A-Isometries and Hilbert-A-Modules Over Product Domains

Abstract

For a compact set K subset {mathbb {C}}^n, let A subset C(K) be a function algebra containing the polynomials {mathbb {C}}[z_1,cdots ,z_n ]. Assuming that a certain regularity condition holds for A, we prove a commutant-lifting theorem for A-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., Mlak (Studia Math. 43(3): 219–233, 1972) and Athavale (J. Oper. Theory 23(2): 339–350, 1990; Rocky Mt. J. Math. 48(1): 2018; Complex Anal. Oper. Theory 2(3): 417–428, 2008; New York J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map for hypo-Shilov-modules (i=1,2). By standard arguments, we obtain an identification where is the minimal C(partial _A)-extension of (i=1,2), provided that is projective and is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra A=A(D)=C({overline{D}})cap {mathcal {O}}(D) over a product domain D = D_1 times cdots times D_k subset {mathbb {C}}^n where each factor D_i is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some {mathbb {C}}^{d_i} (1le i le k). This extends known results from the ball and polydisc-case, Guo (Studia Math. 135(1): 1–12, 1999) and Chen and Guo (J. Oper. Theory 43: 69–81, 2000).