The tetrablock as a spectral set

Abstract

Abstract. We study a commuting triple of bounded operators (A, B, P ) which has the tetrablock as a spectral set.

Explicit and Unique Construction of Tetrablock Unitary Dilation in a Certain Case

“…All known facts about tetrablock contractions, tetrablock isometries and tetrablock unitaries that we quote here are from [4]. There are well known characterizations of a tetrablock isometry and a tetrablock unitary.…”

“…(2) N 3 is a unitary, N 2 is a contraction and N 1 = N * 2 N 3 ; For a Proof of Theorem 5, see [4] (Th. 5.4).…”

“…is a tetrablock isometry on the Hilbert space K, the space K can be decomposed into K 1 ⊕ K 2 in such a way that K 1 and K 2 reduce [4] (Theorem 5.6) for a proof of this fact. Note that if the fundamental operators of the pure tetrablock co-isometry (Ṽ * 1 ,Ṽ * 2 ,Ṽ * 3 ) areG 1 and G 2 , then the fundamental operators of the tetrablock co-isometry (V * 1 , V * 2 , V * 3 ) are 0 ⊕G 1 and 0 ⊕G 2 .…”

“…Recall that the numerical radius of an operator X is not greater than one if and only if the real part of the operator zX is not bigger than identity for every z on the unit circle, see [2]. From Theorem 3.5 of [4], we know that w(G 1 + zG 2 ) ≤ 1, which in turn gives that w(z 1 G 1 + z 2 G 2 ) ≤ 1, for all z 1 , z 2 in the unit circle. Which is equivalent to (z 1 G 1 +z 2 G 2 )+(z 1 G 1 +z 2 G 2 ) * ≤ 2I , which gives after rearranging (z 2 G 2 + z 1 G * 1 ) + (z 2 G 2 + z 1 G * 1 ) * ≤ 2I , which is equivalent to saying that z 2 (G 2 + zG * 1 ) + z 2 (G 2 + zG * 1 ) * ≤ 2I , for every z and z 2 on the unit circle.…”

A generalized Schwarz lemma for two domains related to μ-synthesis

“…We mention here that the domain tetrablock was introduced by these three mathematicians in [1] and that this domain has deep connection with the most appealing and di cult problem of µ-synthesis (see [5] for further details). Since then the tetrablock has attracted considerable attention [5,6,[10][11][12][15][16][17]. So, our rst main result is the following Schwarz lemma for the tetrablock.…”