Classes of regular Dilations
by
Dan Popovici
University of the West, Timisoara

As shown by Douglas and Foia s (A Classification of Multi-Isometries, to appear) every completely non-unitary (c.n.u.) bi-isometry V=(V₁, V₂) is ``modelled'' by a certain isometric pair on H²(E) in terms of two operators on the same Hilbert space E, U unitary and P orthogonal projection, called its unitary invariants. It is our aim in this paper to obtain structure results relative to this model for the minimal regular (respectively *-regular) isometric dilation (introduced by Brehmer and Sz.-Nagy) V_T=(V_T⁽¹⁾, V_T⁽²⁾) (respectively V_T^*) of a given bi-contraction T=(T₁, T₂) (respectively T^*) on a Hilbert space H.

Remark firstly that a minimal isometric dilation of T is c.n.u. iff T₁T₂ in C_·0 (C_{\alpha, \beta},  \alpha, \beta in {·, 0, 1} represent the Sz.-Nagy-Foia s classes of contractions defined in terms of punctual convergence). Consider contractions S_T⁽ⁱ⁾ in L(D_T3-i), S_T⁽ⁱ⁾D_T3-ih:=D_T3-iT_ih and unitary operators R_T⁽ⁱ⁾ in L(D, D_ST⁽ⁱ⁾),  R_T⁽ⁱ⁾\Delta_T^1/2h=D_ST⁽ⁱ⁾D_T3-ih,  h in H,  i=1, 2 (\Delta_T:=I-T₁^*T₁-T₂^*T₂+T₁^*T₂^*T₁T₂, D=[`(\Delta<sub>T</sub>H)], D<sub>Z</sub>=(I-Z<sup>*</sup>Z)<sup>1/2</sup> the defect operator and D<sub>Z</sub>=[`(D_ZH)] the defect space of a contraction Z). The unitary invariants of V_T^* are given by the formulas U=A₁+A₂^*, A_i=(A_jk⁽ⁱ⁾)_{j, k in Z}, A₁₀⁽ⁱ⁾=D_ST⁽ⁱ⁾R_T⁽ⁱ⁾,  A₁₁⁽ⁱ⁾=S_T^(i)*,  A₂₀⁽ⁱ⁾=-S_T⁽ⁱ⁾R_T⁽ⁱ⁾,  A₂₁⁽ⁱ⁾=D_ST^(i)*,  A_j j-1⁽ⁱ⁾=I_DST^(i)* (j >= 3),  A_jk⁽ⁱ⁾=0 (for other (j, k)), i=1, 2 and P=(P_jk)_{j, k in Z}, P_jj=I_DST^(2)* (j <= -2), P_-1 -1=I_DT1,  P_jk=0 (for other (j, k)) on l²_Z-^*(D_ST^(2)*)\oplusD_T1\oplusD\oplusD_T2 \oplusl²_Z+^*(D_ST^(1)*). Having as a starting point a paper of D.Ga spar and N.Suciu (On the geometric structure of regular dilations, Op.Theory: Adv. and Appl., 103(1998), 105-120) we can obtain corresponding results for V_T.

Finally, as applications, we characterize the membership of V_T and V_T^* to some special classes of bi-isometries in terms of T. Thus V_T⁽ⁱ⁾ is unitary iff T_i,  S_T⁽ⁱ⁾ are co-isometries and V_T⁽ⁱ⁾ is pure iff T_i,  S_T⁽ⁱ⁾ in C_·0,  i=1, 2. Moreover V_T^* is bi-shift, V_T is unitary iff T_i are backward shifts and S_T⁽ⁱ⁾ co-isometries i=1, 2, V_T^* is bi-shift, V_T⁽¹⁾ is unitary and V_T⁽²⁾ is shift iff T₁ is backward shift, T₂ in C₀₀,  S_T⁽¹⁾ co-isometry, S_T⁽²⁾ in C_·0. Similar characterizations are obtained for other remarkable parts of V_T and V_T^*.

Date received: June 15, 2000