Hardy Hilbert Space Research Articles

Numerical ranges of composition operators

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol, 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e., a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

A commutant lifting theorem on the polydisc

0. Introduction Interpolation problems for bounded analytic functions in the unit disk have been studied for at least one century. The simplest ones are the Nevanlinna-Pick case, in which the constraints on the functions are the values in a finite number of points, and the Caratheodory-Fejer, where the first finite number of Taylor coefficients of the development of a function are prescribed. In all these cases, one imposes a size constraint on the function: say, its supremum norm should be smaller than 1. Starting with the paper of Sarason ([S]), it has been realized that there exists a natural operatorial frame which unifies all function theoretic problems. The setting is the following: the algebra of bounded analytic functions is identified with the algebra of analytic multiplication (Toeplitz) operators acting on the Hardy Hilbert space H(D), while the interpolation conditions are translated in the existence of a subspace of H, semiinvariant with respect to Toeplitz operators, and the compression of the multiplication operator to this subspace. The most general result in this direction is the intertwining lifting theorem of Sz-Nagy and Foias ([SNF]), which has found subsequently many applications, including applied areas like system theory. The generalization of these interpolation theorems to several variables (that is, to bounded analytic functions on the polydisc) is a relatively new subject. To paraphrase the one-dimensional result, it would imply the consideration of a subspace M ⊂ H(D), semiinvariant to the d operators of multiplication by the variables, and of a contraction X on M commuting with the compression of these multiplications. The problem would be to lift X to a contractive Toeplitz operator on the whole H(D). The most simple case, the Nevanlinna-Pick problem, has been solved only recently ([Ag], [AgMC], [BT]). It also points out that a direct analogue of the onedimensional problem is not possible, and that we have either to restrict the hypothesis or to relax the conclusion. The second alternative has been achieved, most notably by M. Cotlar and C. Sadosky in [CS1], where they obtain two-variable commutant lifting theorems in the more general context of abstract scattering systems. When specialized to the intertwining lifting problem, their result produces two “partial” interpolating Toeplitz

Toeplitz operators in the commutant of a composition operator

If ϕ is an analytic self-mapping of the unit disc D and if $H^2(D)$ is the Hardy-Hilbert space on D, the composition operator $C_ϕ$ on $H^{2}(D)$ is defined by $C_ϕ(f) = f∘ϕ$. In this article, we consider which Toeplitz operators $T_f$ satisfy $T_{f}C_{ϕ}