On the spectral singularities of the truncated shift

Abstract

The spectral singularities of certain linear operators were first studied by J. Schwartz [17], Naimark [13], Ljance [9], [10], B. Pavlov [15], [16], and also by M. Krein and Langer [8]. A general notion of the sets of the spectral singularities of a dosed linear operator in a Banach space has been given by the author in [12]. Deep structure theorems, including generalized spectral decompositions, for the truncated shift have been proved by Vasyunin [19] (of. also [14]). The purpose of this paper is to study the sets of the spectral singularities of the truncated shift operators (hence of contractions of class Coo with one-dimensional defect spaces, see [18]). In Theorem 1 we establish the connection between S-spectrality in the sense of [1], [I 1] and a kind of spectrality in the sense of [19] for the truncated shift. This will enable us to apply several results of Vasyunin [19] to our problems: these will in general be given as propositions. Theorem 2 will extend a result of Foias [5] and show that the support of the singular continuous part of the representing measure of the characteristic function F is contained in the set S(Tr) of the spectral singularities of the corresponding truncated shift. Theorems 3 and 4 will show that there are truncated shifts with arbitrary dosed sets on the unit circle as the sets of the spectral singularities and with prescribed types of characteristic functions. This contrasts with the fact that these operators are all decomposable in the sense of Foias (cf. [4]): thus they have good but not excellent spectral decomposition properties. The results in the last section deal with the set of the spectral singularities in the strict sense, i.e. with the set ~(T~). Lemma 2 shows that the support of the representing singular measure (of the singular factor) of the characteristic function F is contained in ~(Te). Proposition 7 gives necessary and sufficient conditions for an operator TF to be S-scalar. Finally, it is shown that even if F is a Blaschke product, the sets S(Tr) and ~(TF) can be equal to and, in another case, can be very far from each other. Summarizing, the results will show that the uniform spectral behavior of this class of operators (decomposability) gives place to a large variety when finer concepts of spectral decomposition are considered.