Hyponormal Toeplitz Operators And Zeros Of Polynomials

The problem of hyponormality for Toeplitz operators with (trigonometric) polynomial symbols is studied. We give a necessary and sufficient condition using the zeros of the analytic polynomial induced by the Fourier coefficients of the symbol

Interpolation Of Weighted l^q Sequences By H^p Functions

Let (znQ ) be a sequence of points in the open unit disc D and ½n = m6=n j(zn ¡zm)(1¡ ¯zmzn)¡1j > 0. Let a = (aj)1j =1 be a sequence of positive numbers and `s(a) = f(wj) ; (ajwj) 2 `sg where 1 · s · 1. When 1 · p · 1 and 1=p + 1=q = 1, we show that f(f(zn)) ; f 2 Hpg ¾ `s(a) if and only if there exists a finite positive constant ° such that ( 1X n=1 (an½n)¡t(1 ¡ jznj2)tjf(zn)jt )1=t · °kfkq (f 2 Hq), where 1=s+1=t = 1. As results, we show that f(f(zj)) ; f 2 Hpg ¾ `1(a) if and only if sup n (an½n)¡1(1¡jznj2)1=p < 1, and f(f(zn)) ; f 2 H1g ¾ `1(a) if and only if X n (an½n)¡1(1 ¡ jznj2)±zn is finite measure on D. These are also proved in the case of weighted Hardy spaces

Invariant Subspaces In The Bidisc And Wandering Subspaces

Let M be a forward shift invariant subspace and N a backward shift invariant subspace in the Hardy space H2 on the bidisc. We assume that H2 = N M. Using the wandering subspace of M and N, we study the relations between M and N. Moreover we study M and N using several natural operators which are defined by shift operators on H2

Integral Operators on a Subspace of Holomorphic Functions on the Disc

Let H(D) be an algebra of all holomorphic functions on the open nit disc D and X a subspace of H(D). When g is a function in H(D), put g(f)(z) = z ( )g0( )d and Ig(f)(z) = z 0( )g( )d (z 2 D) or f in X. In this paper, we study J[X] = {g 2 H(D) ; Jg(f) 2 X for all f in X} and [X] = {g 2 H(D) ; Ig(f) 2 X for all f in X}. We apply the results to concrete spaces. or example, we study J[X] and I[X] when X is a weighted Bloch space, a Hardy space r a Privalov space

Invariant Subspaces Of Toeplitz Operators And Uniform Algebras

Let T be a Toeplitz operator on the one variable Hardy space H2. We show that if T has a nontrivial invariant subspace in the set of invariant subspaces of Tz then belongs to H1. In fact, we also study such a problem for the several variables Hardy space H2

Exposed points and extremal problems in H1

AbstractIf φ ∈ L∞, we denote by Tφ the functional defined on the Hardy space H1 by Tφ(ƒ) = ∫−ππ ƒ(eiθ) φ(eiθ)dθ2π. Let Sφ be the set of functions in H1 which satisfy Tφ(ƒ) = ∥Tφ∥ and ∥ƒ ∥1 ⩽ 1. It is known that if φ is continuous, then Sφ is weak-∗ compact and not empty. For many noncontinuous φ each Sφ is weak-∗ compact and not empty. A complete descr ption of Sφ if Sφ is weak-∗ compact and not empty is obtained. Sφ is not empty if and only if Sφ = Sψ and ψ = ¦ ƒ¦ƒ for some nonzero ƒ in H1. It is shown that if φ = ¦ƒ ¦ƒ and ƒ = pg, where p is an analytic polynomial and g is a strong outer function, then Sφ is weak-∗ compact. As the consequence, if ƒ = p, then Sφ is weak-∗ compact

Brown-Halmos type theorems of weighted Toeplitz operators II

The spectra of the Toeplitz operators on the weighted Hardy space Hp(Wd =2 ) are studied. For example, the theorems of Brown-Halmos type and Hartman- Wintner type are proved. These generalize results in the previous paper which were proved for p = 2.

Factorizations Of Functions In H^p(T^n )

We are interested in extremal functions in a Hardy space H^p(T^n) (1 <= p <= \infty ). For example, we study extreme points of the unit ball of H^1(T^n) and give a factorization theorem. In particular, we show that any rational function can be factorized