We prove that a composition operator is bounded on the Hardy space H2 of the right half‐plane if and only if the inducing map fixes the point at infinity non‐tangentially, and has a finite angular derivative λ there. In this case the norm, essential norm and spectral radius of the operator are all equal to λ.

2017-08-25T16:31:19

We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

The full Fock space over C d can be identified with the free Hardy space, H 2 (B d N ) -the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball B d N := ∞ n=1 C n×n ⊗ C d 1 . Elements of this space are free or non-commutative functions on B d N . Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of the classical Hardy space of the disk, and many classical function theory results have faithful extensions to this setting. In particular to each contractive (free) multiplier B of the free Hardy space, we associate a Hilbert space H(B) analogous to the deBranges-Rovnyak spaces in the unit disk, and consider the ways in which various properties of the free function B are reflected in the Hilbert space H(B) and the operators which act on it. In the classical setting, the H(b) spaces of analytic functions on the disk display strikingly different behavior depending on whether or not the function b is an extreme point in the unit ball of H ∞ (D). We show that such a dichotomy persists in the free case, where the split depends on whtether or not B is what we call column extreme. k(z, w) := 1 1 − zw * ; z, w ∈ B d , the multi-variable Szegö kernel, and zw * := z1w * 1 +...z d w * d = (w, z) C d . (Here, B d := (C d )1, the multi-variable open unit ball.) The appropriate analogue of the shift in this setting

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse. Mathematics Subject Classification (2000). Primary 47B33; Secondary 47A53, 47L80.