We characterize norm closed subspaces $B$ of $\linf (\partial D)$ such that $C(\partial D) B \subset B$ and maximal ones in the family of proper closed subspaces $B$ of $L^\infty(\partial D)$ such that $A(D) B \subset B$, where $A(D)$ is the disk algebra. Analogously, we characterize closed subspaces of $H^\infty$ that are simultaneously invariant under $S$ and $S^\ast$, the forward and the backward shift operators, and maximal invariant subspaces of $H^\infty$.

Generalizing the notion of invariant subspaces on the 2-dimensional torus ${{T}^{2}}$, we study the structure of ${{A}_{\phi }}$-invariant subspaces of ${{L}^{2}}({{T}^{2}})$. A complete description is given of ${{A}_{\phi }}$-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

Let h∞(Dn) denote the space of all bounded n-harmonic functions on the unit polydisk Dn of Cn. In this paper we prove that the Bourgain algebra h∞(Dn)b and h∞(Dn)bb relative to the Lebesgue space L∞(Dn) are of the following forms:

Here V(Dn) is the space of those functions such that , where denotes the characteristic function of a subset E of Dn.

Analysis on sparse parts of the Banach algebra of bounded analytic functions is given. It is proved that Sarason's theorem for QC-level sets cannot be generalized to general Douglas algebras.

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

A function theoretic proof of Axler's zero multiplier theorem of Bergman spaces is given.

Let D be the open unit disk and let ∂D be its boundary. We denote by C the algebra of continuous functions on ∂d, and by L∞ the algebra of essentially bounded measurable functions with respect to the normalized Lebesgue measure m on ∂D. Let H∞ be the algebra of bounded analytic functions on D. Identifying with their boundary functions, we regard H∞ as a closed subalgebra of L∞. Let A = H∞ Pi C, which is called the disk algebra. The algebras A and H∞ have been studied extensively [5, 6, 7]. In these fifteen years, norm closed subalgebras between H∞ and L∞, called Douglas algebras, have received considerable attention in connection with Toeplitz operators [12]. A norm closed subalgebra between A and H∞ is called an analytic subalgebra. In [2], Dawson studied analytic subalgebras and he remarked that there are many different types of analytic subalgebras. One problem is to study which analytic subalgebras are backward shift invariant. Here, a subset E of H∞ is called backward shift invariant if