Towards a functional calculus for subnormal tuples: the minimal normal extension

Journal of Functional Analysis

Everard

2011

Let T ∈ B(H ) n be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H , and let A T ⊂ B(H ) be the unital dual operator algebra generated by T . In this note we show that every operator S ∈ B(H ) in the essential commutant of A T has the form S = X + K with a T -Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T = (M z 1 , . . . , M z n ) ∈ B(H 2 (σ )) n consisting of the multiplication operators with the coordinate functions on the Hardy space H 2 (σ ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D ⊂ C n . As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H 2 (σ ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.

“…is known to be isometric again (see Conway [5]). Thus γ T defines a weak * continuous isometric algebra homomorphism mapping z i to T i for i = 1, .…”

Section: A-isometries and Inner Functionsmentioning

confidence: 99%

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Didas

Journal of Functional Analysis

Everard

2011

“…which extends the polynomial functional calculus of T in a unique way (see Conway [6], Proposition 1.1). It is an elementary and well-known fact that this isomorphism yields a correspondence between μ-inner functions, that is, functions…”

Section: Introductionmentioning

confidence: 97%

Inner functions and spherical isometries

Didas

2011

Proc. Amer. Math. Soc.

Abstract. A commuting tuple T = (T 1 , . . . , T n ) ∈ B(H) n of bounded Hilbert-space operators is called a spherical isometry ifPrunaru initiated the study of T -Toeplitz operators, which he defined to be the solutions X ∈ B(H) of the fixed-point equationUsing results of Aleksandrov on abstract inner functions, we show that X ∈ B(H) is a T -Toeplitz operator precisely when X satisfies J * XJ = X for every isometry J in the unital dual algebra A T ⊂ B(H) generated by T . As a consequence we deduce that a spherical isometry T has empty point spectrum if and only if the only compact T -Toeplitz operator is the zero operator. Moreover, we show that if σ p (T ) = ∅, then an operator which commutes modulo the finite-rank operators with A T is a finite-rank perturbation of a T -Toeplitz operator.

“…is isometric and weak * continuous if L(H) is equipped with its weak * topology as the dual space of the trace class operators (see Proposition 1.1 in [7]). Since all scalar spectral measures of T , that is, measures µ arising as a scalar spectral measure of some minimal normal extension N of T , are mutually absolutely continuous, both the restriction algebra R(T ) and the algebra homomorphism γ T are independent of the choice of N .…”

Section: Reflexivity Of A-isometriesmentioning

confidence: 99%

On the reflexivity of multivariable isometries

2005

Proc. Amer. Math. Soc.

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.