Maximum Entropy Truncable Positive Definite Sequences of Analytic Toeplitz Operators

Abstract

In this paper we present a geometric approach for studying positive definite sequence of analytic Toeplitz operators following the lines of the general framework of operator extrapolation theory based on choice sequences (cf.[1], [3], [6], [11]). The specific problem here appears is when the positive definite sequence (Tn)n > 1 produced by a choice sequence (Rn)n>i initiated on H2is a sequence of analytic Toeplitz operators. Using special matrix representation based on the choice sequence of the minimal isometric dilation [K, V] of (Tn)n≥1, we prove, in Section 2, that T, is analytic Toeplitz operator for any n if and only if there exists an isometry U on K which commutes with V such that H2, as the subspace of K, reduces U to the shift operator S on H2(Theorem 1). This connects our problem with the topics of Ando dilations (cf.[2], [7], [8]). In Section 3 we write some recurrent formulas appearing in this theory under the specific forms of our context. Section 4 deals with the maximum entropy truncable sequences i.e. the sequences (Tn)n ≥1 of analytic Toeplitz operators whose choice sequence (Rn)n≥1 has the property that for each p ≥ 1, the truncated choice sequence {R1, R2,…, R t 0, 0,…} still produces positive definite sequence of analytic Toeplitz operators. We proved that this happens if and only if, in the matricial form based on the choice sequence, the isometry U is diagonal (Theorem 5). Then we are able to give an algorithm to produce, by a system of free parameters, the set of all maximum entropy truncable positive definite sequences of analytic Toeplitz operators. We end with some considerations concerning a special moment problem on Z2 connected with this results (Sectin 5).KeywordsMaximum EntropyInvariant SubspaceToeplitz OperatorMoment ProblemRecursive ConstructionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.