Operators with small Kreiss constants

This paper investigates matrices and operators satisfying the Kreiss condition with constants arbitrarily close to one, establishing refined lower bounds for power growth and demonstrating that specific variants of the condition guarantee similarity to a contraction when the spectrum touches the unit circle at a single point, utilizing a positivity argument involving the double-layer potential operator.

The Gist

Imagine you are managing a complex machine, like a giant, intricate clockwork toy or a financial portfolio. Every day, you press a "Start" button, and the machine performs a task. You do this over and over again.

The big question mathematicians ask is: Will the machine eventually explode (grow infinitely large), or will it stay under control?

In the world of mathematics, this machine is called an Operator (or a Matrix), and the "Start" button is raising it to a power ($T^n$). If the machine stays under control no matter how many times you press the button, we call it Power-Bounded.

The "Safety Rule" (The Kreiss Condition)

For a long time, mathematicians had a famous safety rule, called the Kreiss Condition. It's like a speed limit sign for your machine.

The rule says: "If the machine's internal resistance (called the resolvent) doesn't get too crazy when you test it just outside its normal operating zone, then the machine will definitely stay under control."

Specifically, the rule has a number attached to it, called $K$ (the Kreiss Constant).

• If $K$ is huge, the machine might still be safe, but we need a lot of room to be sure.
• If $K$ is exactly 1, the machine is perfectly safe (it's a "contraction").
• The tricky part is when $K$ is just a tiny bit bigger than 1 (like 1.0001).

For decades, mathematicians wondered: "If $K$ is very close to 1, is the machine guaranteed to stay small?"

Part 1: The "Almost Safe" Machines (The Lower Bounds)

The authors of this paper, ChalMoukis, Tsikalas, and Yakubovich, decided to test this. They asked: Can we build a machine that follows the safety rule almost perfectly (with $K$ very close to 1) but still grows uncontrollably over time?

The Analogy: Imagine a bank account where you are allowed to withdraw money, but the bank charges a tiny fee if you get too close to the limit. You want to know: "If the fee is tiny, can I still drain the account to zero?"

Their Discovery:
Yes! They built a very specific, clever machine (using something called weighted shifts, which is like a conveyor belt where the weight of the items changes in a specific pattern).

• They showed that even if the safety constant $K$ is incredibly close to 1, the machine can still grow.
• How fast does it grow? It grows very slowly, like the logarithm of the logarithm of time.
  • Imagine: If you wait for the age of the universe, the machine might only grow to the size of a grain of sand. It's slow, but it does grow.
• They proved that for any number of "layers" ($m$) you add to your machine, you can make it grow like $(\log N)^m$. This is a massive improvement over previous guesses, showing that "almost safe" doesn't always mean "safe."

Part 2: When "Almost Safe" Actually Is Safe (The Similarity Problem)

In the second half of the paper, they switch from finite machines (matrices) to infinite ones (operators on infinite spaces). Here, the rules are trickier.

They asked: "If we make the safety rule even stricter—specifically, if the safety margin shrinks to zero as we get closer to the danger zone—does that guarantee the machine is safe?"

The "V-Shape" Curve:
They introduced a new concept called a "V-type curve." Imagine drawing a path on a map that approaches a dangerous cliff (the unit circle) but does so in a specific, sharp "V" shape.

They proved a beautiful result:

• If your machine's behavior follows this "V-shape" path, AND
• If the machine's internal resistance behaves nicely along that path,
• THEN the machine is actually Similar to a Contraction.

What does "Similar to a Contraction" mean?
Think of it like changing your perspective. Maybe the machine looks chaotic and dangerous in its current form. But if you look at it through a special pair of glasses (a mathematical transformation), you realize it's actually just a perfectly safe, shrinking machine. It's not actually shrinking, but it behaves exactly like one.

The Catch:
They also showed that if you relax the rules too much (if the "V" isn't sharp enough, or the safety margin doesn't shrink fast enough), you can still build a machine that looks safe but is actually dangerous (not similar to a contraction). They even used a famous counterexample by a mathematician named Pisier to prove this.

The Big Picture

1. The "Small K" Surprise: Just because a machine follows the safety rules almost perfectly (with $K$ close to 1) doesn't mean it won't eventually grow. It might grow very slowly, but it will grow.
2. The "V-Shape" Solution: However, if you add a little extra structure (the V-type curve) and ensure the machine behaves well in that specific direction, you can prove it is fundamentally safe (similar to a contraction).
3. The Open Mystery: The paper ends by asking: "Is there a magic number or a specific shape that guarantees safety for all machines?" We don't know yet. There is still a gap between what we know is possible and what we think must be true.

In short: The authors built a "slow-growing monster" to show that being almost safe isn't enough, but then they found a special "safety net" (the V-curve) that catches the truly dangerous ones, proving they are actually safe deep down.

Technical Summary

Here is a detailed technical summary of the paper "Operators with Small Kreiss Constants" by Nikolaos Chalmoukis, Georgios Tsikalas, and Dmitry Yakubovich.

1. Problem Statement and Context

The paper investigates the relationship between the Kreiss resolvent condition and the power-boundedness (and similarity to contractions) of linear operators.

• The Kreiss Condition: A matrix or operator $T$ satisfies the Kreiss condition if its spectrum is contained in the closed unit disk $\overline{\mathbb{D}}$ and its resolvent satisfies:
  $$\|(zI - T)^{-1}\| \le \frac{K}{|z| - 1}, \quad \forall |z| > 1$$
  where $K \ge 1$ is the Kreiss constant.
• The Kreiss Matrix Theorem: For finite-dimensional matrices, this condition implies power-boundedness ($\sup_n \|T^n\| < \infty$). However, the bound on the growth of $\|T^n\|$ depends on the dimension $N$ and the constant $K$.
• The Gap: While sharp upper bounds for $\|T^n\|$ are known (e.g., Spijker's bound $e^{KN}$), the behavior of the growth function $P(N, K)$ (the supremum of power bounds for matrices with Kreiss constant $\le K$) when $K$ is close to 1 was poorly understood. Previous lower bounds for $K \approx 1$ were weak (logarithmic in $N$).
• Infinite Dimensions: In infinite-dimensional Hilbert spaces, the Kreiss condition with $K > 1$ does not imply power-boundedness. The authors investigate whether a "perturbed" condition where $K$ approaches 1 (specifically $1 + \varepsilon(|z|-1)$) guarantees similarity to a contraction.

2. Methodology

The authors employ a combination of constructive counterexamples, weighted shift operators, and complex analysis techniques involving potential theory.

A. Construction of Weighted Shifts (Finite Dimensional Case)

To establish lower bounds for power growth when $K \approx 1$, the authors construct a specific family of weighted backward shifts on $\ell_2(\mathbb{C}^{2^m})$.

• Matrix Weights: They define inductive sequences of upper triangular matrices $A_{p,k}(c)$ involving logarithmic terms.
• Iterative Structure: The operators $T_{m,c}$ are built using block matrices where the off-diagonal blocks contain sums of products of these weights.
• Key Inequality: They utilize a known inequality regarding absolutely Cesàro bounded weighted shifts (from Chalmoukis et al. [1]) to control the resolvent and Cesàro bounds while allowing the power norms to grow.
• Projection: They project these infinite-dimensional operators onto finite-dimensional subspaces to derive bounds for $N \times N$ matrices.

B. Double-Layer Potential and Positivity (Infinite Dimensional Case)

To address the similarity problem, the authors use a positivity argument involving the double-layer potential operator.

• v-type Curves: They introduce a geometric class of curves $\gamma$ (v-type curves) that touch the unit circle at a single point (usually $z=1$) and satisfy specific integral conditions.
• Positivity Argument: They decompose the resolvent integral into a positive semi-definite part and a controllable error term. By showing that the error term is integrable under specific growth restrictions on the resolvent along the unit circle, they prove that the operator is similar to a contraction.

C. Pisier-Type Counterexamples

To show that certain conditions do not imply similarity to a contraction, they adapt Pisier's counterexample using Foguel-Hankel operators.

• They construct an operator $F$ of the form $\begin{pmatrix} S^* & X \\ 0 & S \end{pmatrix}$ where $S$ is the shift and $X$ is a Hankel operator.
• By carefully choosing the coefficients of $X$, they ensure the operator satisfies the resolvent condition with a specific $\varepsilon(x)$ but fails to be similar to a contraction.

3. Key Contributions and Results

I. Improved Lower Bounds for Small Kreiss Constants

The authors significantly improve upon McCarthy's (1993) lower bound for the power growth $P(N, K)$ when $K = 1 + \varepsilon$.

• Theorem 1.2: For any $\varepsilon > 0$ and integer $m$, there exists a constant $C$ such that:
  $$P_{AC}(N, 1+\varepsilon) \ge C (\log N)^m$$
  where $P_{AC}$ refers to the power bound for absolutely Cesàro bounded operators (a stronger condition than Kreiss).
• Refined Estimate: They further show that for specific choices of $\varepsilon$, the growth can be as high as:
  $$\exp\left( \frac{\alpha (\log \log N)^2}{\log \log \log N} \right)$$
  This demonstrates that even with a Kreiss constant arbitrarily close to 1, the power growth can be super-polylogarithmic, refuting the idea that $K \approx 1$ implies near-constant power bounds.

II. Similarity to Contractions under Refined Conditions

The paper addresses the question: Does a resolvent condition with $K \to 1$ imply similarity to a contraction?

• Negative Result (Theorem 5.1): There exists a continuous function $\varepsilon(x) \to 0$ such that the condition $\|(zI-T)^{-1}\| \le \frac{1+\varepsilon(|z|-1)}{|z|-1}$ holds, yet $T$ is not power-bounded (and thus not similar to a contraction).
• Positive Result (Theorem 1.5): If the spectrum touches the unit circle at a single point and the resolvent satisfies a specific integral condition along a v-type curve $\gamma$:
  $$\int_{-\delta}^{\delta} \frac{\varepsilon(r(\theta)-1)}{r(\theta)-1} \|(e^{i\theta}-T)^{-1}\|^2 d\theta < \infty$$
  then $T$ is similar to a contraction.
• Corollary for Ritt Operators: For Ritt operators (where $\|(zI-T)^{-1}\| \le \frac{C}{|z-1|}$), a uniform choice of $\varepsilon$ exists that guarantees similarity to a contraction.

III. Counterexamples for Specific Growth Rates

• Theorem 1.3: They construct an operator $F$ such that $\|F^n\| \le 1 + \sqrt{\beta_n}$ (where $\beta_n \to 0$) which is not similar to a contraction. Specifically, taking $\beta_n = 1/\log(n+1)$, the operator satisfies the resolvent condition with $\varepsilon(x) \sim (\log(1/x))^{-1/2}$.

4. Significance and Impact

1. Refining the Kreiss-Spijker Estimate: The paper closes a significant gap in the understanding of the Kreiss Matrix Theorem for small constants. It proves that the "smallness" of the Kreiss constant $K$ does not linearize the power growth bound; rather, the growth can be arbitrarily large (in terms of iterated logarithms) even if $K$ is arbitrarily close to 1.
2. Sharpness of Resolvent Conditions: The results delineate the precise boundary between resolvent conditions that guarantee similarity to a contraction and those that do not. It shows that a mere pointwise bound $1+\varepsilon$is insufficient; the *rate* of decay of$\varepsilon$ and the geometry of the domain (v-type curves) are critical.
3. Methodological Innovation: The use of matrix weights in the construction of weighted shifts and the application of double-layer potential operators to prove similarity results represent novel technical approaches in operator theory.
4. Open Questions: The paper concludes by highlighting that the exact dependence of the power growth bound on $K$ for general matrices remains an open problem, specifically whether $P(N, K)$ grows like $N^{\alpha(K)}$ for some $\alpha(K) \to 0$ as $K \to 1$.

In summary, this work provides a rigorous characterization of the behavior of operators with "almost" contractive resolvents, demonstrating that without additional geometric or integral constraints, such operators can exhibit significant power growth or fail to be similar to contractions.