On the reflexivity of multivariable isometries

Abstract

Abstract. Let A ⊂ C(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K in C n . We define the notion of an A-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov's work on the inner function problem, every A-isometry T ∈ L(H) n is reflexive. This result applies to commuting isometries, spherical isometries, and more generally, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain.

Multivariable isometries related to certain convex domains

“…The preceding observations allow us to bring all the results in [16], [18], [20] and [21] related to a regular A-isometry to bear upon the multiplication tuple M µp,z ; we highlight in Remarks 2.6 and 2.7 below a few implications of the results in those references. We also point out that some of those results are derived exploiting Prunaru's work in [38].…”

“…. , N µp,zn ) associated with L 2 (µ p ) as its minimal normal extension); also, in the light of Proposition 2.5, M µp,z is regular in the sense of [20] (that is, in the sense of [18,Definition 2.6]).…”

“…Let K ⊂ C n be compact and let A be a unital closed subalgebra of C(K) containing n-variable complex polynomials. Following [20], we call a subnormal tuple S an A-isometry if the spectral measure of the minimal normal extension N of S is supported on the Shilov boundary of A and if A is contained in R S . Given a normalized positive regular Borel measure µ p supported on ∂Ω p , we let H 2 (µ p ) be the closure of A(Ω p ) in L 2 (µ p ).…”

“…where ι is the inclusion map and π is a unital * -homomorphism that is in fact a left inverse of the compression map ρ : [20,Corollary 6] that the weak operator topology and the weak * operator topology coincide on A Mµ p ,z and that every unital weak * -closed subalgebra of A Mµ p ,z is reflexive; in particular, M µp,z is reflexive (refer to [20] for the relevant definitions). (b) It is a consequence of [16,Corollary 2] that the set T M µp,z of M µp,z -Toeplitz operators is 2-hyperreflexive with the 2-hyperreflexivity constant κ 2 T M µp,z being less than or equal to 2 (refer to [31] for the relevant definitions).…”

“…. , T n ) of commuting operators T i in B(H) is said to be essentially normal if the operators T * i T j − T j T * i are compact for all i and j, while T is said to be cyclic if there exists a vector f in H (referred to as a cyclic vector for T ) such that the linear span ∨{T k1 1 T k2 2 • • • T kn n f : k i are non-negative integers} is dense in H. There exist several interesting results in the literature on subnormal operator tuples (and in particular on essentially normal and/or cyclic subnormal operator tuples) having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in C n (refer, for example, to [4], [6], [15], [16], [17], [18], [20], [21], [48]). These results are largely manifestations of the functional calculus for subnormal operator tuples thriving upon some elegant function-theoretic results valid in the context of those two types of domains.…”