ABSTRACT
The Fock space, or the so-called Segal-Bargmann space, is the analog of the Bergman space in the context of the complex n-space ℂ n . It is a Hilbert space consisting of entire functions in ℂ n . Let d μ ( z ) = e − | z | 2 / 2 d v ( z ) ( 2 π ) − n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1483.tif"/> be the Gaussian measure on ℂ n (dv is the ordinary Lebesgue measure). The Fock space L a 2 ( ℂ n , d μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1484.tif"/> (in short, L a 2 ( ℂ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1485.tif"/> ), by definition, is the space of all μ-square-integrable entire functions on ℂ n . It is easy to check that L a 2 ( ℂ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1486.tif"/> is a closed subspace of L a 2 ( ℂ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1487.tif"/> with the reproducing kernel K λ ( z ) = e 〈 z , λ 〉 / 2 , here 〈 z , λ 〉 = ∑ i = 1 n λ ¯ i z i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429214226/b8f597ed-99c3-4b0a-af8b-93ab01b4e52f/content/eq1488.tif"/> The Fock space is important because of the relationship between the operator theory on it and the Weyl quantization [Be].
