On commuting operators solving Gleason’s problem

Abstract

Abstract. We prove the uniqueness of commuting operators solving Gleason's problem for certain spaces of functions analytic in the unit ball.

Multivariable backward-shift-invariant subspaces and observability operators

“…A uniqueness result for solutions of the Gleason problem somewhat different from that in Theorem 3.22 was obtained in [3]; rather than assuming that T is a contractive solution of the Gleason problem on M = H(K C,A ) contained isometrically in H Y (k d ) as in Theorem 3.22, Alpay and Dubi in [3] assume instead that T is a commutative solution of the Gleason problem and are then able to conclude that necessarily T = M * λ | M . This latter result can be seen as an immediate consequence of our Theorem 3.17 above since, by the construction in the proof of Theorem 3.21, solutions (C, A) of K a C,A = K are in one-to-one correspondence with solutions T of the Gleason problem.…”

“…, A d ) is not assumed to be commutative, there is no characterization of the abelianized observability gramian as a minimal solution of a generalized Stein equation analogous to the classical case given in Theorem 1.1, but there still is a somewhat more implicit analogue of Theorem 1.2, where the backward shift S * (1.9) in Theorem 1.2 is replaced by a solution of the so-called Gleason problem (see Theorems 3.20 and 3.21 below). The Gleason problem originates in the work of Henkin and Gleason (see [31,36]) and has been studied in the context of the Arveson space (with various formulas for the solution) in [3] with an application to realization questions in [2]. Our analogue of Theorem 1.2 for the Arveson space for the case of commutative d-tuple A has already been given in [19] (with a more general power-series setting worked out in [20]) for the finite-dimensional case.…”

Contractively embedded invariant subspaces

“…We would like to point out that the theory of contractively embedded backward shift invariant subspaces in reproducing kernel Hilbert spaces and the de Branges-Rovnyak models, in the setting of row contractions, are closely related to the Gleason's problem [1]. In this context, the reader should consult the papers by Alpay and Dubi [2], Ball and Bolotnikov [6], Ball, Bolotnikov and Fang [7,9], Ball, Bolotnikov and ter Horst [10,11], Benhida and Timotin [13] and Martin and Ramanantoanina [25].…”

Rational Functions Associated with the White Noise Space and Related Topics

“…The argument follows the one given in the hyper-holomorphic setting in [12]. For an earlier result in the setting of power series in finite number of variables, see [6]. Note that in the theorem we do not assume the operator of multiplication by z k to be bounded.…”