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Additional resources for Toeplitz Centennial: Toeplitz Memorial Conference in Operator Theory, Dedicated to the 100th Anniversary of the Birth of Otto Toeplitz, Tel Aviv, May 11–15, 1981

Example text

For every sequence (:A. ~(k) d) -2Re 1P+, where p(f) = 2 ~lfl G dt is the associated seminorm to K. e) 2i(Hu,vll ~ d[(T 1 u,u) + (T 2v,vl], VuE:emP+' vE:e 1P+' where H, --- ----- T1 ,T 2 are the associated kernels. f) If vis ~measure such that ~(-k) =\, \7 ~ m, then 2ReJ <1> 1~ 2 d11 ::; d r{i <1> 1 i 2Gdt + fi <1> 2 ( 2Gdt] ,V 1EemP +'<1> 2EP _. e. }, REMARK. },then condition g) of theorem 3 says that d(K(m)) can be interpreted as a distance to the subspace e -mP++H 1 in L1 , with respect to Theorem 2 gives conditions for the unicity of the above moment problem, and for the existence of a solution with a positive imaginary part: Let )= {) }oo and 0 ~ GE:L 1 be fixed, and THEOREM 2a.

3. : On a lifting theorem and its relation to some approximation problems,to appear in Proc. International Seminar of Func. , Holo~orphy and Approximation Theory {Brasil 1980). 4. Cotlar, M. : An Introduction to Functional Analysis, North Holland {1974). 55 5. Cotlar, M. : On the Helson-Szego theorem and a related class of modified Toeplitz kernels, Proc. Symp. Pure Math AMS 35: I (1979), 383-407. 6. : Banach Algebra Techniques in Operator Theory, Academic Press (1972). 7. Grenander, V. : Toeplitz forms and their applications, Univ.

Lo9o z . ) ] P J J P J [Lo~(zj) ]p bo(zj) ---=----"''----- This contradiction ends the proof. § 2. PROOF OF THEOREM 1. 1' let E ( z) : L~(r) + «:£, be defined by E ( z)! = Jr B(z,T) _E(T) dT be defined by F(z)~(w) and F ( z) : cr:£,+Ln(r) 2 T ( z) and, therefore, I + F(z)E(z) a(z) = Then dim Ker[I + F(z)E(z)] J I dim Kerrc .. + t~J In the case r r k=l . ) ~ £,x£, J ~ ( 2. 1 A(z,w)~. 1 • When r is smooth this same result about the isolated zeros of the minors follows from Lemma 2. completes the proof.

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