On the Berger-Coburn-Lebow Problem for Hardy Submodules

Abstract

Abstract. In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for C * -algebras on Hardy submodules.

On certain commuting isometries, joint invariant subspaces and C*-algebras

“…In the same paper [6], Berger, Coburn and Lebow asked whether T (S) is isomorphic to T (H 2 (D 2 )) for every finite codimensional invariant subspaces S in H 2 (D 2 ). This question was recently answered positively by Seto in [27]. Here we extend Seto's answer from H 2 (D 2 ) to the general case…”

“…In this section, we extend Seto's result [27] on isomorphic C * -algebras of invariant subspaces of finite codimension in H 2 (D 2 ) to that in H 2 (D n ), n ≥ 2. Given a Hilbert space H, the set of all compact operators from H to itself is denoted by K(H).…”

“…Since T (S) is an irreducible C * -algebra (cf. [27]), it is enough to prove that T (S) contains a non-zero compact operator. As…”

“…To that aim, for (i) and (ii), we consider the initial approach by Berger, Coburn and Lebow [6] from a more modern point of view (due to Bercovici, Douglas and Foias [5]) along with the technique of [20]. For (iii), we will examine Seto's approach [27] more closely from "subspace" approximation point of view.…”

“…We refer to Bercovici, Douglas and Foias [3,4,5] and also [10], [12], [15], [8], [9], [14], [17], [19], [23], [27] and [30,31] for more on n-isometries, n ≥ 2, and related topics.…”