Spherical Isometries Are Hyporeflexive

The result from the title is shown. Let L(H) denote the algebra of all bounded linear operators on a complex Hilbert space H. If M⊂ L(H), then we denote by M′ the commutant of M, M′ = {S ∈ L(H) : T S = S T for every T ∈ M}. The second commutant is denoted by M′′ = (M′)′. Denote further by W(M) the smallest weakly closed subalgebra of L(H) containing M and by AlgLatM the algebra of all operators leaving invariant all subspaces which are invariant for all operators from M. Recall that M is said to be reflexive if W(M) = AlgLatM. For a commutative set M, there is also a weaker version of the reflexivity: M is called hyporeflexive if W(M) = AlgLatM∩M′. Reflexivity and hyporeflexivity have been studied intensely by many authors. Deddens in [D] proved the reflexivity of a single isometry. The result was extended to sets of commuting isometries in [B] (see also [LM]). An analogy and, in some sense, a generalization of commuting N -tuple of isometries are spherical isometries. A spherical isometry is an N -tuple T = (T1, . . . , TN ) of mutually commuting operators on H satisfying T ∗ 1 T1 + · · ·+ T ∗ NTN = IH . The reflexivity of doubly commuting spherical isometries was mentioned in [P]. The aim of the paper is to show the hyporeflexivity of spherical isometries. If μ is a positive Borel measure on the unit sphere ∂BN = {(z1, . . . , zN ) ∈ C : |z1| + · · ·+ |zN |2 = 1}, then denote by H2(μ) the closure of polynomials in L2(μ). We start with the following Lemma 1. Let μ be finite positive Borel measure supported on ∂BN . Then, for all non-negative function h ∈ L1(μ) and ε > 0, there exists a polynomial p such that ‖h− |p|2‖1 < ε. Proof. First note that there is a non–negative continuous function g such that ‖h− g‖1 < ε 2 . Moreover, since μ is finite, we can also assume that g > 0. By Theorem 3.5 of [R], there exists a sequence pn of of polynomials such that |pn| < √ g and |pn(z)| → √ g(z) a.e. μ on ∂BN . Hence g − |pn| ≤ g ∈ L1(μ) and 1991 Mathematics Subject Classification. Primary 47B20, Secondary 47A15.