We consider unitary analogs of 1−dimensional Anderson models on l2(Z) defined by the product Uω = DωS where S is a deterministic unitary and Dω is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of Uω is pure point almost surely for all val...

Let ρ ≥ 1 and w ρ (A) be the operator radius of a linear operator A. Suppose m is a positive integer. It is shown that for a given invertible linear operator A acting on a Hilbert space, one has w ρ (A −m) ≥ w ρ (A) −m. The equality holds if and only if A is a multiple of a unitary operator.

We describe in this note how the "boundary representation" technique introduced in [l ] leads to a complete classification of compact operators on Hubert spaces to unitary equivalence (Theorem 3), in terms of a sequence of invariants related to (and generalizing) the numerical range. These invariants are, we feel, vastly simpler than one might have anticipated in so general a situation. Full de...