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arXiv · 2601.00319

Foguel-type operators similar to contractions

Nilanjan Das·Soma Das·Jaydeb Sarkar
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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.

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Nilanjan Das, Soma Das, Jaydeb Sarkar. 2026-01-01. Foguel-type operators similar to contractions. https://arxiv.org/abs/2601.00319

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