Operator inequalities I. Models and ergodicity

Abstract

We discuss when an operator, subject to an inequality in heridatary form, admits a unitarily equivalent functional model of Agler type in the reproducing kernel Hilbert space associated to the inequality. To the contrary to the previous work, the kernel need not be of Nevanlinna-Pick type. We derive some consequences concerning the ergodic behavior of the operator.

Poisson equation and discrete one-sided Hilbert transform for $(C,α)$-bounded operators

“…Let 0 < β < 1 and let ℓ 2 β (N 0 ) denote the Hilbert space of sequences f such that f 2 2,β := ∞ j=0 |f (j)| 2 k β (j) < ∞. Let T S be the backward shift operator on ℓ 2 β (N 0 ) given by (T S f )(j) = f (j + 1), f ∈ ℓ 2 β (N 0 ), j ∈ N 0 Then T n S 2 ∼ (n + 1) 1−β , so T S is not power-bounded on ℓ 2 β (N 0 ), but T S is (C, α)-bounded for α > (1 − β)/2, see [2].…”

“…Very recently, in connection with operator inequalities and models, it has been shown that the shift operator on weighted Bergman spaces is (C, α)-ergodic, for α > 0 depending on the weight. See [2] and Section 10 below.…”

Functional models up to similarity and $a$-contractions

“…The usual approach, which goes back to Agler, is based on the assumption that the function k(t) := 1/α(t) defines a reproducing kernel Hilbert space. In particular, it is assumed that α does not vanish on the unit disc D. We refer the reader to the introduction of our recent paper [1] and the references therein for more details.…”

“…In Theorem 1.1 we generalize this result. This paper should be seen as a second part of [1], where we focused on unitarily equivalent models for operators T satisfying α(T * , T ) ≥ 0 for certain functions α. We tried to make our exposition independent of [1].…”

“…Given a function α in the Wiener algebra A W and an operator T in L(H) with spectrum σ(T ) ⊂ D, if the series |α n |T * n T n converges in SOT, then we say that α ∈ A T . According to the notation of [1], this means precisely that T ∈ Adm w α . Depending on whether we want to consider fixed the operator T or the function α, we will use one notation or the other.…”