Jingbo Xia

Hyponormal Pairs of Commuting Operators

We analyze the notions of weak and strong joint hyponormality for commuting pairs of operators, with an aim at understanding the gap between hyponormality and subnormality for single operators. We exhibit a commuting pair T = (T1, T2) such that: (i) T is weakly hyponormal; (ii) T is not strongly hyponormal; (iii) T 1 l 1T 2 l 2 is subnormal (all l1, l2 ≥ 0); (iv) T1 + T2 is not subnormal; (v) T1 + T2 is power hyponormal; and (vi) T1 is unitarily equivalent to T2.

Toeplitz operators on flows

Abstract Let R act continuously on a compact Hausdorff space X giving rise to a flow on X , let ϑ ϵ C ( X ), and let T ϑ x denote the Toeplitz operator on H 2 (R) determined by the function ϑ x on R defined by ϑ x ( t ) = ϑ ( x + t ). In this paper, we investigate the relation between the spectral properties of T ϑ x , the dynamical properties of the flow, and the value distribution theory of ϑ. The analysis proceeds by imbedding T ϑ x in a type II ∞ factor and computing the real-valued index of the operator a la Connes. Our sharpest invertibility result asserts that if the flow is strictly ergodic and if the asymptotic cycle determined by the flow is injective on H 1 ( X , Z), then T ϑ x is invertible if and only if ϑ does not vanish on X and determines the zero element in H 1 ( X , Z). This generalizes the classical result of Gohberg and Krein and its extension to Toeplitz operators with almost periodic symbols due to Coburn, Douglas, Schaeffer and Singer. When ϑ is analytic, in the sense that ϑ x belongs to H ∞ (R) for all x , we relate the II ∞ index of the Toeplitz operator determined by ϑ with the density of the zeros of ϑ x in the upper half-plane. Much of our efforts to achieve this result are devoted to generalizing to arbitrary flows the value distribution theory of analytic almost periodic functions developed by Bohr, Jessen, and Tornehave and others.

On the K-theory of some C∗-algebras of Toeplitz and singular integral operators

Abstract We study the C ∗ -algebras GJ (X, R ) are I (X, R ) of singular integral operators and Toeplitz operators (respectively) associated with a strictly ergodic flow (X, R ). We show that the commutator ideals of these algebras, CGJ (X, R ) and CL (X, R ), are simple and are closely related to the transformation group C ∗ -algebra, C ∗ (X, R ). We calculate the K -theory of GJ (X, R ), L (X, R ) and their commutator ideals. The main results of this calculation, Corollary 3.8.4 and Theorem 4.1.1, assert that C ∗ (X, R ) is contained in CGJ (X, R ) and if j denotes the inclusion map, then j ∗ : K 0 (C ∗ (X,R)) → K 0 (CGJ(X,R)) is an order isomorphism and there is a short exact sequence 0 → K 1 (C ∗ (R)) → i i K 1 C ∗ (R)) → j i K 1 (CGJ(X,R)) → 0 where i is the canonical imbedding of C ∗ ( R ) into C ∗ (X R ). We show also that, up to a change of scale, there is a unique trace on each of the commutator ideals. The key ingredient of our analysis is Theorem 3.1.1 which asserts a bijective correspondence between Silov representations of the algebra of analytic functions on the flow and C ∗ -representations of GJ (X, R ) and L (X, R ). This simultaneously generalizes Coburns theorem on the uniqueness of the C ∗ -algebra generated by an isometry and Douglass theorem on the uniqueness of the C ∗ -algebra generated by an isometric representation of a dense subsemigroup of R + .

ON THE CONTRIBUTION OF THE COULOMB SINGULARITY OF ARBITRARY CHARGE TO THE DIRAC HAMILTONIAN

We study the self-adjoint extensions of the Dirac operator α · (p− B)+μ0β−W , where the electrical potential W contains a Coulomb singularity of arbitrary charge and the magnetic potential B is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form v(r)/r where v has a limit at 0, then, for any self-adjoint extension of the Dirac operator, removing the singularity results in a Hilbert-Schmidt perturbation of its resolvent.

Multipliers and essential norm on the Drury-Arveson space

It is well known that for multipliers f of the Drury-Arveson space H 2 n , ∥f∥ ∞ does not dominate the operator norm of M f . We show that in general ∥f∥ ∞ does not even dominate the essential norm of M f . A consequence of this is that there exist multipliers f of H 2 n for which M f fails to be essentially hyponormal; i.e., if K is any compact, self-adjoint operator, then the inequality M* f M f - M f M* f + K ≥ 0 does not hold.

A characterization of compact perturbations of Toeplitz operators

Let X be a bounded operator on the Hardy space H 2 of the unit circle. It has been a longstanding problem to determine whether the condition that T u XT u ― X is compact for every inner function u implies that X is a compact perturbation of a Toeplitz operator. We show that the answer is affirmative.

On the essential commutant of ()

Let T(QC) (resp. T) be the C * -algebra generated by the Toeplitz operators {T φ : φ ∈ QC} (resp. {T φ : φ ∈ L∞}) on the Hardy space H 2 of the unit circle. A well-known theorem of Davidson asserts that T(QC) is the essential commutant of T. We show that the essential commutant of T(QC) is strictly larger than T. Thus the image of T in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of T(QC).

Coincidence of essential commutant and the double commutant relation in the Calkin algebra

Let B be a von Neumann algebra on a separable Hilbert space H. We show that, if the dimension of B as a linear space is infinite, then it has a proper C∗-subalgebra A whose essential commutant in B(H) coincides with the essential commutant of B. Moreover, if π is the quotient map from B(H) to the Calkin algebra B(H)/K(H), then π(A)≠π(B) and {π(A)}″=π(B).

Criteria for Toeplitz operators on the sphere

Let H2(S) be the Hardy space on the unit sphere S in C. We show that a set of inner functions Λ is sufficient for the purpose of determining which A ∈ B(H2(S)) is a Toeplitz operator if and only if the multiplication operators {Mu : u ∈ Λ} on L2(S, dσ) generate the von Neumann algebra {Mf : f ∈ L∞(S, dσ)}.

Commutator ideals and semicommutator ideals of Toeplitz operators associated with flows II

We prove that for a flow with at most one fixed point, the commutator ideal and the semicommutator ideal of the associated Toeplitz algebra coincide. We further show that the situation becomes much more complicated for flows with at least two fixed points.