Toeplitz operators associated to commuting row contractions

Let HH be a complex Hilbert space and let {Tn}n⩾1{Tn}n⩾1 be a sequence of commuting bounded operators on HH such that ∑n⩾1TnTn∗⩽IH. Let F(T¯) denote the space of all operators X   in B(H)B(H) for which ∑n⩾1TnXTn∗=X and suppose that F(T¯)≠{0}. We will show that there exists a triple {K,Γ,{Un}n⩾1}{K,Γ,{Un}n⩾1} where KK is a Hilbert space, Γ:K→HΓ:K→H is a bounded operator and {Un}n⩾1⊂B(K){Un}n⩾1⊂B(K) is a sequence of commuting normal operators with ∑n⩾1UnUn∗=IK such that TnΓ=ΓUnTnΓ=ΓUn for n⩾1n⩾1, and for which the mapping Y↦ΓYΓ∗Y↦ΓYΓ∗ is a complete isometry from the commutant of {Un}n⩾1{Un}n⩾1 onto the space F(T¯). Moreover we show that the inverse of this mapping can be extended to a ∗-homomorphismπ:C∗{IH,F(T¯)}→{Un}n⩾1′ from the unital C∗C∗-algebra generated by F(T¯) onto the commutant of {Un}n⩾1{Un}n⩾1. We also show that there exists a ∗-homomorphismΠ:C∗{IH,{Tn}n⩾1}→C∗{IK,{Un}n⩾1}Π:C∗{IH,{Tn}n⩾1}→C∗{IK,{Un}n⩾1} such that Π(Tn)=UnΠ(Tn)=Un for n⩾1n⩾1. In the particular case when {Tn}n⩾1{Tn}n⩾1 has only a finite number of non-zero components, it turns out that {Un}n⩾1{Un}n⩾1 is unitarily equivalent to the spherical unitary part of the standard commuting dilation of {Tn}n⩾1{Tn}n⩾1.