Homogeneous 2-Shifts

Abstract

The classification of homogeneous scalar weighted shifts is known. Recently, Korányi obtained a large class of inequivalent irreducible homogeneous bi-lateral 2-by-2 block shifts. In this paper, we construct two distinct classes of examples not in the list of Korányi. It is then shown that these new examples of irreducible homogeneous bi-lateral 2-by-2 block shifts, together with the ones found earlier by Korányi, account for every unitarily inequivalent irreducible homogeneous bi-lateral 2-by-2 block shift.2010 Mathematics Subject Classification. Primary 47B37, secondary 20C25. 1 2 SOMNATH HAZRA Korányi. We complete the list by identifying the remaining irreducible homogeneous bi-lateral 2-by-2 block shifts. The main Theorem of this paper is stated at the very end of the paper. PreliminariesThe definition of homogeneous operator while ensuring the existence of a unitary operator U φ intertwining φ(T ) with T does not impose any additional condition on the map φ → U φ . To investigate some of these properties, we recall some basic notions from representation theory of locally compact second countable (lcsc) groups, in particular, the Möbius group. Most of what follows is from [3,2].Definition 2.1. Let G be a locally compact second countable group, H be a Hilbert space and U(H) be the group of unitary operators on H. A Borel function π : G → U(H) is said to be a projective unitary representation of G on the Hilbert space H, ifwhere m : G × G → T is a Borel function. (In this paper, a representation or a projective representation will always mean a projective unitary representation.)The function m associated with a projective representation π is called the multiplier of π and satisfies the equations m(g, 1) = m(1, g) = 1, m(g 1 , g 2 )m(g 1 g 2 , g 3 ) = m(g 1 , g 2 g 3 )m(g 2 , g 3 ) for all g, g 1 , g 2 and g 3 in G. Two multipliers m andm are said to be equivalent if there is a Borel function f :Let π 1 and π 2 be two projective representations of G on Hilbert spaces H 1 and H 2 , respectively. The representations π 1 and π 2 are called equivalent if there exists a unitary operator U : H 1 → H 2 and a Borel function f : G → T such that π 1 (g) = f (g)U * π 2 (g)U holds for all g in G.Definition 2.2. Let T be a homogeneous operator on a Hilbert space H. If there is a projective representation π of Möb on H with the propertythen π is said to be the representation associated with the operator T.A homogeneous operator need not possess an associated representation. However, [3, Theorem 2.2] says that for every irreducible homogeneous operator, there exists a unique (upto equivalence) projective representation associated with it.We fix some notation and terminology that will be used throughout this paper. For any projective representation π of Möb, let π # be the representation of Möb defined by π # (φ) = π(φ * ) where φ * (z) = φ(z), z ∈ D, for every φ in Möb.Proposition 2.3. [3, Proposition 2.1] Suppose T is a homogeneous operator and π is an associated representation of T . Then the adjoint, T * , is also homogeneo...

The irreducibility and strong irreducibility of operators

“…1 2 } n∈Z , 0 < a, b < 1 and a = b. Afterward, S. Hazra [20] also found two classes of irreducible homogeneous bilateral 2by-2 shifts, one of which has the structure shown above. It was also shown that these three classes of operators are mutually unitarily inequivalent when the parameter α > 0.…”

“…When dim W n = 1, T is called a shift operator. In this case, if W n = span{e n }, then {e n } n∈I is the orthonormal basis of H. A. Korányi and S. Hazra found three large classes irreducible bilateral block shifts in [20,31]. B. Bagchi and G. Misra proved that any irreducible homogeneous operator is a block shift in [1].…”

Homogeneous Hermitian Holomorphic Vector Bundles And Operators In The Cowen-Douglas Class Over The Poly-disc

“…The class of homogeneous operators has been studied in a number of articles [2,1,6,7,14,10,18]. It is known that an irreducible homogeneous operator T acting on a Hilbert space H is a block shift, specifically, H is the orthogonal direct sum of subspaces V n , n ∈ I, where I is either the set of a) integers, b) non-negative integers, or c) non-positive integers, such that…”

“…All irreducible homogeneous operators, for which dimV n ≤ 1 as well as dimV n ≤ 2, have been classified in [2] and [18], respectively. The classification, in general, of irreducible homogeneous operators in the Cowen-Douglas class over D has been completed (cf.…”