By Simon A. Levin

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Arveson [l]works out the basic theory of a Hardy-type space H$ of holomorphic functions in several complex variables, and shows that HZ is appropriate for the operator theory of d-contractions. In this paper, we will generalize Arveson’s Hardy space in several complex variables to the infinite complex variables. A Hardy-type space H& of holomorphic functions in infinite complex variables is defined. We show that H& can be identified with the symmetric Fock space over the complex Hilbert space 1 2 , and that N-shift is a contraction of infinite sequences of mutually commuting operators.

371). HARDY SPACE OF HOLOMORPHIC FUNCTIONS IN INFINITE COMPLEX VARIABLES * ZEQIAN CHEN Wuhan Institute of Physics a n d Mathematics, CAS, 30 West District, Xaao-Hong Mountain, Wuhan, P. ac. c n In this paper, we show that some of the function-theoretic properties of Arveson’s Hardy space in several complex variables to the infinite complex variables. The connection is clarified between operator theory of contractions for infinite sequences of mutually commuting operators on a Hilbert space and function theory of holomorphic functions in infinite complex variables.

The second part can be divided into several parts as follows k>l where x2,k := c $([z/(ql x . here m = q1 x . . x pcfl 5 qk < qk-1 ' . 2) satisfies the following minimized condition: m < ' . ' < 4 1 5 (41 x " ' x q k - 1 ) = - 5 (z') < m. 3) qk qk Proof Any m involved in C2 is a composite number such that p(m) does not vanish. So it is necessary and sufficient to prove that, if we relpace &+(z/m,Z(m)- 1) in (2. 4. 2) by its expansion (2. 3), then the original expression of C2 agrees with (2. 4.