ABSTRACT

To generalize the operator-theoretic aspects of function theory on the unit disk to multivariable operator theory, Arveson investigated a new function space H d 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1903.tif"/> on the unit ball B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1904.tif"/> d in the d-dimensional complex space ℂ d (cf. [Arv1, Arv2, Arv3, Arv6]) (for convenience, we call H d 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1905.tif"/> the Arveson space). Indeed, the Arveson space plays an important role in the multi-variable operator theory as shown by Arveson [Arv1, Arv2, Arv3]. Recall that the Arveson space H d 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1906.tif"/> on the unit ball B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1907.tif"/> d is defined by the reproducing kernel K λ(z) = 1/(1 − 〈z, λ〉), where 〈 z , λ 〉 = ∑ j = 1 d z j λ ¯ j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1908.tif"/> and it is easy to verify that the space is invariant under multiplication by polynomials. Therefore, we will regard the Arveson space H d 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1909.tif"/> as a module over the polynomial ring C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1910.tif"/> . Unlike Hardy modules and Bergman modules, the Arveson module H d 2 ( d ≥ 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1911.tif"/> is never associated with some measure on ℂ d as shown by [Arv1]. Therefore, the Arveson module enjoys many properties distinct from those of the Hardy module and the Bergman module.