Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T 1 , . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z 1 , . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B 1 (Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i T i is given. In general, based on this formula, we make a conjecture giving the form of this operator.

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting d-tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a * -algebra representation ρ of C 0 (S) (where S is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on S and is adapted to the * -representation ρ via a unitary representation U of the group G, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation U becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space S ⊂ C d decomposes into finitely many G-orbits. In such cases, the imprimitivity based on S admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of G-orbits.

We consider the family of n-tuples P consisting of polynomials P1, . . . , Pn with nonnegative coefficients such that ∂iPj (0) = δi,j, i, j = 1, . . . , n. With every such P, we associate a Reinhardt domain △ n

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).

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We address the question of the analyticity of a rank one perturbation of an analytic operator. If Mz is the bounded operator of multiplication by z on a functional Hilbert space Hκ and f ∈ H with f (0) = 0, then Mz + f ⊗ 1 is always analytic. If f (0) = 0, then the analyticity of Mz + f ⊗ 1 is characterized in terms of the membership to Hκ of the formal power series obtained by multiplying f (z) byAs an application, we discuss the problem of the invariance of the left spectrum under rank one perturbation. In particular, we show that the left spectrum σ l (T + f ⊗ g) of the rank one perturbation T + f ⊗ g, g ∈ ker(T * ), of a cyclic analytic left invertible bounded linear operator T coincides with the left spectrum of T except the point f, g . In general, the point f, g may or may not belong to σ l (T + f ⊗ g). However, if it belongs to σ l (T + f ⊗ g)\{0}, then it is a simple eigenvalue of T + f ⊗ g.

Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T1, . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for all K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z1, . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B1(Ω). For a bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i Ti is given. In general, based on this formula, we make a conjecture giving the form of this operator.

No abstract