Homogeneous Operators
Abstract
A bounded operator T on a complex separable Hilbert space is said to be homogeneous if '(T ) is unitarily equivalent to T for all ' in M•ob, where M•ob is the M•obius group. A complete description of all homogeneous weighted shifts was obtained by Bagchi and Misra. The first examples of irreducible bilateral homogeneous 2shifts were given by Koranyi. We describe all irreducible homogeneous 2shifts up to unitary equivalence completing the list of homogeneous 2shifts of Koranyi.
After completing the list of all irreducible homogeneous 2shifts, we show that every homogeneous operator whose associated representation is a direct sum of three copies of a Complementary series representation, is reducible. Moreover, we show that such an operator is either a direct sum of three bilateral weighted shifts, each of which is a homogeneous operator or a direct sum of a homogeneous bilateral weighted shift and an irreducible bilateral 2shift.
It is known that the characteristic function T of a homogeneous contraction T with an associated representation is of the form T (a) = L( a) T (0) R( a); where L and R are projective representations of the M•obius group M•ob with a common multiplier. We give another proof of the \product formula".
We point out that the defect operators of a homogeneous contraction in B2(D) are not always quasiinvertible (recall that an operator T is said to be quasiinvertible if T is injective and ran(T ) is dense).
We prove that when the defect operators of a homogeneous contraction in B2(D) are not quasiinvertible, the projective representations L and R are unitarily equivalent to the holomorphic Discrete series representations D+ 1 and D++3, respectively. Also, we prove that, when the defect operators of a homogeneous contraction in B2(D) are quasiinvertible, the two representations L and R are unitarily equivalent to certain known pairs of representations D 1; 2 and D +1; 1 ; respectively. These are described explicitly.
Let G be either (i) the direct product of ncopies of the biholomorphic automorphism group of the disc or (ii) the biholomorphic automorphism group of the polydisc Dn:
A commuting tuple of bounded operators T = (T1; T2; : : : ; Tn) is said to be homogeneous with respect to G if the joint spectrum of T lies in Dn and '(T); defined using the usual functional calculus, is unitarily equivalent to T for all ' 2 G:
We show that a commuting tuple T in the CowenDouglas class of rank 1 is homogeneous
with respect to G if and only if it is unitarily equivalent to the tuple of the multiplication
operators on either the reproducing kernel Hilbert space with reproducing kernel n 1
i=1 (1 ziwi) i
or Q n
i i n; are positive real numbers, according asQG is as in (i)
or 1 ; where ; i, 1 i
i=1 (1 z w )
(ii).
Finally, we show that a commuting tuple (T1; T2; : : : ; Tn) in the CowenDouglas class of rank 2 is homogeneous with respect to M•obn if and only if it is unitarily equivalent to the tuple of the multiplication operators on the reproducing kernel Hilbert space whose reproducing kernel is a product of n 1 rank one kernels and a rank two kernel. We also show that there is no irreducible tuple of operators in B2(Dn), which is homogeneous with respect to the group Aut(Dn):
Collections
 Mathematics (MA) [159]
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