Pratibha G. Ghatage

Sampling sets and closed range composition operators on the Bloch space

We give a necessary and sufficient condition for a composition operator C Φ on the Bloch space to have closed range. We show that when Φ is univalent, it is sufficient to consider the action of C Φ on the set of Mobius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

Composition operators with closed range on the Bloch space

In this note we investigate conditions under which a holomorphic self-map of the unit disk induces a composition operator with closed range on the Bloch space.

Hyperbolic derivatives and generalized Schwarz-Pick estimates

In this paper we use the beautiful formula of Faa di Brune for the nth derivative of composition of two functions to obtain the generalized Scliwarz-Pick estimates. By means of those estimates we show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric.

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH∞ under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).

Duality and multiplication operators

We describe a space of functions contained inxxLx℞(D)⋂C(D ⋃G) but not necessarily inU. We give a representation of these functions as bounded multiplication operators on the Bergman spacexxLxa2 and identify the subspace consisting of functions which induce compact multiplication operators. We also describe a newC*-subalgebra ofxxLx℞(D) which we conjecture to be a proper super-set ofU.

On the spectrum of the Bergman-Hilbert matrix

Abstract The Bergman-Hilbert matrix is the matrix of the Hilbert-Hankel operator on the Bergman space L2a of the disk. We obtain estimates on its norm and prove the existence of eigenvalues.

A remark on Bourgain algebras on the disk

Abstract. It is shown that the Bourgain algebra Xb of the space X = H°°considered as a subalgebra of ^ = alg{//°° , H°° } is #°°(B) + (7C(D) whereUC(B) is the algebra of uniformly continuous functions on the open unit diskD . This uses and extends a recent result of Cima-Janson-Yale on the Bourgain algebra of H°° on dB . Further, (Xb)b = Xb . In [3] Cima and Timoney introduce the concept of Bourgain algebra Xi, of alinear subspace X of a Banach algebra A and show that if X itself is an alge-bra, then X C Xb . In [2] Cima, Janson, and Yale describe the Bourgain algebraof H°°(dH). Since then there has been further study of Bourgain algebras on the bitorus and the polydisk [8].We look at X = H°° as a subalgebra of ^ where ^ = alg{//°°, H00} c^f^CB). We recall that ^ may be identified with C(J?) where JT is the maximal ideal space of H°° . UC(V>) denotes the algebra of uniformly contin-uous functions on D. It is known that H°° + UC(D) = H°°[z], the subalgebraof ^f°°(B) generated by H°° and z (see [1, p. 721]). The only norms we useare the supnorms and essential supnorms; ||p||aD = esssup{|#>(z)|, z e dB}whereas \\