Spectra of Toeplitz Operators and Uniform Algebras

Let A be a uniform algebra on X and P a set of all probability measures on X. For each µ in P, H2 (µ) is the closure of A in L2 (µ) and Tt is a Toeplitz operator on H2 (µ) for a continuous function cf> on X. In this paper we study the invertibility and the spectrum of Tip = L EB Tt. We show that if Tip is invertible then the index of cf> is zero and if the converse is true for an arbitrary continuous function cf> then A is a Dirichlet algebra on X. Moreover we study the spectrum of Tip

Invariant subspaces of finite codimension and uniform algebras

Let A be a uniform algebra on a compact Hausdorff space X and m a probability measure on X. Let Hp(m) be the norm closure of A in Lp(m) with 1 ≤ p < ∞ and H∞(m) the weak * closure of A in L∞(m). In this paper, we describe a closed ideal of A and exhibit a closed invariant subspace of Hp(m) for A that is of finite codimension

The real part of an outer function and a Helson-Szeg weight

Suppose F is a nonzero function in the Hardy space H1. We study the set {f ; f is outer and !Fl ::; Re f a.e. on 8D} where 8D is a unit circle. When F is a strongly outer function in H1 and 'Y is a positive constant, we describe the set {! ; f is outer, IFI ::; 'Y Re f and IF-1 I ::; 'Y Re u-1) a.e. on 8D}. Suppose w is a Helson-Szego weight. As an application, we parametrize real valued functions v in L∞(∂D) such that the difference between log W and the harmonic conjugate function v_*_ of v belongs to L∞(∂D) and llvll∞ is strictly less than π/2 using a contractive function α in H∞ such that (1+α)/(1-α) is equal to the Herglotz integral of W

Compact Toeplitz operators with continuous symbols on weighted Bergman spaces

Let L (n,dud0/21r) be a complete weighted Bergman space on the open unit disc n where du is a positive finite Borel measure on [O, 1). We show the following : When cp is a continuous function on the closed unit disc D, T</J is compact if and only if cp = 0 on an

Generalized Numerical Radius And Unitary p-Dilation

In this paper, we,study an operator A on a Hilbert space H which satisfies one of the following inequalities For some ,\ with O ::; ,\ ::; 1 l(Ay, y)I ::; AIIYll2 + (1 - ,\)IIAYll2 (y EH) or AIIAYll2 + (1 - ,\)l(Ay, y)I ::; IIYll2 (y EH). These two inequalities can be regarded as special cases of generalized numerical ranges. If A has a p-dilation with p > 0, then it satisfies one of them. We show that the operator radii wp(A) of A are calculated using l(Ay,y)I and IIAYII- Several applications are given

On the zeroes of solutions of an extremal problem in H1

For a non-zero function f in H1 , the classical Hardy space on the unit disc, we put Sf= {g E H1 : argf(i8 ) = argg(ei0) a.e. 0}. The intersection of Sf and the unit sphere in H1 is just a set of solutions of some extremal problem in H1 It is known that Sf can be represented in the form Sf = S x g0, where β is a Blaschke product and g0 is a function in H1 with S90 = {Λ· g0 : Λ> O}. Also it is known that the linear span of Sf is of finite dimensional if and only if β is a finite Blaschke product, and when β is a finite Blaschke product, we can describe completely the set Sβ and the zeros of functions in Sβ. In this paper, we study the set of zeros of functions in Sβ when β is an infinite Blaschke product whose set of singularities is not the whole circle. Especially we study the behavior of zeros of functions in Sβ in the sectors of the form: Δ = { reiQ : 0 < r <_ 1, c1 < 0 < c2} on which the zeros of B has no accumulation points, and establish a convergence order theorem of zeros in Δ of functions in Sβ

Some special bounded homomorphisms of a uniform algebra

Let L(H) be the algebra of all bounded linear operators on a Hilbert space H and let A be a uniform algebra. In this paper we study the following questions. When is a unital bounded homomorphism q, of A in L(H) completely bounded ? When is the norm 11-I>II of q, equal to the completely bounded norm 11-I>llcb ? In some special cases we answer this question. Suppose q, is p-contractive (0 is contractive if p == 1. We show that if A is a Dirichlet algebra or dim A/ ker q, == 2 then q, has a p-dilation. If q, is a p-contractive homomorphism then 1\-I>\\ == max(l, p) and if it has a p-dilation then 11-I>llcb == max(l, p). Moreover we give a new example of a hypo-Dirichlet algebra in which a unital contractive homomorphism has a contractive dilation

Toeplitz operators and weighted norm inequalities on the bidisc

Abstract. Let HP be the Hardy space on the bidisc and 1 on HP and the invertibility of the Toeplitz operator T on HP. The latter is strongly related with a weighted norm inequalities on the bidisc

Riesz's functions and Carleson inequalities

Let µ be a finite positive Borel measure on the open unit disc D and H a set of all analytic functions on D. For each a in D, put r(µ, a)= sup lf(a)J2 where .f EH and klfl2dµ ::; 1. Unless the support set ofµ is a finite set, fnr(µ, a)dµ(a) ∞. However zED sup }f Dt(z) r(µ, a)dµ(a) < ∞ may happen where Dt(z) denotes the Bergman disc in D. We study when this is possible. When vis a descrete measure such that dv = I:s(µ,a)8a,zED sup/, D1(z) r(µ, a)dv(a) Under some condition onµ, we show that zsup eDJf D1(z) r(µ,a)dv(a) < ∞ for a finite positive Borel measure v on D if and only if ( v, µ )-Carleson inequality is valid

An outer function and several important functions in two variables

In one variable, an outer function has several important properties and a function with one of the properties is an outer function. In two variables, the situation is very different and we study these functions