Multivariable Operator Theory: A Joint Summer Research by Raul E. Curio; Ronald G. Douglas; Joel D. Pincus; Norberto

By Raul E. Curio; Ronald G. Douglas; Joel D. Pincus; Norberto Salinas

This quantity incorporates a number of papers offered on the summer season examine convention on Multivariable Operator conception, held in July 1993 on the collage of Washington in Seattle. The articles include contributions to numerous parts and issues that could be considered as forming an rising new topic. This topic consists of the research of geometric instead of topological invariants linked to the overall topic of operator thought in numerous variables. advancements have happened in different diverse instructions, via various strategies, and plenty of advances were made via cross-pollination between various components of arithmetic. The target of the convention, and of this quantity, is to spur dialogue of the connections one of the numerous methods and new instructions for examine.

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Then q(A) = r-2 , and ,for 0< j:5 r-2 (k) Mi = E ®Pm . i-jSJa/2 (ii) If D is the Cartan domain of type Ill. then a=1, q(A)=[(n-,2-1)/21+1, and 1 E ®P.. 0< j<_q(A). 1+2J s j (iii) If D is the Cartan domain of type IV,, for n odd, then a=n-2, q(A)=1 for ,2=0,1 and MM= = Cl, Mi = P(O), Mu s) = E. , Mj 2> = p(a/z) M2=0 Recall that N = N. is the (highest) determinant polynomial. 11: Let A E Wd(D), 0 < j < q(A), and let A1,,, denote the kI 00 s e t o f a l l s E N so t h a t No E M f \M j- I . Then (i) S E Aj,a if and only If span (U(a)(1p)N8; rp E G} = MfX' .

27) (f,g)a = As A 1 (n - k)+ , . ( J) n-1 , _dr a+k-rt-1 1 (A+k-n) Jo the probability measure U + k - n)(1 - r)a+k-rt-Idr converges (in the weak topology of measures) to the Dirac measure 61 at 1. 9). 28) (f,g)a = = -1, 0, 1, 2, ... and n. 21). 28). If 2 = 0, 1, 2. 11). 28). A variant of the method described above leads to the following results. 5: Let f,g be analytic functions in a neighborhood of B. 2-1(R+X)f)(z)g(z) 2 \-2 -1 (1-Izl) zl2(rt-B -1) dV(z). 2,3,... 30) respectively by limiting procedures.

1) 7 sup IIU(a)W IIB(H) < VEG (2) For any finite Bore! 2) fKf(kz)dv(k) maps H into itself continuously, and the integral converges in the norm of H. Then H = )CA,o with equivalent norms. e. o. 2w Proof. 3) If(0)111111x:_ 7IIfil,,, fEH. 11,, is a norm, 11111M>0 . 5) thus point evaluations are continuous linear functionals on H. Recall (see §2) that the minimal parabolic subgroup P of G is amenable and transitive on D. 6) , = m (U(a)(V)f,U(a)(V)h),,. e. 7-2N < (f,f)N < 72H , f E H, JONATHAN ARAZY 34 Moreover, is U(a)-P-invariant, namely :

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