Foguel-type operators similar to contractions

TL;DR

The paper investigates seven Foguel-type operators built from 2×2 upper-triangular blocks with diagonal shifts and off-diagonal Toeplitz or Hankel entries, and provides a complete characterization of which of these operators are similar to contractions. It introduces a general criterion equating similarity to contraction with the existence of an auxiliary operator $A$ and a bounded analytic symbol $\Psi$ satisfying $X = A - S_E^* A S_E^*$ (and related formulations), plus a power-boundedness condition expressed via the sequence $X_n$. The results are then specialized to Toeplitz, Hankel, and mixed Toeplitz/Hankel cases, yielding explicit symbol-conditions such as $\Phi \in (1-\bar{z}^2)L^{\infty}$ and $\Phi^* + \widetilde{\Phi}^* \in (1-z^2)(H^\ frac{\infty}{} + \overline{zH^ frac{\infty}{}})$ for Toeplitz, and $H_{\Phi} D_E \in \mathcal{B}(H^2_E(\mathbb{D}))$ with a companion analytic condition for Hankel. The work also provides concrete scalar instances (e.g., BMOA-related Hankel criteria) and a negative example via the Hilbert-Hankel matrix, thereby completing the seven-type classification and connecting to Toeplitz+Hankel operator theory and known results on complete polynomial boundedness.

Abstract

Pisier's celebrated counterexample to Halmos's similarity-to-contractions problem was based on $2 \times 2$ upper triangular block operator matrices involving three classical operators: forward and backward shifts on the diagonal and Hankel operators in the off-diagonal entry. Together with another classical object, namely Toeplitz operators, one can formulate another $2^3 -1 = 7$ types of $2 \times 2$ upper triangular block operator matrices, which we refer to as Foguel-type operators. In this paper, we give a complete characterization of all the seven Foguel-type operators being similar to contractions.

Theorems & Definitions (20)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 4.1
• proof