A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III: The Adjoint

Abstract

This paper contains a further analysis of the Toeplitz-like operators Tω on H p with rational symbol ω having poles on the unit circle that were previously studied in [5,6]. Here the adjoint operator T * ω is described. In the case where p = 2 and ω has poles only on the unit circle T, a description is given for when T * ω is symmetric and when T * ω admits a selfadjoint extension. Also in the case where p = 2, ω has only poles on T and in addition ω is proper, it is shown that T * ω coincides with the unbounded Toeplitz operator defined by Sarason in [12].

A Toeplitz-Like Operator with Rational Matrix Symbol Having Poles on the Unit Circle: Fredholm Properties

“…This paper is a continuation of [11][12][13], where Toeplitz-like operators with rational symbols with poles on the unit circle T were studied. Whilst the aim of [11][12][13] was to Dedicated to our friend an colleague Henk the Snoo, on the occasion of his seventy-fifth birthday. analyze such Toeplitz-like operators with scalar symbols, in this paper we will focus on such Toeplitz-like operators with matrix symbol.…”

“…Also here, if n = 1 we write Rat m (T) and Rat m 0 (T) instead of Rat m×1 (T) and Rat m×1 0 (T), respectively. In the scalar case, i.e., m = n = 1, we simply write Rat, Rat(T) and Rat 0 (T), as was done in [11][12][13].…”

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This paper concerns the analysis of an unbounded Toeplitz-like operator generated by a rational matrix function having poles on the unit circle T. It extends the analysis of such operators generated by scalar rational functions with poles on T found in [11,12,13]. A Wiener-Hopf type factorization of rational matrix functions with poles and zeroes on T is proved and then used to analyze the Fredholm properties of such Toeplitz-like operators.

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