[Paper Review] Operators near completely polynomially dominated ones and similarity problems

Cătălin Badea|arXiv (Cornell University)|Feb 20, 2001

Holomorphic and Operator Theory14 references3 citations

TL;DR

This paper establishes that operators near completely polynomially dominated ones—under a notion of β-quadratic nearness—are similar to operators dominated by the direct sum of the original operator and a weighted unilateral shift. The key contribution is a refined similarity criterion that extends Paulsen’s work and resolves partial cases of a problem on CAR-valued Foguel-Hankel operators, including a Banach space version of Rota’s theorem.

ABSTRACT

Let T and C be two Hilbert space operators. We prove that if T is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by C, then T is similar to an operator which is completely polynomially dominated by the direct sum of C and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pisier in his example of a polynomial bounded operator not similar to a contraction.

Motivation & Objective

• To investigate the similarity of operators near completely polynomially dominated ones to operators dominated by a direct sum of the original and a weighted unilateral shift.
• To provide a refined version of Rota’s similarity theorem for Banach space operators with spectral radius less than one.
• To address a conjecture by K. Davidson and V. Paulsen on the similarity of CAR-valued Foguel-Hankel operators to contractions.
• To extend Paulsen’s completely polynomially bounded similarity criterion to a broader class of near operators.
• To establish stability of similarity to isometries/unitaries under asymptotic nearness, and to analyze failure of similarity under polynomial boundedness alone.

Proposed method

• Introduces the concept of β-quadratic nearness between Hilbert space operators T and C, defined via a sequence β(n) controlling the norm of differences T^n - V₁C^nV₂.
• Uses a weighted norm |x| defined via infimum over decompositions x = Σ T^k x_k, with weights β(k), to construct a new Banach space structure.
• Applies the Cauchy-Schwarz inequality and operator norm estimates to bound the similarity constant between T and an operator dominated by C ⊕ S_w(β).
• Reduces the main similarity result (Theorem 3.3) to a key technical estimate (Theorem 4.1) involving the norm of T^k - V₁C^kV₂ and the weight sequence β.
• Employs the adjoint operator and spectral radius control to show that asymptotically near isometries/unitaries are similar to isometries/unitaries.
• Applies the quotient norm construction on ℓ_p(X) to define a new norm on X, enabling the proof of a Banach space Rota-type theorem.

Experimental results

Research questions

• RQ1 Under what conditions is an operator T similar to a contraction if it is near a completely polynomially dominated operator C?
• RQ2 Can the similarity to a contraction be characterized for CAR-valued Foguel-Hankel operators, as posed by Davidson and Paulsen?
• RQ3 Is the class of operators similar to contractions stable under β-quadratic nearness?
• RQ4 Does asymptotic nearness to an isometry/unitary imply similarity to an isometry/unitary?
• RQ5 Can Rota’s theorem on similarity to contractions be generalized to Banach spaces using weighted norms and direct sums with weighted shifts?

Key findings

• An operator T that is β-quadratically near a completely polynomially dominated operator C is similar to an operator completely polynomially dominated by C ⊕ S_w(β), where S_w(β) is a weighted unilateral shift.
• The similarity constant is bounded by [||V₁||²/γ² + s²]^{1/2} [γ²||V₂||² + β(0)²]^{1/2}, with γ chosen as [β(0)||V₁*|| / (s||V₂||)]^{1/2} when s ≠ 0.
• If s = 0, then T is completely polynomially dominated by C with bound ||V₁||·||V₂||, and the similarity constant is bounded by ||V₁||·||V₂||.
• Asymptotically near isometries are similar to isometries; asymptotically near unitaries are similar to unitaries, even though polynomially bounded operators near contractions may not be similar to contractions.
• A Banach space version of Rota’s theorem is proven: for p > 1 and T on X with spectral radius < 1, T is similar to an operator T₁ on a space isomorphic to ℓ_p(X), completely polynomially dominated by the unilateral shift on ℓ_p(X).
• The paper provides a sufficient condition for CAR-valued Foguel-Hankel operators to be similar to contractions, resolving a partial case of the Davidson-Paulsen problem.