain-independent, asymptotically resolvent-based metric. This suggests future extensions to non-autonomous operator families, further generalizations to spectral splitting, and applications to PDEs with time-dependent domains.

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Summary Table: Key Features of the Yosida Distance

Feature	Description	Source
Formal definition	$$d_Y(U, V) = \limsup_{\mu\to\infty} \|U_\mu - V_\mu\|$$	(Bui et al., 2023, Bui et al., 2023)
Domain relationship	No inclusion of $D(U)$ and $D(V)$ required	Both
Recovering operator norm	$d_Y(A, B) = \|A - B\|$ for $A, B$ bounded	Both
Stability property	Dichotomy, invariant manifolds stable under small $d_Y$	Both