2-Abolutely summable oeprators in certain Banach spaces by Komarchev I.A.

By Komarchev I.A.

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A. Ball, A. Biswas, Q. Fang and S. ter Horst It will be convenient to work also with the analytic Toeplitz operators acting between H 2 (E, σ)-spaces of different multiplicity. For this purpose, we suppose that U and Y are two additional auxiliary Hilbert spaces (to be thought of as an input space and output space respectively). We consider higher multiplicity versions of H 2 (E, σ) by tensoring with an auxiliary Hilbert space (which is to be thought of as adding multiplicity): HU2 (E, σ) := H 2 (E, σ) ⊗C U, HY2 (E, σ) := H 2 (E, σ) ⊗C Y.

The unit element of Fd is the empty word denoted by ∅. 2), we let z α denote the monomial in noncommuting indeterminates z α = ziN · · · zi1 and we let z ∅ = 1. We extend this noncommutative functional calculus to a d-tuple of operators A = (A1 , . . , Ad ) on a Hilbert space K: Av = AiN · · · Ai1 if v = iN · · · i1 ∈ Fd \ {∅}; A∅ = IK . We will also have need of the transpose operation on Fd : α = i1 · · · iN if α = iN · · · i1 . Given a coefficient Hilbert space Y we let Y z denote the set of all polynomials in z = (z1 , .

Let A and B be C ∗ -algebras. By an (A, B)-reproducing kernel correspondence on a set Ω, we mean an (A, B)-correspondence E whose elements are B-valued functions f : (ω, a) → f (ω, a) ∈ B on Ω × A, which is a vector space with respect to the usual pointwise vector space operations and such that for each ω ∈ Ω there is a kernel element kω ∈ E with f (ω, a) = a · f, kω E. 7) is the reproducing kernel for the reproducing kernel correspondence E. 6) of the point evaluation for elements in an (A, B)-reproducing kernel correspondence E on Ω one easily deduces that the left A-action and the right B-action are given by (a · f )(ω , a ) = f (ω , a a) and (f · b)(ω , a ) = f (ω , a )b.

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