2013/10/03 by Bhattacharyya, T., Narayanan, E. K., Sarkar, Jaydeb
#30H05 #46E22 #46M05 #46N99 #47A20 #47A45 #47B32 #47B38 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1310.0950
An n-tuple (n ≥ 2), T = (T1, …, Tn), of commuting bounded linear operators on a Hilbert space H is doubly commuting if Ti Tj^* = Tj^* Ti for all 1 ≤ i < j ≤ n. If in addition, each Ti ∈ C⋅ 0, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple T can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H2_DT^*(\mathbbDn). As a consequence of the dilation theorem, we prove that there exists a closed subspace ST of the form HT := ∑i=1n ΦTi H2_ETi(\mathbbDn), where \ETi\i=1n are Hilbert spaces, ΦTi ∈ H^∞_B(ETi, DT^*)(\mathbbDn) such that each ΦTi (1 ≤ i ≤ n) is either a one variable inner function in zi, or the zero function. Moreover, H ≅ ST^⊥ and (T1, …, Tn) ≅ PST^⊥ (Mz1, …, Mzn)|ST^⊥.