Classes of contractions and Harnack domination

Abstract

Several properties of the Harnack domination of linear operators acting onHilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g. convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it.2010 Mathematics Subject Classification. Primary 47A10, 47A45; Secondary 47A20, 47A35, 47B15.In the sequel T, T ′ ∈ B(H) will be linear contractions acting on the complex Hilbert space H; V acting on K and V ′ acting on K ′ will denote the minimal isometric dilations of T and T ′ respectively. N (T ) and R(T ) stand for the kernel and respectively the range of the operator T . We shall denote by I the identity operator on H and by T λ = (T − λI)(I − λT ) −1

Harnack and Shmul’yan pre-order relations for Hilbert space contractions

“…In order to state the second application, we recall that N (I − S T ) is an invariant subspace for each contraction T ∈ B(H), and T | N (I−S T ) is an isometry. For another contraction T ′ which belongs to the Shmul'yan part of T it is necessary that N (I − S T ) is also invariant (even N (I − S T ) = N (I − S T ′ ) because T ′ will be in the Harnack part of T ) and T ′ = T on N (I − S T ); see [2,Lemma 5.1]. More precisely we have the following Corollary 3.6.…”

“…Both pre-order relations have nice geometric and analytic interpretations. Although these two pre-orders have been around since 1970s and 1980s [1,5,[15][16][17]20,24], their structure is to date not completely understood, and in recent years there has been an increase in interest for this topic [2,7,8,10,12,13,18,19,23].…”

“…The canonical triangulation is not preserved by the Harnack domination, more precisely it is possible to have T H ≺ T ′ and N (S T ) = N (S T ′ ). However, the classes C 0• , C •0 and C 00 are preserved in both senses by the Harnack domination (see [2,Theorem 5.5]). On the other hand, the class C 11 (hence C 1• and C •1 ) are not preserved by the Harnack equivalence (see [2, Propositions 5.9 and 5.10]).…”

Harnack parts for some truncated shifts

“…It was proved in [20] that the Harnack part of a contraction T is trivial if and only if T is an isometry or a coisometry (the adjoint of an isometry), this a response of the question posed by Ando et al in the class of contractions. The authors of [4] proved that maximal elements for the Harnack domination in C 1 (H) are precisely the singular unitary operators and the minimal elements are isometries and coisometries.…”