Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

Abstract

Let B d be the open Euclidean ball in C d and T := (T1, . . . , T d ) be a commuting tuple of bounded linear operators on a complex separable Hilbert space H. Let U(d) be the linear group of unitary transformations acting on C d by the rule: z → u • z, z ∈ C d , and u • z is the usual matrix product. Consequently, u • z is a linear function taking values in C d . Let u1(z), . . . , u d (z) be the coordinate functions of u • z.We define u • T to be the operator (u1(T ), . . . , u d (T )) and say that T is U(d)-homogeneous if u • T is unitarily equivalent to T for all u ∈ U(d). We find conditions to ensure that a U(d)-homogeneous tuple T is unitarily equivalent to a tuple M of multiplication by coordinate functions acting on some reproducing kernel Hilbert space HK (B d , C n ) ⊆ Hol(B d , C n ), where n is the dimension of the joint kernel of the d-tuple T * . The U(d)-homogeneous operators in the case of n = 1 have been classified under mild assumptions on the reproducing kernel K. In this paper, we study the class of U(d)-homogeneous tuples M in detail for n = d, or equivalently, kernels K quasi-invariant under the group U(d). Among other things, we describe a large class of U(d)-homogeneous operators and obtain explicit criterion for (i) boundedness, (ii) reducibility and (iii) mutual unitary equivalence of these operators. Finally, we classify the kernels K quasi-invariant under U(d), where these kernels transform under an irreducible unitary representation c of the group U(d).