$$\mathbb{K}$$-homogeneous tuple of operators on bounded symmetric domains

Abstract

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.

Commuting Tuple of Multiplication Operators Homogeneous under the Unitary Group

“…, s r ) ∈ Z r + , s 1 ≥ s 2 ≥ • • • ≥ s r and P s are the irreducible components under the action of K. The invariant kernel K is then of the form: K a (z, w) = s∈ Z r + a s E s (z, w), where E s is the reproducing kernel of P s equipped with the Fischer-Fock inner product defined by p, q F := 1 π d C d p(z)q(z)e − z 2 2 dm(z). The results of [9] also show that the properties of M like boundedness, membership in the Cowen-Douglas class B 1 (Ω), unitary and similarity orbit etc. can be determined from the properties of the sequence a := {a s } s∈ Z r + .…”

“…It would be convenient for us to let AK(Ω) denote the class A 1 K(Ω). A classification, modulo unitary equivalence, of the d-tuples in AK(Ω) was obtained in [9]. In this paper, we continue the investigation initiated in [9], now for the class A n K(Ω), n ∈ N. We describe below the results of this paper.…”

“…It follows from [9,Theorem 2.3] that the commuting d-tuples in the class AK(Ω) introduced earlier in [9] coincides with the class A 1 K(Ω). It would be convenient for us to let AK(Ω) denote the class A 1 K(Ω).…”

“…(1) dim ker D T * = 1, (2) ker D T * is cyclic for T , and (3) Ω ⊆ σ p (T * ); was introduced in the recent paper [9], see also [17]. Among other things, it is shown in [9] that any d-tuple T in AK(Ω) must be unitarily equivalent to the d-tuple M of multiplication by the coordinate functions on a reproducing kernel Hilbert space H K (Ω) ⊆ Hol(Ω) for some invariant kernel K. Recall that the Hilbert space H K (Ω) has a direct sum decomposition ⊕ s∈ Z r + P s , where Z r + is the set of signatures: s := (s 1 , . .…”

“…If the d-tuple M of multiplication by the coordinate functions on some Hilbert space H K (Ω) is in AK(Ω), then the kernel K is invariant under the action of the group K, that is, K(z, w) = s∈ Z r + a s E s (z, w) with a 0 = 1, see [1,Proposition 3.4] and [9,Theorem 2.3]. But if n > 1 and the d-tuple M acting on H K is in A n K(Ω), then we can only assume that the kernel K is merely quasiinvariant, not necessarily invariant.…”