Abstract
Abstract In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕd with the property that ℕ\X + ej ⊂ ℕ\X for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.
Highlights
We begin by xing some notation and delimiting the framework we work in
In the Drury-Arveson space, we consider the subspace of functions whose Taylor coe cients are supported in a set Y ⊂ Nd with the property that N X + ej ⊂ N X for all j =, . . . , d
This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for speci c choices of X
Summary
We begin by xing some notation and delimiting the framework we work in. Let H be an abstract Hilbert space and for d ≥ consider a d-tuple of operators A = (A , . . . , Ad) ∶ H → Hd. Let H be an abstract Hilbert space and for d ≥ consider a d-tuple of operators A = Ad) ∶ H → Hd. It is not di cult to see that the formal adjoint operator A∗ ∶ Hd → H acts as follows. Kd) ∈ Hd. Following Drury, we will relate A to an operator acting on a Hilbert space of holomorphic functions of several variables on the unit ball. The Drury-Arveson space is the space Hd of functions f (z) = ∑n∈Nd a(n)zn holomorphic on the unit ball Bd ⊂ Cd, such that f Hd ∶=.
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